Part QM: Quantum Mechanics

Transcription

1 Stoy Brook Uiversity Academic Commos Essetial Graduate Physics Departmet of Physics ad Astroomy 3 Part Kostati Likharev SUNY Stoy Brook, kostati.likharev@stoybrook.edu Follow this ad additioal works at: Part of the Physics Commos Recommeded Citatio Likharev, Kostati, "Part " (3). Essetial Graduate Physics This Book is brought to you for free ad ope access by the Departmet of Physics ad Astroomy at Academic Commos. It has bee accepted for iclusio i Essetial Graduate Physics by a authorized admiistrator of Academic Commos. For more iformatio, please cotact darre.chase@stoybrook.edu.

2 Kostati K. Likharev Essetial Graduate Physics Lecture Notes ad Problems Beta versio Ope olie access at ad Part QM: Quatum Mechaics Last correctios: 7/6/6 K. Likharev

3 Table of Cotets Chapter. Itroductio (6 pp.).. Eperimetal motivatios.. Wave mechaics postulates.3. Postulates discussio.4. Cotiuity equatio.5. Eigestates ad eigevalues.6. Dimesioality reductio.7. Eercise problems () Chapter. D Wave Mechaics (76 pp.).. Probability curret ad ucertaity relatio.. Free particle: Wave packets.3. Particle motio i simple potetial profiles.4. The WKB approimatio.5. Trasfer matri, resoat tuelig, ad metastable states.6. Coupled quatum wells.7. D bad theory.8. Effective mass ad the Bloch oscillatios.9. Ladau-Zeer tuelig.. Harmoic oscillator: Brute force approach.. Eercise problems (43) Chapter 3. Higher Dimesioality Effects (56 pp.) 3.. Quatum iterferece, the AB effect, ad magetic flu quatizatio 3.. Ladau levels ad quatum Hall effect 3.3. Scatterig ad diffractio 3.4. Eergy bads i higher dimesios 3.5. Aially-symmetric systems 3.6. Spherically-symmetric systems: The brute force approach 3.7. Atoms 3.8. Eercise problems (36) Chapter 4. Bra-ket Formalism (4 pp.) 4.. Motivatio 4.. States, state vectors, ad liear operators 4.3. State basis ad matri represetatio 4.4. Chage of basis, uitary operators, ad matri diagoalizatio 4.5. Observables: Epectatio values, ucertaities, ad ucertaity relatios 4.6. Quatum dyamics: Three pictures 4.7. Eercise problems (3) Table of Cotets Page of 4

4 Chapter 5. Some Eactly Solvable Problems (5 pp.) 5.. Two-level systems, a.k.a. spi-½ systems, a.k.a. qubits 5.. Revisitig wave mechaics 5.3. Feyma s path itegral 5.4. Revisitig harmoic oscillator 5.5. The Glauber ad squeezed states 5.6. Revisitig spherically-symmetric problems 5.7. Spi ad its additio to orbital agular mometum 5.8. Eercise problems (4) Chapter 6. Perturbatio Theories (4 pp.) 6.. Eigevalue/eigestate problems 6.. The Stark effects 6.3. Fie structure 6.4. The Zeema effect 6.5. Time-depedet perturbatios 6.6. Quatum-mechaical Golde Rule 6.7. Golde Rule for step-like perturbatios 6.8. Eercise problems (3) Chapter 7. Ope Quatum Systems (58 pp.) 7.. Ope systems, ad the desity matri 7.. Coordiate represetatio ad the Wiger fuctio 7.3. Ope system dyamics: Dephasig 7.4. Fluctuatio-dissipatio theorem 7.5. The Heiseberg-Lagevi approach 7.6. Desity matri approach 7.7. Quatum measuremets 7.8. Eercise problems (9) Chapter 8. Multiparticle Systems (46 pp.) 8.. Distiguishable ad idistiguishable particles 8.. Siglets, triplets, ad the echage iteractio 8.3. Secod quatizatio 8.4. Perturbative approaches 8.5. Quatum computatio ad cryptography 8.6. Eercise problems (6) Chapter 9. Itroductio to Relativistic Quatum Mechaics (36 pp.) 9.. Electromagetic field quatizatio 9.. Photo statistics 9.3. Spotaeous ad stimulated emissio 9.4. Cavity QED 9.5. The Klie-Gordo ad relativistic Schrödiger equatios 9.6. Dirac s theory 9.7. Low eergy limit Table of Cotets Page 3 of 4

5 9.8. Eercise problems (9) Chapter. Makig Sese of Quatum Mechaics (6 pp.).. Hidde variables, Bell s theorem, ad local reality.. Iterpretatios of quatum mechaics * * * Additioal files (available upo request): Eercise ad Test Problems with Model Solutios ( = 33 problems; 48 pp.) Table of Cotets Page 4 of 4

6 Chapter. Itroductio This itroductory chapter briefly reviews the maor motivatios for quatum mechaics. The its simplest formalism - Schrödiger s wave mechaics - is described, ad its mai features are discussed Much of this material (perhaps ecept for the last sectio) may be foud i udergraduate tetbooks... Eperimetal motivatios By the begiig of the 9s, physics (which by that time icluded what we ow kow as orelativistic classical mechaics, classical statistics ad thermodyamics, ad classical electrodyamics icludig geometric ad wave optics) looked as a almost completed disciplie, with a lot of eperimetal observatios eplaied, ad ust a couple of mysterious dark clouds o the horizo. However, the rapid techological progress ad the resultig fast developmet of eperimetal techiques have led to a fast multiplicatio of observed pheomea that could ot be eplaied o the classical basis. Let me list the most cosequetial of those eperimetal fidigs. (i) Blackbody radiatio measuremets, started by G. Kirchhoff i 859, have show that the i the thermal equilibrium, the power of electromagetic radiatio by a fully absorbig ( black ) surface per uit frequecy iterval drops epoetially at high frequecies. This is ot what could be epected from the combiatio of the classical electrodyamics ad statistics, which predicted a ifiite growth of the radiatio desity with frequecy. Ideed, the classical electrodyamics shows 3 that electromagetic field modes i free space evolve i time as harmoic oscillators, ad that the desity of these modes i a large volume V >> 3 per small frequecy iterval is dvk 4k dk dn V V V d, (.) 3 3 c 3 where c 3 8 m/s is the free-space speed of light, its frequecy, k = /c the free-space wave umber, ad = /k is the radiatio wavelegth. O the other had, classical statistics 4 predicts that i the thermal equilibrium at temperature T, the average eergy E of each D harmoic oscillator should equal k B T, where k B is the Boltzma costat. 5 Combiig these two results, we readily get the so-called Rayleigh-Jeas formula for the average electromagetic wave eergy per uit volume: For remedial readig, I ca recommed, for eample, D. Griffith, Quatum Mechaics, d ed., Cambridge U. Press, 6. This epressio was used i a 9 talk by Lord Kelvi (bor W. Thomso) i referece to the blackbody radiatio measuremets ad Michelso-Morley eperimet results, i.e. the precursors of the quatum mechaics ad relativity theory. 3 See, e.g., EM Sec The degeeracy factor i Eq. () is due to two possible polarizatios of trasverse electromagetic waves. For waves of other physical ature, which obey with the liear ( acoustic ) dispersio law, similar relatios are also valid, though possibly with a differet degeeracy factor - see, e.g., CM Sec See, e.g., SM Sec... 5 I the SI uits, used through these otes, k B.38-3 J/K. Note that i may theoretical papers (ad i the SM part of my otes), k B is take for, i.e. temperature is measured i eergy uits. K. Likharev

7 de k BT dn u kbt, (.) 3 V d V d c that diverges at. O the other had, the blackbody radiatio measuremets, improved by O. Lummer ad E. Prigsheim, ad also H. Rubes ad F. Kurlbaum to reach a %-scale accuracy, were compatible with the pheomeological law suggested i 9 by Ma Plack: u. (.3a) 3 c ep( / k T ) The law may be recociled with the fudametal Eq. () if the followig replacemet is made for the average eergy of each field oscillator: kbt, (.3b) ep( / k T ) with a costat factor B B J s, (.4) ow called Plack s costat. 6 At low frequecies ( << k B T), the deomiator i Eq. (3) may be approimated as /k B T, so that the average eergy (3b) teds to its classical value k B T, ad the Plack law (3a) reduces to the Rayleigh-Jeas formula (). However, at higher frequecies ( >> k B T), Eq. (3) describes the eperimetally observed rapid decrease of the radiatio desity see Fig.. Plack radiatio law Plack s costat u u... /k B T Fig... Blackbody radiatio desity u, epressed i uits of u (k B T) 3 / c 3, as a fuctio of frequecy, accordig to: the Rayleigh-Jeas formula (blue lie) ad the Plack law (red lie). (ii) The photoelectric effect, eperimetally discovered i 887 by H. Hertz, shows a sharp lower boud o the frequecy of light that may kick electros out from metallic surfaces, regardless of the light itesity. Albert Eistei, i the first of his three famous 95 papers, oticed that this 6 M. Plack himself wrote as h, where = / is the cyclic frequecy, measured i Hz (periods per secod), so that i early tets the term Plack s costat referred to h, while was called the Dirac costat for a while. Chapter Page of 6

8 Eergy vs frequecy threshold mi could be readily eplaied assumig that light cosisted of certai particles (ow called photos) with eergy E h, (.5) with the same Plack s costat that participates i Eq. (3). 7 Ideed, with this assumptio, at the photo absorptio by the surface, its eergy E = is divided betwee a fied eergy W (ow called the workfuctio) of electro bidig iside the metal, ad the residual kietic eergy mv / > of the freed electro see Fig.. I this picture, the frequecy threshold fids a atural eplaatio as mi = W/. 8 Moreover, as was show by S. Bose i 94, Eq. (5) readily eplais 9 Plack s law (3). E -e mv E W Fig... Eistei s eplaatio of the photoelectric effect s frequecy threshold. (iii) The discrete frequecy spectra of radiatio by ecited atomic gases, kow sice the 6s, could ot be eplaied by classical physics. (Applied to the plaetary model of atoms, proposed by E. Rutherford, it predicts the collapse of electros o uclei i ~ - s due to electric dipole radiatio of electromagetic waves. ) Especially challegig was the observatio by J. Balmer (i 885) that the radiatio frequecies of simple atoms may be described by simple formulas. For eample, for the simplest atom, hydroge, all radiatio frequecies may be umbered with ust two positive itegers ad :, ', (.6) ' with,.7 6 s -. The Balmer series, icludig the value of, have foud its first eplaatio i the famous 93 theory by Niels Bohr, which was a semi-pheomeological precursor for quatum mechaics. I this theory,, is iterpreted as the frequecy of a photo that obeys the Eistei s formula (5), with its eergy E, beig the differece betwee two quatized (discrete) eergy levels of the atom (Fig. 3): E E E. (.7), ' ' 7 As a remider, A. Eistei received his oly Nobel Prize (i 9) for eactly this work, which essetially started quatum mechaics, rather tha for his relativity theory. 8 For most metals, W is betwee 4 ad 5 electro-volts (ev), so that the threshold correspods to ma = c/ mi = ch/w 3 m approimately at the border betwee the visible light ad ultraviolet radiatio. 9 See, e.g., SM Sec..5. See, e.g., EM Sec. 8.. Chapter Page 3 of 6

9 E ' E, ' E ' E Fig..3. Electromagetic wave radiatio at system s trasitio betwee its two quatized eergy levels. Bohr showed that the correct epressio for the levels (relative to the free electro eergy), E E H, (.8) ad the correct value of the so-called Hartree eergy Hydroge atom s eergy levels me e EH 7. ev 4, (.9) (where e.6-9 C is the fudametal electric charge, ad m e.9-3 kg is electro s rest mass) could be obtaied, with a virtually oe-lie calculatio, from the classical mechaics plus ust oe additioal postulate, equivalet to the assumptio that the agular mometum L = m e vr of the electro movig o a circular traectory of radius r about hydroge s uclei (i.e. proto, assumed to stay at rest), is quatized as L, (.) where is agai the same Plak s costat (4), ad is a iteger. Ideed, i order to derive Eq. (8), it is sufficiet to solve Eq. () together with the d Newto s law for the rotatig electro, v e m e, (.) r 4 r for the electro velocity v ad radius r, ad the plug the results ito the o-relativistic epressio for the full electro s eergy mev e E. (.) 4 r (This o-relativistic approach to the problem is ustified a posteriori by the fact the relevat eergy scale E H is much smaller tha electro s rest eergy, m e c ~.5 MeV.) By the way, the value of r, correspodig to =, i.e. to the smallest possible electro orbit, r B m e e / 4.53 m, (.3) Hartree eergy costat Agular mometum quatizatio Bohr radius Besides very small correctios due to the fiite ratio of the electro mass m e to that of the uclei, ad mior spi-orbital ad relativistic effects - see Secs. 6.3 ad 9.7 below. Ufortuately, aother mae, Rydberg costat is also frequetly used for either this atomic eergy uit or its half, E H / 3.6 ev. To add to the cofusio, the same term Rydberg costat is sometimes used for the reciprocal free-space wavelegth (/ = /c) correspodig to frequecy = E H /. Chapter Page 4 of 6

10 Mometum vs wave umber ad called the Bohr radius, defies the most importat spatial scale of pheomea i atomic, molecular ad codesed matter physics - as well as i chemistry ad biochemistry. Now ote that the quatizatio postulate () may be preseted as the coditio tha a iteger umber () of certai waves 3 fits the circular orbit s perimeter r =. Dividig both parts of this relatio by, we see that for this statemet to be true, the wave umber k / of the (the hypothetic) de Broglie waves should be proportioal to electro s mometum p = mv: p k. (.4) (iv) The Compto effect 4 is the reductio of frequecy of X-rays at their scatterig o free (or early-free) electros see Fig. 4. / c ' / c m e p Fig..4. Compto effect. The effect may be eplaied assumig that the X-ray photo also has a mometum that obeys the vector-geeralized versio of Eq. (4): p photo k, (.5) c where k is the wavevector (whose magitude is equal to the wave umber k, ad directio coicides with that,, of the wave propagatio), ad that mometa p of both the photo ad the electro are related to their eergies E by the classical relativistic formula 5 E ( cp) ( mc ). (.6) (For a photo, the rest eergy is zero, ad this relatio is reduced to Eq. (5): E = cp = ck =.) Ideed, a straightforward solutio of the followig system of three equatios, ( cp) ( m c ) /, mec ' e (.7) ' cos p cos c c, (.8) ' si psi, (.9) c 3 This fact was oticed ad discussed i detail i 93 by L. de Broglie, so that istead of discussig wavefuctios, especially of free particles, we are still frequetly speakig of de Broglie waves. 4 This effect was observed (i 9) ad eplaied a year later by A. Compto. 5 See, e.g., EM Sec Chapter Page 5 of 6

11 (which describe, respectively, the coservatio of the full eergy of the photo-electro system, ad of two relevat Cartesia compoets of its full mometum, at the scatterig evet see Fig. 4), yields the followig result, ( cos ), (.a) ' m c e which is traditioally represeted as the relatio betwee the iitial ad fial values of photo s wavelegth = /k = /(/c): ' ( cos ) c ( cos ), with c, (.b) m c m c e ad is i agreemet with eperimet. 6 (v) De Broglie wave diffractio. I 97, followig the suggestio by W. Elassger (who was ecited by de Broglie s coecture of matter waves ), C. Davisso ad L. Germer, ad idepedetly G. Thomso succeeded to observe diffractio of electros o crystals (Fig. 5). Specifically, they have foud that the itesity of the elastic reflectio from a crystal icreases sharply whe agle betwee the icidet beam of electros ad crystal s atomic plaes, separated by distace d, satisfies the followig relatio: d si, (.) where = /k = /p is the de Broglie wavelegth of electros, ad is a iteger. As Fig. 5 shows, this is ust the well-kow coditio 7 that the optical path differece l = dsi betwee the de Broglie waves reflected from two adacet crystal plaes coicides with a iteger umber of, i.e. of the costructive iterferece of the waves. 8 e Compto effect Bragg coditio d si d d si Fig..5. Electro scatterig from a crystal lattice. 6 The costat c, which participates i this relatio, is close to.46 - m ad is called the Compto wavelegth of the electro. This term is somewhat misleadig: as the reader ca see from Eqs. (7)-(9), o wave i the Compto problem has such a wavelegth either before or after the scatterig. 7 Frequetly called the Bragg coditio, due to the pioeerig eperimets by W. Bragg with X-ray scatterig from crystals (that started i 9). 8 Later, spectacular eperimets with diffractio ad iterferece of heavier particles, e.g., eutros ad eve C 6 molecules, have also bee performed see, e.g., a review by A. Zeiliger et al., Rev. Mod. Phys. 6, 67 (988) ad a later publicatio by O. Nairz et al., Am. J. Phys. 7, 39 (3). Nowadays, such iterferece of heavy particles is used for ultrasesitive measuremets of gravity see, e.g., a popular review by M. Ardt, Phys. Today 67, 3 (May 4), ad recet advaced eperimets by P. Hamilto et al., Phys. Rev. Lett. 4, 45 (5). Moreover, quatum iterferece betwee differet parts ad differet quatum states of such macroscopic obects as supercoductig codesates of millios Cooper pairs has bee observed see Sec. 3. below for details. Chapter Page 6 of 6

12 To summarize, all the listed effects may be eplaied startig from two very simple (ad similarly lookig) formulas: Eq. (5) for photos, ad Eq. (5) for both photos ad electros - both relatios ivolvig the same Plack s costat. This might give a impressio of sufficiet eperimetal evidece to declare light cosistig of discrete particles (photos), ad, o the cotrary, electros beig some matter waves rather tha particles. However, by that time (the mid 9s) physics has accumulated overwhelmig evidece of wave properties of light, such as iterferece ad diffractio. I additio, there was also a strog evidece for lumped-particle ( corpuscular ) behavior of electros. It is sufficiet to metio the famous oil-drop eperimets by R. Millika ad H. Fletcher (99-93) i that oly sigle (ad whole!) electros could be added to a oil drop, chagig its total electric charge by multiples of electro s charge (-e) ad ever its fractio. It was apparetly impossible to recocile these observatios with a purely wave picture, i which a electro ad hece its charge eed to be spread over the wave, so that its arbitrary part of it could be cut out usig appropriate eperimetal setups. Thus the foudig fathers of quatum mechaics faced a formidable task of recocilig the wave ad corpuscular properties of electros ad photos - ad other particles. The decisive breakthrough i that task has bee achieved i 96 by Ervi Schrödiger ad Ma Bor who formulated what is ow kow as either the Schrödiger picture of o-relativistic quatum mechaics i the coordiate represetatio, or simply as wave mechaics. I will ow formulate that picture, somewhat disregardig the actual history of its developmet... Wave mechaics postulates Let us cosider a spiless, 9 o-relativistic poit-like particle whose classical dyamics may be described by a certai Hamiltoia fuctio H(r, p, t), where r is particle s radius-vector ad p is coordiate. Wave mechaics of such Hamiltoia particles may based o the followig set of postulates that are comfortigly elegat - though their fial ustificatio is give oly by the agreemet of all their corollaries with eperimet. (i) Wavefuctio ad probability. Such variables as r or p caot be always measured eactly, eve at perfect coditios whe all eteral ucertaities, icludig measuremet istrumet imperfectio, macroscopic fluctuatios of the iitial state preparatio, ad uiteded particle iteractios with its eviromet, have bee removed. 3 Moreover, r ad p of the same particle ca 9 Actually, i wave mechaics, the spi of the described particle has ot to be equal zero. Rather, it is assumed that the spi effects are egligible - as they are, for eample, for a o-relativistic electro movig i a regio without a appreciable magetic field. As a remider, for may systems (icludig those whose kietic eergy is a quadratic-homogeeous fuctio of geeralized velocities, like mv /), H coicides with the total eergy E see, e.g., CM Sec..3. Note that this restrictio is very importat. I particular, it ecludes from our curret discussio the particles whose iteractio with eviromet is irreversible, for eample it is the viscosity leadig to particle s eergy decay. Such systems eed a more geeral quatum-mechaical descriptio that will be discussed i Chapter 7. Geerally, quatum mechaics, as ay theory, may be built o differet sets of postulates ( aioms ) leadig to the same coclusios. I this tet, I will ot try to beat dow the umber of postulates to the absolute miimum, ot oly because this would require loger argumetatio, but chiefly because such attempts typically result i makig certai implicit assumptios hidde from the reader the practice as commo as regrettable. 3 I will imply such perfect coditios util the discussio of particle s iteractio with eviromet, ad realistic ( physical ) measuremets i Chapter 7. Chapter Page 7 of 6

13 ever be measured eactly simultaeously. Istead, eve the most detailed descriptio of the particle s state, allowed by Nature, 4 is give by a certai comple fuctio (r, t), called the wavefuctio, that geerally eables oly probabilistic predictios of measured values of r, p, ad other directly measurable variables (i quatum mechaics, called observables). Specifically, the probability dw of fidig a particle iside a ifiitesimal volume dv d 3 r is proportioal to this volume ad may be characterized by the probability desity w dw/d 3 r that i tur is related to the wavefuctio as * w ( r, t) ( r, t) ( r, t), (.a) where sig * meas the comple cougate. As a result, the total probability of fidig the particle somewhere iside a volume V may be calculated as Probability via wavefuctio W V wd r 3 * V 3 d r. (.b) I particular, if the volume V cotais the particle defiitely (i.e. with the % probability, W = ), Eq. (b) is reduced to the so-called ormalizatio coditio V * 3 d r. (.c) (ii) Observables ad operators. To each observable A, quatum mechaics associates a certai liear operator Â, such that, i the perfect coditios metioed above, the average measured value (also called the epectatio value) of A is epressed as 5 Normalizatio coditio A V A 3 d r, (.3) * Observable s epectatio value where meas the statistical average, i.e. the result of averagig the measuremet results over a large esemble (set) of macroscopically similar eperimets, ad is the ormalized wavefuctio see Eq. (c). For Eqs. () ad (3) to be compatible, the idetity ( uit ) operator Î, defied by relatio Î, (.4) has to be associated with a particular type of measuremet, amely with particle s detectio. (iii) Hamiltoia operator ad the Schrödiger equatio. Aother particular operator, the Hamiltoia Ĥ, whose observable is the particle s eergy E, also plays i wave mechaics a very special role, because it participates i the Schrödiger equatio, i H, (.5) t Idetity operator Schrödiger equatio 4 This is oe more importat caveat. As we will see i Chapter 7, i may cases eve the Hamiltoia particles caot be described by a certai wavefuctio, ad allow oly a more geeral (ad less precise) descriptio, e.g., by the desity matri. 5 This key measuremet postulate is sometimes called the Bor rule. Chapter Page 8 of 6

14 Operators of coordiate ad mometum Free particle s Hamiltoia Free particle s Schrödiger equatio Plae wave solutio that determies wavefuctio s dyamics, i.e. its time evolutio. (iv) Radius-vector ad mometum operators. I the coordiate represetatio accepted i wave mechaics, the (vector) operator of particle s radius-vector r ust multiples the wavefuctio by this vector, while the operator of particle s mometum 6 is represeted by the spatial derivative: p i, (.6a) where is the del (or abla ) vector operator. 7 Thus i the Cartesia coordiates, r r, y, z, p i,,. (.6b) y z (v) Correspodece priciple. I the limit whe quatum effects are isigificat, e.g., whe the characteristic scale of actio S 8 (i.e. the product of the relevat eergy ad time scales of the problem) is much larger tha Plack s costat, all wave mechaics results have to ted to those give by classical mechaics. Mathematically, the correspodece is achieved by duplicatig the classical relatios betwee observables by similar relatios betwee the correspodig operators. For eample, for a free particle, the Hamiltoia (that i this case correspods to the kietic eergy aloe) has the form p H, (.7a) m m so that, takig ito accout Eq. (6b), i the Cartesia coordiates, H. (.7b) m y z Eve before a discussio of physics of the postulates (offered i the et sectio), we may immediately see that they ideed provide a way toward the resolutio of the apparet cotradictio betwee the wave ad corpuscular properties of particles. For a free particle, the Schrödiger equatio (5), with the substitutio of Eq. (7), takes the form i, (.8) t m whose particular (but most importat) solutio is a plae, moochromatic wave, 9 i( krt ) ( r, t) ae, (.9) 6 For a electrically charged particle i magetic field, this relatio is valid for its caoical mometum see Sec. 3. below. 7 See, e.g., Secs. 8- of the Selected Mathematical Formulas appedi (below, referred to as MA). Note that accordig to these formulas, the del operator follows all the geometric rules of the usual (c-umber) vectors. This is, by defiitio, true for other vector operators of quatum mechaics to be discussed below. 8 See, e.g., CM Sec See, e.g., CM Sec. 7.7 ad/or EM Sec. 7.. Chapter Page 9 of 6

15 where a, k ad are costats. Ideed, pluggig Eq. (9) ito Eq. (8), we immediately see the plae wave, with a arbitrary amplitude a, is ideed a solutio of the Schrödiger equatio, provided a specific dispersio relatio betwee wavevector k ad frequecy : ( k). (.3) m Costat a may be calculated, for eample, assumig that solutio (9) is eteded over a certai volume V, while beyod it, =. The from the ormalizatio coditio (c) ad Eq. (9), we get 3 a V. (.3) Now we ca use Eqs. (3), (6) ad (7) to calculate the epectatio value of particle s mometum p ad eergy E (which, for a free particle, coicides with its Hamiltoia fuctio H), The result is ( k) p k, E H ; (.3) m accordig to Eq. (3), the last equality may be rewritte as E =. Net, Eq. (3) eables oe to calculate ot oly the statistical average (i the math speak, the first momet) of a observable, but also its higher momets, otably the secod momet (i physics, usually called either the variace or dispersio): ad hece its root mea square (r.m.s.) fluctuatio, ~ that characterizes the scale of deviatios A A ~ A A A A A, (.33) ~ / A A, (.34) A of measuremet results from the average, i.e. the ucertaity of observable A. I applicatio to wavefuctio (9), these relatios yield E =, p =, while the particle coordiate r (at V ) is completely ucertai. This meas that i the plae-wave, moochromatic state (9), the eergy ad mometum of the particle are eactly defied, so that the sigs of statistical average i Eqs. (3) might be removed. Thus, these relatios are reduced to the eperimetally-iferred Eqs. (5) ad (5), though the relatio of frequecy of wavefuctio s evolutio i time to eperimetal observatios still has to be clarified. Hece the wave mechaics postulates may ideed eplai the observed wave properties of orelativistic particles. (For photos, we would eed a relativistic formalism see Ch. 9 below.) O the other had, due to the liearity of the Schrödiger equatio (5), ay sum of its solutios is also a solutio the so-called liear superpositio priciple. For a free particle, this meas that a set of plae waves (9) is also a solutio of this equatio. Such sets, with close values of k ad hece p = k (ad, accordig to Eq. (3), of as well), may be used to describe spatially localized pulses, called wave packets see Fig. 6. I Sec.., I will prove (or rather reproduce H. Weyl s proof :-) that the wave Free particle s dispersio relatio Observable s variace Observable s ucertaity 3 For ifiite space (V ), Eq. (3) yields a, i.e. wavefuctio (9) vaishes. This formal problem may be readily resolved cosiderig sufficietly log wave packets see Sec.. below. Chapter Page of 6

16 Heiseberg s ucertaity relatio packet etesio i ay directio (say, ) is related to the width k of the correspodig compoet of its wave vector distributio as k ½, ad hece, accordig to Eq. (5), to the width p of the mometum compoet distributio as p. (.35) Re Im (a) a k k (b) the particle is (somewhere :-) here! k k p / Fig..6. (a) Sapshot of a typical wave packet propagatig alog ais, ad (b) the correspodig distributio of wave umbers k, i.e. mometa p. This is the famous the famous Heiseberg s ucertaity priciple, which quatifies the first postulate s poit that coordiate ad mometum caot be defied eactly simultaeously. However, sice the Plack s costat is etremely small o the huma scale of thigs, it still allows for the particle s localizatio i a very small volume eve if the mometum spread i the wave packet is also small o that scale. For eample, accordig to Eq. (35), a.% spread of mometum of a kev electro (p ~.7-4 kgm/s) allows a wave packet to be as small as ~3 - m. (For a heavier particle such as a proto, the packet would be eve tighter.) As a result, wave packets may be used to describe particles that are poit-like from the macroscopic poit of view. I a utshell, this is the mai idea of the wave mechaics, ad the first part of this course (Chapters -3) will be essetially a discussio of various maifestatios of this approach. Durig this discussio, we will ot oly evidece wave mechaics may triumphs withi its applicability domai, but will also gradually accumulate evidece for its hadicaps, which force the evetual trasfer to a more geeral formalism to be discussed i Chapter 4 ad beyod..3. Postulates discussio The postulates listed i the previous sectio look very simple, ad they are hopefully familiar to the reader from his or her udergraduate studies. However, the physics of these aioms are very deep, they lead to several couter-ituitive coclusios, ad their i-depth discussio requires solutios of several key problems usig these aioms. This is why i this sectio I will give oly a iitial, admittedly superficial discussio of the postulates, ad will be repeatedly returig to the coceptual foudatios of quatum mechaics throughout the course, especially i Secs. 7.7,., ad.. First of all, the fudametal ucertaity of observables, which is i the core of postulate (i), is very foreig to the basic ideas of classical mechaics, ad historically has made quatum mechaics so hard to swallow for may star physicists, otably icludig A. Eistei despite his 95 work which essetially lauched the whole field! However, this fact has bee cofirmed by umerous eperimets, Chapter Page of 6

17 ad (more importatly) there have ot bee a sigle cofirmed eperimet which would cotradict to this postulate, so that quatum mechaics was log ago promoted from a theoretical hypothesis to the rak of a reliable scietific theory. Oe more remark i this cotet is that Eq. (5) itself is determiistic, i.e. coceptually eables a eact calculatio of wavefuctio s distributio i space at ay istat t, provided that its iitial distributio, ad particle s Hamiltoia, are kow eactly. I classical kietics, the probability desity distributio w(r,t) may be also calculated from determiistic differetial equatios, e.g., the Fokker- Plack equatio or the Boltzma equatio. 3 The quatum-mechaical descriptio differs from those situatios i two importat aspects. First, i the perfect coditios outlied above (eact iitial state preparatio, o irreversible iteractio with eviromet, the best possible measuremet), the Fokker- Plack equatio reduces to the d Newto law, i.e. the statistical ucertaity disappears. I quatum mechaics this is ot true: the quatum ucertaily, such as Eq. (35), persists eve i this limit. Secod, the wavefuctio (r, t) gives more iformatio tha ust w(r, t), because besides the modulus of, ivolved i Eq. (), this comple fuctio also has phase arg, ad may affect some observables, describig, i particular, the iterferece ad diffractio of the de Broglie waves. Net, it is very importat to uderstad that the relatio betwee the quatum mechaics to eperimet, give by postulate (ii), ecessarily ivolves aother key otio: that of the correspodig statistical esemble. Such esemble may be defied as a set of may eperimets carried out at apparetly (macroscopically) similar coditios, which evertheless may lead to differet measuremet results (outcomes). Ideed, the probability of a certai (-th) outcome of a eperimet may be oly defied for a certai esemble, as the limit W N M lim M, with M M, (.36) M where M is the total umber of eperimets, M is the umber of outcomes of the -th type, ad N is the umber of differet outcomes. It is clear that a particular choice of a esemble may affect probabilities W very sigificatly. For eample, if we pull out playig cards at radom from a pack of 5 differet cards of 4 suits, the probability W of gettig a certai card (e.g., the quee of spades) is /5. However, if cards of a certai suit (say, hearts) had bee take out from the pack i advace, the probability of gettig the quee of spades is higher, /39. It is importat that we would also get the last umber for probability eve if we had used the full 5-card pack, but by some reaso igored results of all eperimets givig us ay rak of hearts. Similarly, i quatum mechaics, the probability distributios (ad hece epectatio values of particle coordiate ad other observables) deped ot oly o the eperimet setup, but also o the set of outcomes we cout. Because of the fudametal relatio () betwee w ad, this meas the wavefuctio also depeds o those factors, i.e. o both the eperimet set preparatio ad the subset of outcomes take ito accout. The isistece o the attributio of the wavefuctio to a sigle eperimet, both before ad after the measuremet, may lead to very uphysical iterpretatios of some eperimets, icludig wavefuctio s evolutio ot described by the Schrödiger equatio (the socalled wave packet reductio), sublumial actio o distace, etc. Later i the course we will see that midig the statistical ature of the quatum mechaics, ad i particular the depedece of the 3 See, e.g., SM Secs. 5.8 ad 6., respectively. Defiitio of probability Chapter Page of 6

18 Defiitio of statistical average wavefuctio o statistical esemble s specificatio, may readily eplai some apparet paradoes of quatum measuremets. Let me also emphasize that statistics is itimately related to the iformatio theory - ad ot oly via their commo mathematical backgroud, the probability theory. For eample, the questio, What subset of eperimetal results we will cout? may be replaced by the questio, What subset of results will we use iformatio about? As a result, the reader has to be prepared to the use of iformatio theory otios for the discussio quatum mechaics, or at least its relatio to eperimet - i.e. to the physical reality. This feature of quatum mechaics makes some physicists ucomfortable, because much of classical mechaics ad electrodyamics may be discussed without ay referece to iformatio. I quatum mechaics (as i statistical mechaics), such a abstractio is impossible. Proceedig to postulate (ii) ad i particular Eq. (3), a better feelig of this defiitio may be obtaied by its compariso with the geeral defiitio of the epectatio value (i.e. the statistical average) i the probability theory. Namely, let each of N possible outcomes i a set of M macroscopically similar eperimets give a certai value A of observable A; the A N N lim M A M AW. (.37) M Takig ito accout Eq. (), which relates W ad, the structure of Eq. (3) ad the fial form of Eq. (37) is similar. Their eact relatio will be further discussed i Sec Cotiuity equatio The wave mechaics postulates survive oe more saity check: they satisfy the atural requiremet that the particle does ot appear or vaish i the course of the quatum evolutio. 3 Ideed, let us use Eq. () to calculate the rate of chage of the probability W to fid the particle withi a certai volume V: dw d * 3 d r dt dt. (.38) V Assumig for simplicity that the boudaries of volume V do ot move, it is sufficiet to carry out the partial differetiatio of the product * iside the itegral. Usig the time-depedet Schrödiger equatio (5), together with its comple cougate, we get dw dt V t * * * i ( H), (.39) t * H H d r. 3 3 * 3 d r V * t t d r i V * (.4) 3 Note that this requiremet is ot eteded to the relativistic quatum theory see Chapter 9 below. Chapter Page 3 of 6

19 Let the particle move i a field of eteral forces (ot ecessarily costat i time), so that its classical Hamiltoia fuctio H is a sum of particle s kietic eergy p /m ad its potetial eergy U(r, t). 33 Accordig to the correspodece priciple, the Hamiltoia operator may be preseted as the sum 34, p H U ( r,t) m m U ( r, t). (.4) At this stage we should otice that such operator, whe actig o a real fuctio, returs a real fuctio. 35 Hece, the result of its actio o a arbitrary comple fuctio = a + ib (where a ad b are real) is H H ( a ib) Ha ihb, (.4) Hamiltoia of a particle i a field where Ĥa ad Ĥb are also real, while This meas that Eq. (4) may be rewritte as dw dt i * * ( ) ( Ha ihb )* Ha ihb H ( a ib) H V H. (.43) H H d r m i * * 3 * * V Now, let us use geeral rules of vector calculus 36 to write the followig idetity: 3 d r. (.44) * * * * Ψ Ψ ΨΨ Ψ Ψ Ψ Ψ, (.45) A compariso of Eqs. (44) ad (45) shows that we may write dw dt 3 ( ) d r, (.46) V where vector is defied as i * * ΨΨ c.c. Im Ψ Ψ, (.47) m m where c.c. meas the comple cougate of the previous epressio i this case, (*)*, i.e. *. Now usig the well-kow divergece theorem, 37 Eq. (46) may be rewritte as the cotiuity equatio dw I, with I d r, (.48) dt S Probability curret desity Cotiuity equatio: itegral form 33 As a remider, such descriptio is valid ot oly for potetial forces (i that case U has to be timeidepedet), but also for ay force F(r, t) which may be preseted via the gradiet of U(r, t) see, e.g., CM Chapters ad. (A good eample whe such a descriptio is impossible is give by the magetic compoet of the Loretz force see, e.g., EM Sec. 9.7, ad also Sec. 3. of this course.) 34 Historically, this was the mai step made (i 96) by E. Schrödiger o the backgroud of L. de Broglie s idea. The probabilistic iterpretatio of the wavefuctio was put forward, almost simultaeously, by M. Bor. 35 I Chapter 4, we will discuss a more geeral family of Hermitia operators, which have this property. 36 See, e.g., MA Eq. (.4a), combied with the del operator s defiitio. 37 See, e.g., MA Eq. (.). Chapter Page 4 of 6

20 Cotiuity equatio: differetial form where is the proectio of vector o the outwardly directed ormal to surface S that limits volume V, i.e. the scalar product, where is the uit vector alog this ormal. Equatios (47) ad (48) show that if the wavefuctio o the surface vaishes, the total probability W of fidig the particle withi the volume does ot chage, providig the required saity check. I the geeral case, Eq. (48) says that dw/dt equals to flu I of vector through the surface, with the mius sig. It is clear that this vector may be iterpreted as the probability curret desity - ad I, as the total probability curret through surface S. This iterpretatio may be further supported by rewritig Eq. (47) for a wavefuctio preseted i the polar form = ae i, with real a ad : a, (.49) m - evidetly a real quatity. Note that for a real wavefuctio, or eve for that with a arbitrary but spacecostat phase, the probability curret desity vaishes. O the cotrary, for the travelig wave (9), with a costat probability desity w = a, Eq. (49) yields a ovaishig (ad physically very trasparet) result: p w k w wv, (.5) m m where v = p/m is particle s velocity. If multiplied by the particle s mass m, the probability desity w turs ito the (average) mass desity, ad the probability curret desity ito the mass flu desity v, while if multiplied by the total electric charge q of the particle, with w turig ito the charge desity, becomes the electric curret desity, both satisfyig the classical cotiuity equatios similar to Eq. (48). 38 Fially, let us recast the cotiuity equatio, rewritig Eq. (46) as w 3 d r. (.5) t V Now we may argue that this equality may is true for ay choice of volume V oly if the epressio uder the itegral vaishes everywhere, i.e. if w. (.5) t This differetial form of the cotiuity equatio is sometimes more coveiet tha its itegral form (48)..5. Eigestates ad eigevalues Now let us discuss importat corollaries of wave mechaics liearity. First of all, it uses oly liear operators. This term meas that the operators must obey the followig two rules: See, e.g., respectively, CM 7. ad EM Sec By the way, if ay equality ivolvig operators is valid for a arbitrary wavefuctio, the latter is frequetly dropped from otatio, resultig i a operator equality. I particular, Eq. (53) may be readily used to prove that the operators are commutative: A A A A, ad associative: A A A A A A. 3 3 Chapter Page 5 of 6

21 A A A, A (.53) c c A c A c c A c A A, (.54) where are arbitrary wavefuctios, while c are arbitrary costats (i quatum mechaics, frequetly called c-umbers, to distiguish them from operators ad wavefuctios). Most importat eamples of liear operators are give by: (i) the multiplicatio by a fuctio, such as for operator r i wave mechaics, ad (ii) the spatial or temporal differetiatio of the wavefuctio, such as i Eqs. (5)-(7). Net, it is of key importace that the Schrödiger equatio (5) is also liear. (We have already used this fact whe we discussed wave packets i the last sectio.) This meas that if each of fuctios are (particular) solutios of Eq. (5) with a certai Hamiltoia, the a arbitrary liear combiatio c (.55) is also a solutio of the same equatio. 4 Now let us use the liearity of wave mechaics to accomplish a apparetly impossible feat: immediately fid the geeral solutio to the Schrödiger equatio for the most importat case whe system s Hamiltoia does ot deped o time eplicitly for eample, like i Eq. (7), or i Eq. (4) with time-idepedet U = U(r). First of all, let us prove that the followig product, T ( t) ( r), (.56) qualifies as a (particular) solutio to the Schrödiger equatio. Ideed, pluggig Eq. (56) ito Eq. (5), usig the fact that for a time-idepedet Hamiltoia H T ( t) ( r) T ( t) H ( r), (.57) ad dividig both parts of the equatio by = T, we get it H, (.58) T where (here ad below) the dot deotes the differetiatio over time. The left had side of this equatio may deped oly o time, while the right had oe, oly o coordiates. These facts may be oly recociled if we assume that each of these parts is equal to (the same) costat of the dimesio of eergy, which I will deote as E. 4 As a result, we are gettig two separate equatios for the temporal ad spatial parts of the wavefuctio: i T E T, (.59) Variable separatio 4 It may seem strage that the liear Schrödiger equatio correctly describes quatum properties of systems whose classical dyamics is described by oliear equatios of motio (e.g., a aharmoic oscillator see, e.g., CM Sec. 4.). Note, however, that equatios of classical physical kietics (see, e.g., SM Chapter 6) also have this property, so it is ot specific to quatum mechaics. 4 This argumetatio, leadig to variable separatio, is very commo i mathematical physics see, e.g., its discussio i EM Sec..5. Chapter Page 6 of 6

22 Statioary Schrödiger equatio Statioary state: time evolutio Statioary Schrödiger equatio for static potetial H. (.6) E The first of these equatios is readily itegrable, givig E T cost ep i t, with ω, (.6) ad thus substatiatig the fudametal relatio (5) betwee eergy ad frequecy. Pluggig Eqs. (56) ad (6) ito Eq. (), we see that i such a state, the probability w of fidig the particle at a certai locatio does ot deped o time. Doig the same with Eq. (3) shows that the same is true for the epectatio value of ay operator that does ot deped o time eplicitly: A 3 * A d r = cost. (.6) Due to this property, the states described by Eqs. (56), (6), ad (6), are called statioary. I cotrast to the simple ad uiversal time depedece (6), the spatial distributios (r) of the statioary states are ofte hard to fid, ad the solutio of the statioary (or time-idepedet ) Schrödiger equatio (6), 4 which describes the distributios, for various situatios is a maor focus of wave mechaics. The statioary Schrödiger equatio (6), with time-idepedet Hamiltoia (4), U ( r) m E, (.63) falls ito the mathematical category of liear eigeproblems, 43 i which eigefuctios ad eigevalues E should be foud simultaeously - self-cosistetly. 44 Mathematics tells us that for the such problems with space-cofied eigefuctios, tedig to zero at r, the spectrum of eigevalues is discrete. It also proves that the eigefuctios correspodig to differet eigevalues are orthogoal, i.e. that space itegrals of the products * vaish for all pairs with. Moreover, due to the Schrödiger equatio liearity, each of these fuctios may be multiplied by a costat coefficiet to make this set orthoormal: * 3, if ', ' d r, ' (.64), if '. Also, the eigefuctios form a full set, meaig that a arbitrary fuctio (r), i particular the actual wavefuctio of the system i the iitial momet of its evolutio (which I will take for t =, with a few eceptios), may be preseted as a uique epasio over the eigefuctio set: ( r,) c ( r). (.65) The epasio coefficiets c k may be readily foud by multiplyig both parts of Eq. (65) by *, itegratig the result over the space, ad usig Eq. (64). The result is 4 I cotrast, the iitial Eq. (4) is frequetly called the time-depedet or ostatioary Schrödiger equatio. 43 From Germa root eige meaig particular or characteristic. 44 Eigevalues of eergy are frequetly called eigeeergies, ad it is ofte said that eigefuctio ad eigeeergy E together characterize -th statioary eigestate of the system. Chapter Page 7 of 6

23 c * 3 ( r) ( r,) d r. (.66) Now let us cosider the followig wavefuctio E ( r, t) cak ( t) k ( r) c ( r)ep i t. (.67) Sice each term of the sum has the form (56) ad satisfies the Schrödiger equatio, so does the sum as the whole. Moreover, if coefficiets c are derived i accordace with Eq. (66), the solutio (67) satisfies the iitial coditios as well. At this momet we ca agai use oe more help by mathematicias who tell us that the partial differetial equatio of type (8) with the Hamiltoia operator (4) with fied iitial coditios, may have oly oe (uique) solutio. This meas that i our case of motio i a time-idepedet potetial U = U(r), Eq. (67) gives the geeral solutio of the timedepedet Schrödiger equatio (5) for our case: i U ( r). (.68) t m We will repeatedly use this key fact through the course, though i may cases, followig the physical sese of particular problems, will be more iterested i certai specific particular solutios of Eq. (68) rather i the whole liear superpositio (67). I order to get some feelig of fuctios, let us cosider perhaps the simplest eample, which evertheless will be the basis for discussio of may less trivial problems: a particle cofied i a rectagular quatum well 45 with a flat bottom ad sharp ad ifiitely high hard walls :, for a, y a y, ad z az, U ( r ) (.69), otherwise. The oly way to keep the product U i Eq. (68) fiite outside the well, is to have = i these regios. Also, the fuctio have to be cotiuous everywhere, to avoid the divergece of its Laplace operator. Hece, we may solve the statioary Schrödiger equatio (63) oly iside the well, where it takes a simple form 46 m E, (.7a) with zero boudary coditios o all the walls. For our particular geometry, it is atural to epress the Laplace operator i the Cartesia coordiates {, y, z} aliged with the well sides, so that we get the followig boudary problem: 45 By usig the term quatum well for what is essetially a potetial well I bow to a commo, but a very ufortuate covetio. Ideed, this term seems to imply that the particle s cofiemet i such a quatum well is a pheomeo specific for quatum mechaics, while as we will repeatedly see i this course, that the opposite is true: quatum effects do as much as they oly ca to overcome particle s cofiemet i a potetial well, lettig the particle to partly peetrate i the classically forbidde regios. 46 Rewritte as f + k f =, this is the Helmholtz equatio, which describes scalar waves of ay ature (with wave vector k) i a uiform, liear media see, e.g., CM Sec. 5.5 ad/or EM Secs Chapter Page 8 of 6

24 m, E y z, for ad a ; for a y ad a y ;, y a y z ad a, ad z. z a z, (.7b) Rectagular quatum well: partial eigefuctios This problem may be readily solved usig the same variable separatio method which was used earlier i this sectio to separate the spatial ad temporal variables, ow to separate Cartesia spatial variables from each other. Let us look for a particular solutio i the form ( r) X ( ) Y ( y) Z( z). (.7) (It is coveiet to postpoe takig care of proper idices for a miute.) Pluggig this epressio ito the Eq. (7b) ad dividig by = XYZ, we get d X d Y d Z E m X d Y dy Z dz. (.7) Now let us repeat the stadard argumetatio of the variable separatio method: sice each term i the paretheses may be oly a fuctio of the correspodig argumet, the equality is possible oly if each term is a costat - with the dimesioality of eergy. Callig them E, etc., we get three D equatios d X d Y E, E m X d m Y dy with Eq. (7) turig ito the eergy-matchig coditio E y, d Z m Z d E z, (.73) E E E. (.74) y All three ordiary differetial equatios (73), ad their solutios, are similar. For eample, for X(), we have a D Helmholtz equatio d X me k, with X k, (.75) d ad simple boudary coditios: X() = X(a ) =. Let me hope that the reader kows how to solve this well-kow D boudary problem - describig, for eample, usual mechaical waves o a guitar strig, though with a very much differet epressio for k. The problem allows a ifiite umber of siusoidal stadig-wave solutios, 47 X a correspodig to eigeeergies / si k a / z si a, E k m ma E with,,..., (.76). (.77) 47 The frot coefficiet is selected i a way that esures the (ortho)ormality coditio (64). Chapter Page 9 of 6

25 Figure 7 shows this result usig a somewhat odd but very graphic ad hece commo way whe the eigeeergy values (frequetly called eergy levels) are used as horizotal aes for plottig eigefuctios, despite their differet dimesioality. Due to the similarity of all Eqs. (73), Y (y) ad Z(z) are similar fuctios of their argumets, ad may also be umbered by itegers (say, y ad z ) idepedet of, so that the spectrum of the total eergy (74) is y z E, y,. (.78) z m a a y az E E X ( ) / a Fig..7. Eigefuctios (solid lies) ad eigevalues (dashed lies) of the D wave equatio (75) o a fiitelegth segmet. Solid black lies show the potetial eergy profile of the problem. Rectagular quatum well: eergy levels Thus, i this 3D problem, the role of ide i Eq. (67) is played by a set of 3 idepedet itegers {, y, z }. I quatum mechaics, such itegers play a key role, ad thus have a special ame, quatum umbers. Now the geeral solutio (67) of our simple problem may be preseted as the sum Ψ( r, t),, y y z z E y z c,, si si si ep y i t z, (.79) a a y az,, y z Rectagular quatum well: geeral solutio with the coefficiets which may be readily calculated from the iitial wavefuctio (r, ), usig Eq. (66), agai with the replacemet {, y, z }. This simplest problem is a good illustratio of the basic features of wave mechaics for a spatially-cofied motio, icludig the discrete eergy spectrum, ad (i this case, evidetly) orthogoal eigefuctios. A eample of the opposite limit of a cotiuous spectrum for ucofied motio of a free particle is give by plae waves (9) which, with the accout of relatios E = ad p = k, may be viewed as the product of the time-depedet factor (46) by eigefuctio i a ep k r (.8) k k that is the solutio to the statioary Schrödiger equatio (7a) if it is valid i the whole space. 48 The reader should ot be worried too much by the fact that the fudametal solutio (8) i free space is a travelig wave (havig, i particular, ovaishig value (5) of the probability curret ), Free particle: eigefuctios 48 I some systems (e.g., a particle iteractig with a fiite-depth quatum well), a discrete eergy spectrum withi a certai iterval of eergies may coeist with a cotiuous spectrum i a complemetary iterval. However, the coceptual philosophy of eigefuctios ad eigevalues remais the same i this case as well. Chapter Page of 6

26 Free particle: eigeeergies 3D umber of states Summatio over 3D states while those iside a quatum well are stadig waves, with =, eve though the free space may be legitimately cosidered as the ultimate limit of a quatum well with volume V = a a y a z. Ideed, due to the liearity of wave mechaics, two travelig-wave solutios (8) with equal ad opposite values of mometum (ad hece with the same eergy) may be readily combied to give a stadigwave solutio, for eample ep{ikr} + ep{-ikr} = cos(kr), with the et curret =. Thus, depedig o coveiece for solutio of a particular problem, we ca preset the geeral solutio as a sum of either travelig-wave or stadig-wave eigefuctios. Sice i the free space there are o boudary coditios to satisfy, Cartesia compoets of the wave vector k i Eq. (8) ca take ay real values. (This is why it is more coveiet to label the wavefuctios ad eigeeergies, k Ek, (.8) m by their wave vector k rather tha a iteger ide.) However, oe aspect of systems with cotiuous spectrum requires a bit more math cautio: summatio (67) should be replaced by itegratio over a cotiuous ide or idices (i this case, 3 compoets of vector k). The mai rule of such replacemet may be readily etracted from Eq. (76): accordig to this relatio, for stadig-wave solutios, the eigevalues of k are equidistat, i.e. separated by equal itervals k = /a (with the similar relatios for other two Cartesia compoets of vector k). Hece the umber of differet eigevalues of the stadig wave vector k (with k, k y, k z ), withi a volume d 3 k >> /V of the k space is ust dn = d 3 k/(k k k ) = V/ 3. Sice i cotiuum it is more coveiet to work with travelig waves, we should take ito accout that, as was ust discussed, there are two differet travelig wave vectors (k ad k = -k) correspodig to each stadig wave vector k. Hece the same umber of physically differet states correspods to 3 = 8-fold larger k space (which ow is ifiite i all directios) or, equivaletly, to a smaller umber of states per uit volume d 3 k: V dn 3 d k 3. (.8) For dn >>, this epressio is idepedet o the boudary coditios, 49 ad is frequetly preseted as the followig summatio rule V 3 lim f f dn f d k k3 ( k) ( k) ( k), (.83) V k where f(k) is a arbitrary fuctio of k. This rule is very importat for statistical physics. Note also that if the same wave vector k correspods to several iteral quatum states (such as spi see Chapter 4), the right-had part of Eq. (83) requires multiplicatio by the correspodig degeeracy factor Dimesioality reductio To coclude this itroductory chapter, let me discuss the coditios whe the spatial dimesioality of a wave mechaics problem may be reduced. 5 For eample, followig our discussio 49 For a more detailed discussio of this poit, the reader may be referred, e.g., to CM Secs. 5.4 (i the cotet of D mechaical waves), because it is valid for waves of ay ature. Chapter Page of 6

27 of the 3D rectagular, flat-bottom quatum well i Sec. 5, let us cosider a ifiitely deep quatum well whose bottom is flat oly i oe directio, say z: U (, y), for z a, U ( r ) z (.84), otherwize. I this case, we ca separate variables oly partly, by presetig the eigefuctio as (,y)z(z). Pluggig such solutio ito the correspodig form of the statioary Schrödiger equatio (63), we see that fuctios Z(z) are agai similar to those give by Eq. (76), while fuctio (,y) satisfies the followig D statioary Schrödiger equatio: where U ef, y U ef (, y) E, y, (.85) m (, y) U (, y) Ez U (, y). (.86) ma Thus, we have arrived at the boudary problem similar to the iitial oe, but with the spatial dimesioality reduced from 3 to, due to what is called the partial cofiemet 5 i directio z. If all partial fuctios Z(z) are ormalized to uity, the wavefuctio ormalizatio coditio (c) becomes z z * W (, y) (, y) ddy, (.87) A where A is the total area of the system o the [, y] plae, ad is formally similar to the iitial 3D ormalizatio coditio. However, the effective D potetial eergy U ef (,y) icludes term E z depedig o quatum umber z, 5 makig the physical relevace of such variable separatio much less geeral tha might be aively epected. There are three possible cases: (i) If there is o strog relatio betwee the eergy scale E,y of potetial U ef (,y) ad E z, the solutio of a typical problem has to be preseted as a (typically, large) sum of partial solutios (,y)z(z), each with its ow z, U ef, ad E z. I this geeral case, the variable separatio may ot provide much relief at all, because eigeeergies of solutios with differet z may be close, so that several of them would simultaeously participate i realistic processes. (ii) E z is much smaller tha E,y ad may be eglected. This may be the case, for eample, if the potetial profile is more steep alog aes ad y, tha alog directio z. Notice, however, that coditio, a z, does ot guaratee the smalless of E z, because it may be compesated by large values of z. I this case (typical for solid state problems), either summatio or itegratio over z still D statioary Schrödiger equatio Effective potetial eergy 5 May tetbooks o quatum mechaics ump to solutio of D without such discussio, ad most of my begiig graduate studets did ot uderstad that i realistic physical systems, such dimesioality restrictio is oly possible uder very specific coditios. 5 The term quatum cofiemet, sometimes used to describe this pheomeo, is as ufortuate as the quatum well, because of the same reaso: the cofiemet is a purely classical effect, ad as we will repeatedly see i this course, quatum mechaics reduces it, allowig a partial peetratio of the particle ito the classically forbidde regios with E > U(r). 5 The last term i Eq. (86) is frequetly referred to as the (partial) cofiemet eergy; despite its iclusio to U ef, it is importat to remember about the kietic-eergy origi of this cotributio. Chapter Page of 6

28 may be eeded, though sometimes may be carried out aalytically, because fuctios Z(z) are simple siusoidal waves. (iii) Couter-ituitively, the most robust dimesioality reductio is possible i the opposite limit, whe a z is much smaller tha the characteristic scale of motio withi the [, y] plae (Fig. 8a). Ideed, i this case the distace betwee adacet levels of the cofiemet eergy E z is much larger tha the characteristic eergy E,y of motio withi the plae. As a result, if the system was iitially prepared to be o the lowest, groud level of E z,, a soft motio alog ad y caot ecite the system to higher levels of E z. 53 Hece, the system keeps the fied quatum umber z =, through the motio, so that the cofiemet eergy E z is costat ad, accordig to Eq. (86), may be treated ust as a fied potetial eergy offset. The last coclusio is true eve if the quatum well s profile i directio z is ot rectagular (provided that E z is still much larger tha E,y ). For eample, may D quatum pheomea, such as the quatum Hall effect, 54 have bee studied eperimetally usig electros cofied at semicoductor heterouctios (e.g., epitaial iterfaces GaAs/Al Ga - As) where the potetial well i the directio perpedicular to the iterface has a early triagular shape, with the splittig of eergies E z is the order of - ev. 55 This splittig eergy correspods to k B T at temperature ~ K, so that careful eperimetatio at liquid helium temperatures (4K ad below) may keep the electros performig purely D motio i the lowest subbad ( z = ). z y (a) z y (b) Fig..8. Partial cofiemet i: (a) oe dimesio, ad (b) two dimesios. Now, if a quatum well is formed i two dimesios (say, y ad z, see Fig. 8b), 56 U ( ), for y a y ad z a, U ( r ) z (.88), otherwize. the repeatig the variable separatio procedure we see that the 3D Schrödiger equatio (68) may be satisfied with particular solutios of the type (7), agai with siusoidal stadig waves Y(y) ad Z(z), but geerally a more comple fuctio X(), which has to satisfy the followig D Schrödiger equatio D statioary Schrödiger equatio d X U ef ( ) X m d E X, (.89) 53 I the frequet case whe motio i the [, y] plae is free (or almost free), the set of quatum states with the same quatum umber z is frequetly called a subbad, because their eergies form a (quasi-) cotiuum of eigeeergies E,y. 54 To be discussed i Sec See, e.g., P. Harriso, Quatum Wells, Wires, ad Dots, 3 rd ed., Wiley,. 56 This is a reasoable first approimatio, for eample, for electro motio potetial i so-called quatum wires, for eample i the ow-famous carbo aotubes see, e.g., the same moograph by P. Harriso. Chapter Page 3 of 6

29 with the effective potetial eergy U ef ( ) U ( ) E y E z. (.9) Agai, if the particle stays i the lowest subbad, y = z =, both E y ad E z retai their costat values E y ad E z. Repeatig the above discussio of the oe-dimesioal partial cofiemet, we ca epect that a wave mechaics problem may be substatially simplified if E y ad E z are much larger tha the eergy scale E of the motio i directio. Namely, if: (i) the potetial profile withi the D partial cofiemet plae [y, z] is arbitrary (provided that it provides partial cofiemet scales a y ad a z much smaller the spatial scale of the motio i directio ), ad (ii) the potetial eergy U is either costat i time or chages relatively slowly, at a time scale >> /E yz (where E yz is the lowest eigeeergy of motio withi the [y, z] plae), the a large rage of eperimets may be adequately described by lookig for solutio of the geeral (time-depedet, 3D) Schrödiger equatio i the form of the followig product E yz (, t) YZ( y, z) ep i t, (.9) where YZ is the lowest (groud-state) eigefuctio of the D problem i the [y, z] plae. Substitutig this solutio to the equatio, ad separatig variables {y, z} from {, t}, we obtai the followig timedepedet, D equatio Ψ(, t) Ψ(, t) i U (, t)ψ(, t). (.9) t m The et chapter will be devoted to a detailed discussio of the wave mechaics described by this D equatio, because it allows to study most basic pheomea ad cocepts of wave mechaics without ivolvig overly comple math. I that chapter, for the otatio simplicity, eergy E D motio will be referred to ust as E. However, oe should always remember that each D problem has two hidde degrees of freedom ad that the geuie eergy of the particle also icludes a costat shift E yz which is typically much larger tha E. The Uiverse is (at least :-) 3-dimesioal, ad it shows! Fially, ote that i systems with reduced dimesioality, Eq. (8) for the umber of states at large k (i.e., for a essetially free particle motio) should be replaced accordigly: i a D system of area A >> /k, A dn d k, (.93) while i a D system of legth l >> /k, l dn dk, (.94) with the correspodig chages of the summatio rule (83). This chage has importat implicatios for the desity of states o the eergy scale, dn/de: it is straightforward (ad hece left for the reader :-) to use Eqs. (8), (93), ad (94) to show that for free 3D particles the desity icreases with E Effective potetial eergy D timedepedet Schrödiger equatio D umber of states D umber of states Chapter Page 4 of 6

30 (proportioally to E / ), for free D particles it does ot deped o eergy, while for free D particles it scales as E -/, i.e. decreases with eergy..7. Eercise problems.. The actual postulate made by N. Bohr i his origial 93 paper was ot directly Eq. (), but a assumptio that at quatum leaps betwee adacet large (quasiclassical) orbits with >>, hydroge atom either emits or absorbs eergy E =, where is its classical radiatio frequecy - accordig to classical electrodyamics, equal to the agular velocity of electro s rotatio. Prove that this postulate is ideed compatible with Eqs. (8)-(). A.. Use Eq. (53) to prove that liear operators of quatum mechaics are commutative: A A A A A A A A A., ad associative: 3 3 g(r),.3. Prove that for ay Hamiltoia operator Ĥ ad two arbitrary comple fuctios f(r) ad 3 3 rhg rd r Hf rg rd r f..4. Prove that the Schrödiger equatio (.5) with Hamiltoia (.4) is Galilea-ivariat, provided that the wave fuctio is trasformed as mv r mv t ' r ', t' r, t ep i i, where the prime sig deotes the variables measured i the referece frame O that moves, without rotatio, with a costat velocity v relatively to the lab frame O. Give a physical iterpretatio of this trasformatio..5. * Prove the so-called Hellma-Feyma theorem: 57 E H where is some parameter, o whom the Hamiltoia Ĥ, ad hece its eigeeergies E deped..6. Calculate, p,, ad p for eigestate {, y, z } of a rectagular, ifiitely deep quatum well (69). Compare product p with Heiseberg s ucertaity relatio., 57 Despite the theorem s ame, H. Hellma (i 937) ad R. Feyma (i 939) were ot the first i the log list of physicists who have (apparetly, idepedetly) discovered this fact. Ideed, it may be traced back at least to a 9 paper by W. Pauli, ad was carefully proved by P. Güttiger i 93. Chapter Page 5 of 6

31 .7. A particle, placed i a hard-wall, rectagular bo with sides a, a y, ad a z, is i its groud state. Calculate the average force actig o each face of the bo. Ca the forces be characterized by a certai pressure?.8. A D quatum particle was iitially i the groud state of a very deep, rectagular quatum well of width a:, for a / a /, U ( ), otherwise. At some istat, the well s width is abruptly icreased to value a > a (leavig the well symmetric about poit = ), ad the left costat. Calculate the probability that after the chage, the particle is still i the groud state of the system..9. At t =, a D particle of mass m is placed ito a hard-wall, flat-bottom potetial well, for a, U ( ), otherwise, i a 5/5 liear superpositio of the lowest (groud) ad the first ecited states, so that its wavefuctio at that istat is (,) C, where C is the ormalizatio costat which esures that the particle is (somewhere) i the well with probability W =. Calculate: (i) the ormalized wavefuctio (, t) for arbitrary time t, ad (ii) the time evolutio of the epectatio value of particle s coordiate... Fid the potetial profile U() for which the followig wavefuctios, (i) c ep a ibt (ii) c ep a ibt,, ad (with real coefficiets a > ad b), satisfy the Schrödiger equatio for a particle with mass m. For each case, calculate, p,, ad p, ad compare the product p with Heiseberg s ucertaity relatio... Calculate the eergy desity dn/de of travelig wave states i large rectagular quatum wells of various dimesios: d =,, ad 3... * Use the fiite differece method with steps a/ ad a/3 to fid as may eigeeergies as possible for a particle i the ifiitely deep, hard-wall quatum well of width a. Compare the results with each other, with the eact formula. 58 g e 58 You may like to start from readig about the fiite-differece method - see, e.g., CM Sec. 8.5 or EM Sec..8. Chapter Page 6 of 6

32 Chapter. D Wave Mechaics The mai goal of this chapter is the solutio ad discussio of a few coceptually most importat problems of wave mechaics for the simplest, D case. This lowest dimesioality, ad a wide use of potetial profiles approimatio by sets of Dirac s delta-fuctios, simplify the ecessary calculatios cosiderably without sacrificig the physical essece of the described pheomea. The reader is advised to pay special attetio to Sectios 6-9, which cover some importat material ot usually discussed i tetbooks. Schrödiger equatio Probability Normalizatio Epectatio value Probability curret Cotiuity equatio.. Probability curret ad ucertaity relatios As was discussed i the ed of Chapter, i several cases (most importatly, at strog cofiemet withi the [y, z] plae), the geeral (3D) Schrödiger equatio may be reduced to the D equatio (.9): (, t) (, t) i U (, t) (, t). (.) t m If the trasversal factor say, the fuctio YZ (y, z) that participates i Eq. (.9), is ormalized to uity, the the itegratio of Eq. (.a) over a segmet [, ], gives the probability to fid the particle o this segmet: W ( t) Ψ(, t)ψ (, t) d. (.) If the particle uder aalysis is defiitely iside the system, the ormalizatio of its D wavefuctio (, t) is provided by etedig itegral () to the whole ais : * w(, t) d, where w(, t) Ψ(, t)ψ * (, t). (.3) A similar itegratio of Eq. (.3) shows that the epectatio value of ay operator depedig oly o coordiate (ad possibly time), may be epressed as * A ( t) Ψ (, t) A Ψ(, t) d. (.4) It is also useful to itroduce the probability curret alog the -ais (a scalar): * I(, t) dydz ImΨ Ψ Ψ(, t), (.5) m m where is -compoet of the probability curret desity vector (r,t). The the cotiuity equatio (.48) for the segmet [, ] takes the form dw dt I ( ) I( ). (.6) K. Likharev

33 The above formulas are the basis for the aalysis of D problems of wave mechaics, but before proceedig to particular cases, let me deliver o my earlier promise to prove that Heiseberg s ucertaity relatio (.35) is ideed valid for ay wavefuctio (,t). For that, let us cosider a evidetly positive (or at least o-egative) itegral J d, (.7) where is a arbitrary real costat, ad assume that at the at the wavefuctio vaishes, together with its first derivative. The left-had part of Eq. (7) may be recast as d d * * * d d * Accordig to Eq. (4), the first term i the last form of Eq. (8) is ust. The secod ad the third itegrals may be worked out by parts: * d. * * * d d d * * * d (.8), (.9) * d * * d As a result, Eq. (7) takes the followig form: * * d p d p. (.) J p, i.e. a b, with a ad b p p. (.) This iequality should be valid for ay real, i.e. the correspodig quadratic equatio, + a + b =, ca have either oe (degeerate) real root - or o real roots at all. This is oly possible if its determiat, Det = a 4b, is o-positive, leadig to the followig requiremet: p. (.) 4 I particular, if = ad p =, the accordig to Eq. (.33), Eq. () takes the form ~ ~ p, (.3) 4 which, accordig to the defiitio (.34) of r.m.s. ucertaities, is equivalet to Eq. (.35). Heiseberg s ucertaity relatio Eq. (3) may be proved eve if ad p are ot equal to zero, by makig the followig replacemets, -, / / + ip/, i Eq. (7), ad the repeatig all the calculatios which become rather bulky. We will re-derive the ucertaity relatios, i a more efficiet way, i Chapter 4. Chapter Page of 76

34 Coordiate/ mometum operators commutator Now let us otice that the Heiseberg s ucertaity relatio looks very similar to the commutatio relatio betwee the correspodig operators:, p p p i i i. (.4a) Sice this relatio is valid for arbitrary wavefuctio (, t), we may preset it as a operator equality:, p i. (.4b) I Sec. 4.5 we will see that the relatio betwee Eqs. (3) ad (4) is ust a particular case of a geeral relatio betwee the epectatio values of o-commutig operators ad their commutators. Iitial Gaussia wave packet.. Free particle: Wave packets Let us start our discussio of particular problems with free the D motio, with U(,t) =. From our discussio of Eq. (.9) i Chapter, it is clear that i the D case, a similar fudametal (i.e. a particular but the most importat) solutio of the Schrödiger equatio () is a moochromatic wave i( kt) (, t) cost e. (.5) Accordig to Eqs. (.3), it correspods to a particle with a eactly defied mometum p = k ad eergy E = = k /m. However, for this wavefuctio, product * does ot deped o either or t, so that the particle is completely delocalized, i.e. its probability is spread all over ais, at all times. (As a result, such state is still compatible with Heiseberg s ucertaity relatio (3), despite the eact value p of mometum p.) I order to describe a space-localized particle, let us form, at the iitial momet of time (t = ), a wave packet of the type show i Fig..6, by multiplyig the siusoidal waveform (5) by some smooth evelope fuctio A(). As the most importat particular eample, cosider a Gaussia packet ik (,) Ae, with A ep /4 / -. (.6) ( ) ( ) ( ) (By the way, Fig..6 shows eactly such a packet.) The pre-epoetial factor i this evelope fuctio has bee selected i the way to have the iitial probability desity, * * w(,) (,) (,) A ( ) A( ) ep ormalized accordig to Eq. (3), for ay parameters ad k. 3 / ( ), (.7) I order to eplore the evolutio of this packet i time, we could try to solve Eq. () with the iitial coditio (6) directly, but i the spirit of the discussio i Sec..5, it is easier to proceed From this poit o, i this chapter I will drop ide i otatio for -compoet of vectors k ad p. 3 This may be readily prove usig the well-kow itegral of the Gaussia fuctio ( bell curve ) give by Eq. (7) see, e.g., MA Eq. (6.9b). It is also straightforward to use MA Eq. (6.9c) to prove that for wave packet (6), parameter is ideed the r.m.s. ucertaity (.34) of coordiate, thus ustifyig its otatio. Chapter Page 3 of 76

35 differetly. Let us first preset the iitial wavefuctio (6) as a sum (.65) of eigefuctios k () of the correspodig statioary D Schrödiger equatio (.6), i our curret case d k m d that are simply moochromatic waves, E, k k with E k k m, (.8) ik k a k e, (.9) with a cotiuum spectrum of possible wave umbers k. For that, sum (.65) should be replaced with a itegral: 4 ik (,) a ( ) dp a e dk. (.) k k Now let us otice that from the poit of view of mathematics, Eq. () is ust the usual Fourier trasform from variable k to the cougate variable, ad we ca use the well-kow formula of the reciprocal Fourier trasform to calculate a k ik (,) e d /4 ( ) ( ) / k ep- () ~ ikd, ~ where k k k, (.) This Gaussia itegral may be worked out by the followig stadard method. Let us complemet the epoet to the full square of a liear combiatio of ad k, plus a term idepedet of : ~ ik ~ ( ) k ( ) - ~ ik () (). (.) Sice the itegratio i the right-had part of Eq. () should be performed at costat k ~, i the ifiite limits, its result would ot chage if we replace d by d d[ + i() k ~ ]. 5 As a result, we get, / ~ ~ ep ep - ' ep. /4 / ( ) ( ) ' k a k k /4 / d ( ) ( ) ( k) ( k (.3) ) so that a k also has a Gaussia distributio, ow alog ais k, cetered to value k (Fig..6b), with costat k defied as Thus we may preset the iitial wave packet (6) as / k /. (.4) ( k k ) (,) k ep /4 / ( ) ( ) (k ) e ik dk. (.5) From compariso of this formula with Eq. (6), it is evidet that the r.m.s. ucertaity of the wave umber k i this packet is ideed equal to k defied by Eq. (4), thus ustifyig the otatio. The 4 For otatio s brevity, from this poit o the ifiite limit sigs will be dropped i all D itegrals. 5 The fact that the argumet shift is imagiary is ot importat, because fuctio uder the itegral is aalytical, ad teds to zero at Re. Chapter Page 4 of 76

36 Gaussia wave packet at arbitrary time compariso of that relatio with Eq. (.35) shows that the Gaussia packet presets the ultimate case i which the product p = (k) has the lowest possible value (/); for ay other evelope s shape the ucertaity product may oly be larger. We could of course get the same result for k from Eq. (6) usig defiitios (.3), (.33), ad (.34); the real advatage of Eq. (4) is that it ca be readily geeralized to t >. Ideed, we already kow that the time evolutio of the wavefuctio is give by Eq. (.67), for our case givig 6 / ( k k ) ik k (, t) e i tdk k ep ep. (.6) /4 / ( ) ( ) (k ) m Fig. shows several sapshots of the real part of wavefuctio (6), for a particular case k =. k. Re ) t v ph v gr t. v / Fig... Time evolutio of the wave D wave packet evolutio o: (a) a smaller ad (b) larger time scales. Dashed lies show packet s evelope, i.e.. Re t t 3 v t v / The plots clearly show the followig effects: (i) the wave packet as a whole (as characterized by its evelope) moves alog the ais with a certai group velocity v gr, 6 Note that this packet is equivalet to Eq. (6) ad hece is properly ormalized to see Eq. (3). Hece the wave packet itroductio offers a atural solutio to the problem of ifiite wave ormalizatio, which was metioed i Sec... Chapter Page 5 of 76

37 (ii) the carrier wave iside the packet moves with a differet, phase velocity v ph, which may be defied as the velocity the spatial poits where wave s phase (, t) arg takes a certai fied value (say, = /, where Re vaishes), ad (iii) the packet s spatial width gradually icreases with time - the packet spreads. All these effects are commo for waves of ay physical ature. 7 Ideed, let us cosider a D wave packet of the type (6), i( kt), t a e dk k, (.7) propagatig i a media with a arbitrary (but smooth!) dispersio relatio (k), ad assume that the wave umber distributio a k is arbitrary but arrow: k << k k - see Fig..6b. The we may epad fuctio (k) ito the Taylor series ear the cetral poit k, ad keep oly two of its leadig terms: d ~ d ~ ~ ( k) k k, where k k k, k, (.8) dk dk ad both derivatives are also evaluated at poit k = k. I this approimatio, 8 the epressio i paretheses i the right-had part of Eq. (7) may be rewritte as ~ d ~ d ~ ~ d d ~ k t k k k k t dk dk dk dk so that Eq. (7) is reduced to itegral k t k t k t i ( k t ~ ~ d dk d ~ dk, t e a ep i k t k t dk ) k, (.9). (.3) First, let eglect the last term i square brackets (which is much smaller tha the first term if the dispersio relatio is smooth eough ad/or the time iterval t is sufficietly small), ad compare the result with the iitial form of the wave packet (7) ik, ake dk Ae, with A The compariso shows that Eq. (3) is reduced to ik ~ ik ~ a e dk. (.3) ik v t) ( ph, t A( vgr t) e, (.3) k Arbitrary wave packet where v gr ad v ph are two costats with the dimesio of velocity: v gr d dk kk v k, ad ph kk. (.33) It is clear that Eq. (3) describes effects (i) ad (ii) listed above. Let us calculate the group ad phase velocities for the particular case of de Broglie waves whose dispersio law is give by Eq. (.3): Group ad phase velocities 7 See, e.g., brief discussios i CM Sec. 5.3 ad EM Sec By the way, i the particular case of de Broglie wave described by dispersio relatio (.3), Eq. (8) is eact, because = E/ is a quadratic fuctio of k = p/, ad all higher derivatives of over k vaish for ay k. Chapter Page 6 of 76

38 k d k k v gr, vgr k k v, v ph k k. (.34) m dk m k m We see that (very fortuately!) the velocity of the wave packet evelope is costat ad equals to that of the classical particle movig by iertia, i accordace with the correspodece priciple. The remaiig term i the square brackets of Eq. (3) describes effect (iii), the wave packet s spread. It may be readily evaluated if the packet (7) is iitially Gaussia, as i our eample (5): ~ k a k cost ep. (.34) k I this case itegral (3) is Gaussia, ad may be worked out eactly as itegral (), i.e. mergig the epoets uder the itegral, ad presetig them as a full square of liear combiatio of ad k: k ~ ~ i d ~ v t v t ~ gr gr i d ik ( v t k t t k i ik k gr ) ( ) t (k ) dk ( t), (.35) 4( t) dk where I have itroduced the followig comple fuctio of time: i d i d ( t) t ( ) t, (.36) 4( k) dk dk ad have used Eq. (4) i the secod equality. Now itegratig over k ~, we get ( vgrt) (, t) ep 4( t) i k d k t. (.37) dk Wave packet s spread with time The imagiary part of ratio /(t) i the epoet gives ust a additioal cotributio to wave s phase, ad does ot affect the resultig probability distributio * ( vgrt) w (, t) ep Re. (.38) ( t) This is agai a Gaussia bell curve spread over ais, cetered to poit = v gr t, with the r.m.s. width d ( t) dk I the particular case of de Broglie waves, d /dk = /m, so that ' Re t ' t m ( ) ( ). (.39a). (.39b) The physics of the spreadig is very simple: if d /dk, the group velocity d/dk of each small group dk of moochromatic compoets of the wave packet is differet, resultig i the gradual (evetually, liear) accumulatio of the differeces of the distaces traveled by the groups. The most curious feature of Eq. (39) is that the packet width at t > depeds o its iitial width () = i a Chapter Page 7 of 76

39 o-mootoic way, tedig to ifiity at both ad. Because of that, for a fied t, there is a optimal value of with miimizes : ' / t mi opt. (.4) m This epressio may be used for spreadig effect estimates. Due to the smalless of the Plack costat o the huma scale of thigs, for macroscopic bodies this effect is etremely small eve for very log time itervals; however, for light particles it may be very oticeable: for the electro (m = m e -3 kg), ad t = s, Eq. (4) yields ( ) mi ~ cm! Note also that for ay t, the wave packet retais its Gaussia evelope, but the ultimate relatio (4) is ot satisfied, p > / - due to a gradually accumulated phase shift betwee the compoet moochromatic waves. The last remark o this topic: i quatum mechaics, the wave packet spreadig is ot a ubiquitous effect! For eample, i Chapter 5 we will see that i a quatum oscillator, the spatial width of a Gaussia packet (for that system, called the Glauber state) does ot grow mootoically but rather either stays costat or oscillates i time. Now let us briefly discuss the case whe the iitial wave packet is ot Gaussia, but is described by a arbitrary iitial wavefuctio. I order to make the forthcomig result more appealig, it is beeficial to geeralize out calculatios to a arbitrary iitial time t ; it is evidet that if U does ot deped o time eplicitly, it is sufficiet to replace t with (t t ) i all above formulas. With this replacemet, Eq. (7) becomes ik ( t t ) t a e (, ) dk, (.4) ad the reciprocal trasform () reads ak (, t ) e k ik d. (.4) If we wat to epress these two formulas with oe relatio, i.e. plug Eq. (4) ito Eq. (4), we should give the itegratio variable some other ame, e.g.,. The result is ik t t (, t) dk d (, t ) e. (.43) Chagig the order of itegratio, this epressio may be rewritte i the followig geeral form:, t;, t (, t) (, t) G d, (.44) where fuctio G, usually called kerel i mathematics, i quatum mechaics is called the propagator. 9 Accordig to Eq. (43), i our particular case of a free particle the propagator is equal to D propagator: defiitio 9 Its stadard otatio by letter G stems from the fact that the propagator is essetially the spatial-temporal Gree s fuctio of Eq. (.8), defied very similarly to Gree s fuctios of other ordiary ad partial differetial equatios describig various physics systems see, e.g., CM Sec. 4. ad/or EM Sec..7 ad 7.3. Chapter Page 8 of 76

40 G, t;, t ik t t e dk, (.45) The physical sese of the propagator may be uderstood by cosiderig the followig special iitial coditios: (, t ) ( ' ), (.46) where is a certai poit withi the domai of particle s motio. I this particular case, Eq. (44) evidetly gives (, t) G(, t; ', t ). (.47) Ψ Hece, the propagator, cosidered as a fuctio of ad t oly, is ust the solutio of the liear differetial equatio with -fuctioal iitial coditios. Thus while Eq. (4) may be uderstood as a mathematical epressio of the liear superpositio priciple i the mometum (i.e., reciprocal) space domai, Eq. (44) is a epressio of this priciple i the direct space domai: the system s respose (,t) to a arbitrary iitial coditio (,t ) is ust a sum of its resposes to its thi spatial slices, with propagator G(,t;,t ) represetig the weight of each slice i the fial sum. Calculatig itegral (45), oe should remember that is ot a costat but a fuctio of k, give by the dispersio relatio for particular waves. I particular, for the de Broglie waves k G (, t;, t ) epik ( t t ) dk. (.48) m This is a Gaussia itegral agai, ad may be readily calculated ust it was doe (twice) above, by completig the epoet to the full square. The result is Free particle s propagator G(, t;, t m ) i( t t ) / m( ) ep. (.49) i( t t ) Please ote the followig features of this comple fuctio (plotted i Fig. ):.5 Re Im G(, t; m / ( t t ) /, t ).5 ( t t ) / ( m ) / / Fig... Real (solid lie) ad imagiary (dashed lie) parts of the D free particle s propagator. Note that this iitial coditio is ot equivalet to a -fuctioal iitial probability desity (). Chapter Page 9 of 76

41 (i) It depeds oly o differeces ( - ) ad (t t ). This is atural, because the free-particle propagatio problem is uiform (traslatio-ivariat) both i space ad time. (ii) The fuctio shape does ot deped o its argumets they ust rescale the same fuctio: its sapshot (Fig. ), if plotted as a fuctio of u-ormalized, ust becomes broader ad lower with time. It is curious that the spatial broadeig scales as (t t ) / ust as at the classical diffusio, as a result of a deep aalogy betwee quatum mechaics ad classical statistics to be discussed further i Chapter 7. (iii) I accordace with the ucertaity relatio, the ultimately compressed wave packet (46) has a ifiite width of mometum distributio, ad the quasi-siusoidal tails of the free-particle propagator, clearly visible i Fig., are the results of the free propagatio of the fastest (highestmometum) compoets of that distributio, i both directios from the packet ceter. I the followig sectios, we will mostly focus o the spatial distributio of statioary, moochromatic wavefuctios (that, for ucofied motio, may be iterpreted as wave packets of very large spatial width ), oly rarely comig back to the wave packet discussio. Our ecuse is the liear superpositio priciple, i.e. our coceptual ability to restore the geeral solutio from that of moochromatic waves of all possible eergies. However, the reader should ot forget that, as the above discussio has illustrated, mathematically this restoratio is ot always trivial..3. Particle motio i simple potetial profiles Now, let us proceed to the cases i which the potetial eergy U(,t) is ot idetically equal to zero. The easiest case is that of spatially-uiform but time-depedet potetial: U = U(t) = cost. Ideed, the correspodig Schrödiger equatio (.5) with Hamiltoia p H U ( t) U ( t), (.5) m m allows the variable separatio similar to that performed i Sec..5, besides that the time-depedet fuctio T(t) obeys a equatio of motio that is slightly more geeral tha Eq. (.59): i T E U ( t) T, (.5) whose solutio may be epressed as a evidet geeralizatio of Eq. (.6): i t ( t) E d U ( t) T ( t) T () e, with ad. (.5) dt Lookig at the basic relatios (.) ad (.3) of wave mechaics, it seems that this additioal phase factor does ot affect the particle probability distributio, or eve ay observable (icludig eergy it is referred to the istat value of U), ad hece the phase icremet, associated with U(t), is ust a mathematical artifact. This is certaily true for a sigle particle, however, the situatio chages as soo as we recall that the Uiverse cosists of more that oe of them. For eample, cosider two similar, idepedet particles, each i the same (say, groud) eigestate, but with the potetial eergies (ad hece eigeeergies E, ) differet by a costat U U U. The, the differece - of their wavefuctio phases evolves i time as Chapter Page of 76

42 Quatum phase differece s evolutio d U. (.53) dt If the particles are i differet worlds (or at least i differet laboratories :-), this evolutio is uobservable; however, it should be ituitively clear that a very weak couplig of a certai detector to each particle may allow it to observe phase, while keepig the particle dyamics virtually uperturbed, i.e. Eq. (53) itact. Perhaps the most dramatic demostratio of this pheomeo is the Josephso effect i supercoductors. Eperimetally, the easiest way to observe the effect is by coectig two bulk supercoductor samples with a weak, short electric cotact (called either the weak lik or the Josephso uctio) ad bias them with a costat (dc) voltage V, typically i a few-microvolt rage see Fig. 3. I si( ) epi i ep V Fig..3. Josephso effect i a weak lik betwee two bulk supercoductor electrodes. Josephso effect: basic equatios Supercoductivity may be eplaied by a specific couplig betwee its coductio electros, that leads, at low temperatures, to formatio of the so-called Cooper pairs. Such pairs, each cosistig of two electros with opposite spis ad mometa, behave as Bose particles, ad form coheret Bose- Eistei codesate. Most properties of such a codesate may be described by a sigle wavefuctio, evolvig i time as that of a free particle with the effective potetial eergy U = q = -e, where is the electrochemical potetial, 3 ad q = -e is the total charge of the Cooper pair. As a result, for the situatio show i Fig. 3, Eq. (53) takes the form d e V, (.54) dt where V = - is the applied voltage. B. Josephso has predicted that, i a particular case whe a weal lik is a tuel uctio, electric curret I of Cooper pairs through it should have a simple form: 4 I si, (.55) I c It was predicted theoretically by B. Josephso (the a graduate studet!) i 96 ad observed eperimetally i less tha a year. More recetly, aalogs of this effect were also observed i superfluid helium ad atomic Bose- Eistei codesates. See, e.g., SM Sec For more o this otio see, e.g. SM Sec Later, Eq. (55) has bee show to be valid for other weak lik types as well, though deviatios from have also bee foud. These deviatios, however, do ot affect the fudametal -periodicity of fuctio I() see, e.g., EM Sec As a result, o deviatios from the fudametal relatios (56)-(57) have bee foud (yet :-). Chapter Page of 76

43 where I c is some costat (scalig as the weak lik stregth). Combiig Eqs. (53) ad (54), we see that if the applied voltage is costat i time, the curret oscillates with the so-called Josephso frequecy J e f J, where J V, (.56) as high as ~ 484 MHz per each microvolt of applied dc voltage. This effect is ow well documeted, though a direct detectio of the Josephso radiatio is tricky; it is much easier to observe the phase lockig (sychroizatio) 5 of the radiatio by eteral microwave sigal, which results i formatio of early flat dc curret steps at dc voltages V, (.57) e where is the eteral sigal frequecy ad is a iteger. 6 This effect is ow beig used i highly accurate stadards of dc voltage. 7 Now, let us move o to a discussio of the opposite case, whe a D particle modes i various potetial profiles U() that are costat i time. Coceptually, the simplest of such profiles is a potetial step see Fig. 4. classically accessible classically forbidde E U () c classical turig poit Fig..4. Classical D motio i a potetial profile U(). As I am sure the reader kows, i classical mechaics, if a particle is icidet o such a step (i Fig. 4, from the left), its kietic eergy p /m caot be egative, so that it ca oly travel through the classically accessible regio where its (coserved) full eergy, p E U ( ), (.58) m is larger tha the local value U(). Let the iitial velocity v = p/m be positive, i.e. directed toward the step. Before it has reached the classical turig poit c, defied by equatio U ( c ) E, (.59) 5 See, e.g., CM Sec If is ot too high, this effect may be adequately described combiig Eqs. (54)-(55). Let me leave this task for the reader. 7 The most precise proof that the Josephso frequecy-to-voltage ratio f J /V does ot deped o supercoductig material (to at least 5 decimal places!) has bee carried out by the group led by J. Lukes here at Stoy Brook see J.-S. Tsai et al., Phys. Rev. Lett. 5, 36 (983). Chapter Page of 76

44 kietic eergy p /m ever turs to zero, so that the particle cotiues to move i the iitial directio. O the other had, the particle caot peetrate that classically forbidde regio > c, because there its kietic eergy would be egative there. At the poit = c, particle s velocity chages sig, i.e. it is reflected back from the classical turig poit. I order to see what the wave mechaics says about this situatio, let us start from the simplest, sharp potetial step show with bold black lies i Fig. 5:, at, U ( ) U ( ) (.6) U, at. For this choice, ad ay eergy withi the iterval < E < U, the classical turig poit is c =. E A B U( ), E ( ) U C Fig..5. Reflectio of a moochromatic wave from a potetial step U > E. (This particular wavefuctio s shape is for U = 5E.) The wavefuctio is plotted with the same schematic vertical offset by E, as those i Fig..7. Let us represet a icidet particle with a wave packet so log that the spread k ~ / of its wave umber spectrum, ad hece the eergy ucertaity E = = (d/dk)k is egligible i compariso with its average value E < U, as well as with (U E). I this case, E may be cosidered a give costat, ad the time depedece of the solutio is give by Eq. (.6), ad we ca limit ourselves to the solutio of the D versio of the statioary Schrödiger equatio (.63), i this case d U ( ) E, (.6) m d for the spatial part () of the wavefuctio. 8 At <, i.e. at U =, the equatio is reduced to the Helmholtz equatio (.75), ad may be satisfied with two travelig waves, proportioal to ep{+ik} ad ep{-ik} correspodigly, with k satisfyig the dispersio equatio (.3): me k. (.6) Thus the geeral solutio of Eq. (6) i this regio may be preseted as 8 Note that this is ot the eigeproblem like the oe we have solved i Sec..4 for a quatum well. Ideed, ow eergy E is cosidered fied e.g., by the iitial coditios that lauch a log wave packet upo the potetial step, from the left. Chapter Page 3 of 76

45 ik ik Ae Be. (.63) The secod term i the right-had part evidetly describes a (ifiitely log) wave packet travelig to the left, which represets particle s reflectio from the potetial step. If B = -A, this solutio is reduced to Eq. (.76) for the potetial well with ifiitely high walls, but as we will see i a miute, for our curret case of fiite step height U, the relatio betwee coefficiets B ad A may be differet. To show this, let us solve Eq. (6) for >, where U = U > E. I this regio the equatio may be rewritte as d, (.64) d where is a real costat defied by the relatio similar to Eq. (6): m( U E). (.65) The geeral solutio of Eq. (64) is the sum of ep{+} ad ep{-}, with arbitrary coefficiets. However, the wavefuctio should be fiite at, so oly the latter epoet is acceptable: Ce. (.66) This peetratio of the wavefuctio ito the classically forbidde regio, ad hece a fiite probability to fid the particle there, is oe of the most fasciatig predictios of quatum mechaics, ad has bee repeatedly observed i eperimet, e.g., via tuelig eperimets see below. From Eq. (66), it is evidet that the costat, defied by Eqs. (65), may be iterpreted as the reciprocal peetratio depth. Eve for the lightest particles this depth is usually very small. Ideed, for E << U that equatio yields E. (.67) mu / For eample, for a coductio electro i a typical metal, that rus, at its surface, ito a sharp potetial step U, whose height equals to metal s workfuctio W 5 ev (see the discussio of the photoelectric effect i Sec..), is close to. m, i.e. is close to a typical size of a atom. For heavier elemetary particles (e.g., protos) the peetratio depth is correspodigly lower, ad for macroscopic bodies it is hardly measurable. Returig to our problem, we still should fid coefficiets A, B, ad C from the boudary coditios at =. Sice E is a fiite costat, ad U() is a fiite fuctio, Eq. (6) says that d /d should be fiite as well. This meas that the first derivative should be cotiuous: lim d d d m lim d lim d d d U ( ) E d. (.68) Repeatig such calculatio for fuctio () itself, we see that it also should be cotiuous at all poits, icludig =, so that Icidet ad reflected waves Decayig wave i classically forbidde regio Chapter Page 4 of 76

46 d d () (), () (). (.69) d d Pluggig solutios (63) ad (66) ito these two boudary coditios, we get a system of two liear equatios A B C, ika ikb C, (.7) whose (elemetary) solutio eables us to epress B ad C via A : k i k B A, C A. (.7) k i k i We immediately see that sice the omiator ad deomiator i the first of these formulas have equal moduli, so that B = A. This meas that, as we could epect, a particle with eergy E < U is totally reflected from the step. As a result, at < our solutio (63) may be preseted by a stadig wave k iae i si( k ), with ta. (.7) Notice that the shift /k = (ta - k/)/k of the stadig wave to the right, due to the partial peetratio of the wavefuctio uder the potetial step, is commesurate with, but geerally ot equal to /. Figure 5 shows the full behavior of the wavefuctio, for a particular case E = U /5, at which k/ = [E/(U -E)] / = /. Accordig to Eq. (65), as the particle s eergy E is icreased to approach U, the peetratio depth / diverges. This raises a importat issue: what happes at E > U, i.e. if there is o classically forbidde regio i the problem? Agai, i classical mechaics the icidet particle would cotiue to move to the right, though with a reduced velocity, correspodig to the ew kietic eergy E U, so there would be o reflectio. I quatum mechaics, however, the situatio is differet. I order to aalyze it, it is ot ecessary to re-solve the whole problem; it is sufficiet to ote that all our calculatios, ad hece Eqs. (7) are still valid if we take 9 m( E U ) ik', with k'. (.73) With this replacemet, Eq. (7) becomes k k' k B A, C A. (.74) k k' k k' The most importat result of this chage is that ow the reflectio is ot complete: B < A. I order to evaluate this effect qualitatively, it is more fair to use ot the B/A or C/A ratios, but rather that 9 Our earlier discardig of the particular solutio ep{}, ow becomig ep{-ik }, is still valid, but ow o a differet grouds: this term would describe a wave packet icidet o the potetial step from the right, ad this is ot the problem uder our cosideratio. These formulas are completely similar to those for the partial reflectio of classical waves from a sharp iterface betwee two uiform media, at ormal icidece (see, e.g., CM Sec. 5.4 ad EM Sec. 7.4), with the effective impedace Z of de Broglie waves proportioal to their wave umber k. Chapter Page 5 of 76

47 of the probability currets (5) correspodig to travelig waves with amplitudes C ad A, i the correspodig regios (respectively, > ad < ): / 4E( E U) E / E U / I k' C C 4kk' T. (.75) I k A ( k k' ) A (T so defied is called the trasparecy of the ihomogeeity, i our curret case of the potetial step.) The result give by Eq. (75) is plotted i Fig. 6a. Notice its most importat features: (i) At U =, the trasparecy is full, T = aturally, for havig o step at all. (ii) At U E, the trasparecy teds to zero - givig a proper coectio with the case E < U. (iii) We ca use result (75) eve for U <, i.e. for the step-dow (or cliff ) profile see Fig. 6b. Very couter-ituitively, the particle is (partly) reflected eve from such a cliff, ad the trasmissio dimiishes (rather slowly) at U -. Potetial step s trasmissio.8 (a) E U A B C (b) T.6.4 U. U / E The most importat coceptual coclusio of our aalysis is that the quatum particle is partly reflected from a potetial step with U < E, i the sese that there is a ovaishig probability T < to fid it passed over the step, while there is also probability ( T) to have it reflected. The same property is ehibited, for ay relatio betwee E ad U, by aother simple potetial profile U(), the famous tuel barrier. Figure 7 shows its simple, rectagular versio: U U Fig..6. (a) Trasparecy of a potetial step with U < E as a fuctio of its height, accordig to Eq. (75), ad (b) the potetial profile at U <., for d /, U ( ) U, for d / d /, (.76), for d /. E A B C D F U d / d / Fig..7. Rectagular tuel barrier. Chapter Page 6 of 76

48 Rectagular tuel barrier s trasparecy I order to aalyze this problem, it is sufficiet to look for the solutio to the Schrödiger equatio i the form (63) at -d/. At > +d/, i.e., behid the barrier, we may use the argumets preseted above (o wave packet source o the right!) to keep ust oe travelig wave, ik ( Fe. (.77) ) However, uder the barrier, i.e. at -d/ +d/, we should geerally keep both epoetial terms, ( ) Ce De, (.78) b because our previous argumet, used i the potetial step problem s solutio, is o loger valid. (Here k ad are still defied, respectively, by Eqs. (6) ad (65).) I order to fid the relatio betwee coefficiets A, B, C, D, ad F, we eed to plug i the solutios ito the boudary coditios similar to Eqs. (69), but ow at two boudary poits, = d/. Solvig the resultig system of 4 liear equatios for five amplitudes (A, B, C, D, ad F), we ca readily calculate four ratios B/A, C/A, etc., i particular, ad hece barrier s trasparecy T F A F A ep ikd (.79a) i k coshd sihd k cosh k d sih d. (.79b) Figure 8a shows the trasparecy as a fuctio of particle eergy E, for several characteristic values of the barrier thickess d, or rather of the ratio d/, where is defied by Eq. (67). k (a) (b) T d /U T E / E / U Fig..8. Trasparecy of the rectagular tuel barrier as a fuctio of particle s eergy E. d 3 / 3. Chapter Page 7 of 76

49 The plots show that for a thi barrier (d < ) the trasparecy grows gradually with particle s eergy. This growth is atural, because the peetratio costat decreases with the growth of E, i.e., the wavefuctio peetrates more ad more ito the barrier, so that more ad more of it is picked up at the secod iterface ( = +d/) ad trasferred ito the wave Fep{ik} propagatig behid the barrier. As Eq. (79b) shows, for thick barriers (d >> ), this depedece is domiated by a epoet, 4k d T e, (.8) k that may be clearly see as a straight segmets i semi-log plots (Fig. 8b) of T as a fuctio of the combiatio ( E/U ) / which is proportioal to - see Eq. (65). Equatio (8) also clearly shows the epoetial depedece of the barrier trasparecy of its thickess at d >>. This depedece is the most importat factor for various applicatios of the quatum-mechaical tuelig from the field emissio of electros to scaig tuelig microscopy. Also oted should be substatial egative implicatios of the effect for moder electroic egieerig, most importatly imposig a limit for scalig dow of field effect trasistors i semicoductor itegrated circuits (ad hece the circuit desity icrease accordig to the well-kow Moore s law), due to icrease of tuelig both through the gate oide ad alog trasistor s chael. 3 Aother iterestig effect visible i Fig. 8a (for case d =.3) are the oscillatios of T at E > U. This is our first glimpse at oe more iterestig quatum effect: resoat tuelig. I will discuss this effect i detail i Sec. 5 below. Thick tuel barrier s trasparecy.4. The WKB approimatio Before movig o to eplorig more comple potetials, let us see whether the results discussed i the previous sectio hold o i the opposite limit of so-called soft, gradual potetial profiles, like that sketched i Fig. 4. (The quatitative coditios of the softess will be derived below). The most efficiet aalytical tool i this limit is the WKB (or quasiclassical ) approimatio developed by H. Jeffrey, G. Wetzel, A. Kramers, ad L. Brilloui i I order to derive its D versio, let us rewrite the Schrödiger equatio (6) as d k ( ) d where the local value of wave umber k() is defied similarly to Eq. (73), k m E U ( ) ( ) (.8) ; (.8) but ow it may be a fuctio of. We already kow that for k() = cost, the fudametal solutios of this equatio have form Aep{+ik} ad Bep{-ik}. Ay of them may be preseted i a simple form Local wave umber See, e.g., G. Fursey, Field Emissio i Vacuum Microelectroics, Kluwer, New York, 5. See, e.g., G. Biig ad H. Rohrer, Helv. Phys. Acta 55, 76 (98). 3 See, e.g., V. Sverdlov et al., IEEE Tras. o Electro Devices 5, 96 (3). Chapter Page 8 of 76

50 i( ) ( ) e, (.83) where () is a comple fuctio, i this simplest case equal to either (k ila) or (-k ilb). This is why we may try use Eq. (83) to look for solutio of Eq. (8) eve i the geeral case, k() cost. Differetiatig Eq. (83) twice, we get d d i d d d i i e,. i e (.84) d d d d d Pluggig the last epressio ito Eq. (8) ad requirig the factor before ep{i()} to vaish, we get d d i ( ) k. (.85) d d This is still a eact, geeral result. At the first sight, it looks worse tha the iitial equatio (8), because Eq. (85) is oliear. However, it is more ready for simplificatio i the limit whe the potetial profile is very smooth, du/d. Ideed, we kow that for a uiform potetial, =. Hece, i the th approimatio, () (), we may try to keep that result, so that Eq. (85) yields d k d Just as i the uiform case, this equatio has two roots, so that its geeral solutio is d d ( ). (.86a) k( ), (.86b) ( ) Aep i k( ' ) d' B ep i k( ' ) d', (.87) where is the lower limits of itegratio affect oly costats A ad B. The physical sese of this result is simple: it is a sum of forward- ad back-propagatig waves, with the coordiate-depedet local wave umber k() that self-adusts to the potetial profile. Let me emphasize the o-trivial ature of this approimatio. 4 First, ay attempt to address the problem with a stadard perturbatio approach (say, = + +, with proportioal to th power of some small parameter, 5 i this case scalig d U/d ) would fail for most potetials, because eve a slight but persistig deviatio of U() from a costat leads to a gradual accumulatio of phase, impossible to describe by ay small perturbatio of. Secod, the droppig of term d /d i Eq. (85) is ot too easy to ustify. Ideed, sice we are committed to the soft potetial limit du/d, we should be ready to assume the characteristic legth a of spatial variatio of to be large, ad eglect 4 Philosophically, this space-domai method is very close to the time-domai rotatig wave approimatio (RWA) used, for eample, i the classical ad quatum theory of oscillatios see, e.g., CM Secs , ad Secs. 6.5, 7.6, 7.7, 9., ad 9.4 of this course. 5 Such perturbatio theories will be discussed i Chapter 6. Chapter Page 9 of 76

51 the terms that are the smallest oes i the limit a. However, both first terms i Eq. (85) are apparetly of the same order i a, amely O(a - ); why have we eglected ust oe of them? The price we have paid for such a sloppy treatmet is high: Eq. (87) does ot satisfy the fudametal property of the Schrödiger equatio, the probability curret coservatio. Ideed, sice Eq. (8) describes a fied-eergy (statioary) spatial part of the geeral Schrödiger equatio, its probability desity w = * =*, ad should ot deped o time. Hece, accordig to Eq. (6), we should have I() = cost. However, this is ot true for each compoet of Eq. (87); for eample for the forward-propagatig compoet of its right-had part, Eq. (5) yields I ( ) A k( ), (.88) m evidetly ot a costat if k() cost. The brilliace of the WKB theory is that the problem may be fied without revisig the th approimatio. Ideed, let us eplore the et, st approimatio istead: ) ( ) ( ), (.89) WKB ( where still obeys Eq. (85), while describes a small correctio to the th approimatio, i the followig sese: 6 d d d d k( ). (.9) Pluggig Eq. (89) ito Eq. (85), with the accout of the defiitio (86), we get d d d d d i. (.9) d d d d d Usig coditio (9), we may eglect d /d i compariso with d /d i the first parethesis, ad d /d i compariso with d /d i the secod parethesis. As a result, we get the followig approimate result: d d i d d d / d i d d l d d Φ i d d d / l k( ) i l k ( ) d, (.9) iφ iφ i k( ' ) d' l, (.93) / k ( ) Φ WKB i a b WKB ( ) ep ( ) ep ( ), for. / i / ( ) k ' d' i ( ) k ' d' k (.94) k k WKB wavefuctio (Agai, the lower itegratio limit is arbitrary, but its choice may be icorporated ito comple costats a ad b.) This modificatio of the th approimatio (87) overcomes the problem of curret cotiuity; for eample, for the forward-propagatig wave, Eq. (5) gives 6 For certaity, I will use the discretio give by Eq. (8) to defie k() as the positive root of its right-had part. Chapter Page of 76

52 WKB probability curret WKB: first coditio of validity I WKB ( ) a m cost. (.95) Physically, factor k / i the deomiator of the WKB wavefuctio s pre-epoet is easy to uderstad. The smaller the local group velocity (34) of the wave packet, v gr () = k()/m, the easier (more probable) it should be to fid the particle withi a certai iterval d. This is eactly the result that WKB gives: dw/d = w() = * /k() /v gr. Aother value of the st approimatio is a clarificatio of WKB theory s validity coditio: it is give by Eq. (9). Pluggig ito this relatio the first form of Eq. (9), ad estimatig as /a, where a is the spatial scale of a substatial chage of = k(), we ca rewrite the coditio as ka. (.96) I plai Eglish, this meas that the regio where U(), ad hece k(), chage substatially should cotai may de Broglie wavelegths = /k. So far I have implied that k () E U() is positive, i.e. particle moves i the classically accessible regio. Now let us eted the WKB approimatio to the situatio where the differece E - U() may chage sig, for eample to the reflectio problem sketched i Fig. 4. Just as we did for the sharp potetial step, we first eed to fid the appropriate solutio for the classically forbidde regio, i this case > c. For that, there is o eed to redo our calculatios, because they are still valid if we, ust as i the sharp step problem, take k() = i(), where m U ( ) E, for c, (.97) ad keep ust oe of two possible solutios (with > ), i aalogy with Eq. (66). The result is c WKB ( ) ep ( ), for i.e., / ( ) ' d' k, κ (.98) with the lower limit at some poit with > as well. This is a really woderful formula! It describes the quatum-mechaical peetratio of the particle ito the classically forbidde regio, ad provides a atural geeralizatio of Eq. (66) - leavig itact, of course, our estimates of the depth ~ / of such peetratio. Now we have to do what we have doe for the sharp-step problem i Sec. : use the boudary coditios i the iterface poit = c to relate costats a, b, ad c. However, ow this operatio is a tad more comple, because both WKB fuctios (94) ad (98) diverge, albeit weakly, at the classical turig poit, were both k() ad () ted to zero. This coectio problem may be however, solved i the followig way. 7 Let us use the commitmet of potetial softess, assumig that it allows us to keep ust two leadig terms i the Taylor epasio of fuctio U() at poit c : du du U ( ) U ( c ) ( c ) E ( c ) c. (.99) c d d 7 A alterative way to solve the coectio problem, without ivolvig the Airy fuctios but usig a aalytical etesio of WKB formulas to the plae of comple argumet, may be foud, e.g., i Sec. 47 of tetbook by L. Ladau ad E. Lifshitz, Quatum Mechaics, No-Relativistic Theory, 3rd ed. Pergamo, 977. Chapter Page of 76

53 Usig this trucated epasio, ad itroducig a dimesioless variable for coordiate s deviatio from the classical turig poit, c, ( / ) m du d, (.) we reduce the Schrödiger equatio (6) to the simple Airy equatio d. d (.) As for all liear, ordiary differetial equatios of the secod order, the geeral solutio of Eq. () may be preseted as a liear combiatio of two fudametal solutios, i this case called Airy fuctios Ai( ) ad Bi( ), show i Fig. 9a. / 3 Airy equatio Bi( ) (a) Ai WKB ( ) (b) Ai( ) ) Ai( ) 3 3 Fig..9. (a) Airy fuctios Ai ad Bi, ad (b) the WKB approimatio for fuctio Ai(). The latter fuctio diverges at, ad thus is ot suitable for our curret problem (Fig. 4), while the former fuctio has the followig asymptotic behaviors at >> : 8 3 / ep, for, 3 Ai( ) (.) / / 4 3 / si, for. 3 4 Now let us apply the WKB approimatio to the Airy equatio (). Takig the classical turig poit ( = ) for the lower limit, for > we get (i dimesioless uits) 8 The followig (eact!) itegral formulas, Ai( ) cos d, Bi( ) ep si, d are ofte coveiet for practical calculatio of Airy fuctios at itermediate values of the argumet, ~. Chapter Page of 76

54 WKB: secod coditio of validity ( ), ( ) /, 3 / ( ' ) d', (.3) 3 i.e. eactly the epoet i the first lie of Eq. (). Makig a similar calculatio for <, with the atural assumptio b = a (full reflectio from the potetial step), we arrive at the followig result: Ai WKB / 4 c ep 3 a si 3 3 / 3 /, for,, for. (.4) This approimatio differs from the eact solutio at small values of, i.e. close to the classical turig poit see Fig. 9b. However, at >>, Eqs. (4) describe the Airy fuctio eactly if a ad c. (.5) 4 Hece we ca use these coectio formulas to epress the relatios betwee coefficiets a, b, ad c of the geeral WKB solutios (94) ad (98). I particular, the first of them yields b = -a ep{i/}, so that Eq. (94) becomes a' WKB ( c ) ep i k( ' ) d' ep i k( ' ) d' i /. (.6) k c c This result may be also described by a simple memoic rule: reflectig from a soft potetial step, the wavefuctio acquires a additioal phase shift = /, if compared with the reflectio from a hard (vertical) potetial wall located at = c, for which, accordig to Eq. (.76), we would have b = -a. Let us quatify the coditio of validity of the coectio formulas (5) - i other words, the criterio of the step softess. For that, withi the regio where the WKB approimatio differs from for the eact Airy equatio ( ~, i.e. - c ~ ), the deviatio from the liear approimatio (99) of the potetial profile should be relatively small. This deviatio may be estimated usig the et term of the Taylor epasio, d U/d = c ( c ) /. As a result, the softess coditio may be epressed as d U/d =c << du/d =c. With the accout of Eq. () for, the coditio becomes 3 d U m du, (.7) d d c c As a eample of a very useful applicatio of the WKB approimatio, let us use it to calculate the eergy spectrum of D particle i a soft D quatum well (Fig. ). As was discussed above, we may always cosider the stadig wave describig a eigestate (correspodig to eigeeergy E ) as a travelig wave goig back ad forth betwee the walls, beig sequetially reflected by each of them. Let us apply the WKB approimatio to such a travelig wave. First, accordig to Eq. (94), propagatig from the left classical turig poit L to the right poit R, it acquires phase chage R k ( ) d. (.8) L 4 Chapter Page 3 of 76

55 At the reflectio from the soft wall at R, accordig to the coectio formula (6), the wave acquires a additioal shift /. Now, travelig back from R to L the wave gets a shift similar to oe give by Eq. (8): =. Fially, at the reflectio from L it gets oe more /. Summig up all these cotributios, we may write the self-cosistecy coditio (that the wavefuctio catches its ow tail with its teeth ), i the form R total ( ), with,,... k d (.9) L Rewritig this result i terms of particle s mometum p() = k(), we arrive at the famous D Bohr- Sommerfeld quatizatio rule p( ) d, (.) C Bohr- Sommerfeld quatizatio rule where the closed path C meas the full period of classical motio. 9 U() E L R Fig... Quasiclassical treatmet of eigestates i a soft D potetial well. Let us see what does this rule give for the very importat particular case of a quadratic potetial profile of a harmoic oscillator of frequecy. I this case, m U ( ) ad the classical turig poits are the roots of a simple equatio, (.) m c E, (.) so that R = (E /m) / / >, L = - <. Due to potetial s symmetry, the itegratio required by Eq. () is also simple: R L p( ) d / / E c c ( ) / c / m E U d me d me c, (.3) 9 Note that at motio i more tha oe dimesio, a closed classical traectory may have o turig poits. I this case, the costat ½ i the paretheses of Eq. (9), arisig from the turs, should be dropped. The simplest eample is the circular motio of the electro about the proto i Bohr s picture of the hydroge atom, for which the modified quatizatio (9) coditio takes form (.) postulated by N. Bohr. (A similar relatio for the radial motio is sometimes called the Sommerfeld-Wilso quatizatio rule.) Chapter Page 4 of 76

56 Harmoic oscillator s eergy levels so that Eq. () is satisfied if E ', with ',,,... (.4) I order to estimate the validity of this result, we have to check coditio (96) at all poits of the classically allowed regio, ad Eq. (7) at the turig poits. A straightforward calculatio shows that both coditios are valid for >>. However, we will see below that Eq. (4) is actually eactly correct for all eergy levels thaks to special properties of potetial profile (). Now, let us look at the secod of coectio formulas (5), c = a/. Agai, it differs from the result (7) for a sharp potetial step, that may be rewritte as k C A A / k i ( / k) ep i, (.5) by both the modulus ad phase factor. (I the WKB approimatio, the latter factor always equals /4.) Hece, agai, the WKB approimatio s predictio is ot eact for sharp potetials; evertheless, it is broadly used for practical calculatios. Oe of the most importat of them is the trasparecy of a arbitrary but smooth potetial barrier (Fig. ). U () d ' c c U ma E a b c c m d ' c f Fig... D potetial barrier of a arbitrary (but smooth) shape. Here, ust as i the case of a rectagular barrier, we eed to take uto cosideratio five partial waves (or rather fudametal solutios of the Schrödiger equatio): 3 a b epi k( ' ) d' ep i k( ' ) d', for, / / c k ( ) k ( ) c d WKB ep ( ' ) d' ep( ' ) d', for, / / c c' (.6) ( ) ( ) f epi k( ' ) d', for ', / c k ( ) where lower limits of itegrals are arbitrary (each withi the correspodig rage of ). Sice o the right of the left classical poit we have two epoets rather tha oe, ad o the right of the secod 3 Sorry, but the same letter, d, is used here for the barrier thickess (defied i this case as the classically forbidde regio legth, c c ), ad the costat i oe of the wave amplitudes see Eq. (6). Let me hope that the differece betwee these uses is absolutely evidet from the cotet. Chapter Page 5 of 76

57 poit, oe travelig waves rather tha two, the coectio formulas (5) have to be geeralized, usig asymptotic formulas ot oly for Ai( ), but also for the secod Airy fuctio, Bi( ). The aalysis, absolutely similar to that carried out above (though aturally a bit more bulky), 3 gives a remarkably simple result: c' c' f / TWKB ep ( ) d ep mu ( ) E d, (.7) a c c with o pre-epoetial factor. This formula is broadly used i applied quatum mechaics, despite the approimate character of its pre-epoetial coefficiet for isufficietly soft barriers that do ot satisfy Eq. (7). For eample, Eq. (8) shows that for a thick rectagular barrier with k =, i.e. U = E, the WKB approimatio (7) uderestimates T by a factor of 4. However, o the logarithmic scale of Fig. 8b, such factor, about half a order of magitude, still looks as a small correctio. Notice that whe E approaches the barrier top U ma (Fig. ), poits c ad c merge, so that, accordig to Eq. (7), T WKB, i.e. the particle reflectio vaishes at E = U ma. However, this coclusio is icorrect eve for smooth barriers where oe could aively epect the WKB approimatio to work perfectly. Ideed, ear poit = m where the potetial reaches maimum (i.e. U( m ) = U ma ), we may always approimate a smooth fuctio U() by a iverted parabola, m U ( ) U m ma. (.8) Calculatig the derivatives du/d ad d U/d of this fuctio ad pluggig them ito coditio (7), we see that the WKB approimatio is oly valid if U ma - E >>. A eact aalysis 3 of tuelig through barrier (8) gives the followig Kemble formula: ma ) / T ep ( E U, (.9) valid for ay sig of differece (E U ma ). This formula describes a gradual approach of T to, i.e. a gradual reductio of reflectio at particle eergy s icrease, with T = ½ (rather tha ) at E = U ma. Now the last remark of this sectio: our discussios of the propagator ad the WKB approimatio ope a straight way toward a alterative formulatio of quatum mechaics, based o the Feyma path itegral, but I will postpoe its discussio util a more compact ( bra-ket ) otatio has bee itroduced i Chapter 4. Soft tuel barrier s trasparecy Kemble formula.5. Trasfer matri, resoat tuelig, ad metastable states Let us ow eplore motio i more comple potetial profiles. The piecewise-costat ad smooth-potetial models of U() are ot too coveiet here, because they both require stitchig local 3 Note, however, that i the most importat case T WKB <<, Eq. (7) may be simply derived from Eqs. (5) a eercise left for the reader. 3 It was carried out by E. Kemble i 935. Notice that mathematically the Kemble formula is similar to the Fermi distributio i statistical physics, with effective temperature T ef = /k B. This similarity has some iterestig implicatios for the statistics of Fermi gas tuelig. Chapter Page 6 of 76

58 solutios i each classical turig poit, which may lead to very cumbersome calculatios. However, we may get a very good isight of the physics pheomea i such profiles, usig their approimatio by a set of Dirac s delta-fuctios. For that, let us have a look at what our old result (79) gives i the limit of a very thi ad high rectagular barrier, d <<, E << U (givig k << << /d): where parameter is defied as F T, (.) A i k d m d U d k. (.) k k The last product, U d, is ust the area W U ( ) d (.) U ( ) E of the barrier. This fact implies that the very simple result () for the trasparecy may be correct for a barrier of ay shape, provided that it is sufficietly thi ad high. Ideed, let us cosider the tuelig problem for a very thi barrier with d, kd << (Fig. ), approimatig it by Dirac s -fuctio: U ( ) W ( ). (.3) U ( ) W ( ) E A B F Fig... Delta-fuctioal tuel barrier. We already kow the solutios i all poits but = see Eqs. (63) ad (77) so we oly eed to aalyze boudary coditios i that poit to fid coefficiets A, B, ad F - or rather the ratios B/A ad F/A. However, due to the special character of the -fuctio, we should be careful here. Ideed, istead of Eq. (68) we ow get d d d d d d m m lim d lim ( ) (). U E d W O the other had, the wavefuctio itself is still cotiuous: d d lim d. Usig these boudary coditios, we readily get the followig system of two liear equatios, (.4) (.5) Chapter Page 7 of 76

59 whose solutio yields mw B F, ikf ( ika ikb) F, (.6) A B A i, i F A, i mw where. k (.7) For the barrier trasparecy T F/A, this result agai gives Eq. (). That formula may be recast to give a simple epressio (valid oly for E << U ma ) for the trasmissio coefficiet, E mw T, where E, (.8) E E that shows that as eergy becomes larger tha parameter E, the barrier s trasparecy approaches uity. However, the most importat applicatio of Eqs. (6) is for derivig trasparecy of more comple potetial profiles. For that, let us first itroduce very geeral otios of the scatterig ad trasfer matrices, curretly for the D case. Cosider a arbitrary but fiite-legth potetial bump (more formally called a scatterer), localized somewhere betwee poits ad, o the flat potetial backgroud, say U = (Fig. 3). We kow the geeral solutio, with a certai eergy E, outside the iterval are a set of two siusoidal waves. Let us preset them i the form ik ( ) ik( ) A e B e, (.9) where (for ow) = or, ad (k) /m = E. Note that each of the wave pairs (9) has, i this otatio, its ow referece poit, because this is very coveiet for the calculatios which follow. U () Thi barrier s trasparecy E B A B A Fig..3. A sigle D scatterer. As we have already discussed, if the wave/particle is icidet from the left, the liear Schrödiger equatio withi the scatterer rage ( < < ), ca provide oly liear epressios of the trasmitted (A ) ad reflected (B ) wave amplitudes via the icidet wave amplitude A : A S A, B S, (.3) A where S ad S are certai (geerally, comple) coefficiets. I this case, B =. Alteratively, if a wave, with amplitude B, is icidet from the right, it also may iduce a trasmitted wave (B ) ad reflected wave (A ) with amplitudes B S B, A S, (.3) B where coefficiets S ad S are geerally differet from S ad S. Now we ca use the liear superpositio priciple to argue that if waves A ad B are simultaeously icidet o the scatterer (say, Chapter Page 8 of 76

60 Scatterig matri: defiitio Trasfer matri: defiitio because wave B has bee partly reflected back by some other scatterer located at > ), the resultig scattered wave amplitudes A ad B are ust the sums of their values for separate icidet waves: B A S S A S A S B B,. (.3) These liear relatios may be coveietly preseted by the so-called scatterig matri (frequetly called ust S-matri ): B A S S S, S. A B S S (.33) Scatterig matrices, duly geeralized, are a importat tool for the aalysis of wave scatterig i more tha oe dimesios; for D problems, however, aother matri is more coveiet to preset the same liear relatios (3). Ideed, let us solve this system for A ad B. The result is A TA TB, A A i.e. T, B TA TB, B B (.34) where T is the trasfer matri with elemets SS S S T S, T, T, T S S S S. (.35) Oe ca woder whether matrices S ad T obey ay uiversal properties that would be valid for a arbitrary (but time-idepedet) scatterer. Such uiversal equatios may be readily foud from the probability curret coservatio ad the time-reversal symmetry of the Schrödiger equatio. Let me leave fidig these relatios for reader s eercise. The results show, i particular, that the scatterig matri may be rewritte i the followig form: i i re t S e i, (.36a) t re where 4 real parameters r, t,, ad satisfy ust oe uiversal relatio: r t (.36b) (so that oly 3 of the parameters are idepedet). As a result of this symmetry, T may be also preseted i a simpler form, similar to T : T = ep{i}/t = /S * = /S *. The last form allows a ready epressio of scatterer s trasparecy via ust oe coefficiet of the trasfer matri: T A S A B T. (.37) I our curret cotet, the most importat property of D trasfer matrices is that i order to fid the total trasfer matri T of a system cosistig of several (say, N) sequetial arbitrary scatterers (Fig. 4), it is sufficiet to multiply their matrices. Ideed, etedig the defiitio (34) to other poits ( =,,, N + ), we ca write Chapter Page 9 of 76

61 A B A T, B A3 A T B3 B A TT, B etc. (where the matri idices idicate the scatterers order o ais ), so that A B N N T N T N A...T. B (.38) (.39) A A A 3 A N B B B 3 B N 3 N Fig..4. A sequece of several D scatterers. But we ca also defie the total trasfer matri similarly to Eq. (34), i.e. as AN A T, BN B so that fially T T (.4) N TN... T. (.4) This formula is valid eve if the flat-potetial gaps betwee compoet scatterers vaish, so that it may be applied to a scatterer with a arbitrary profile U(), by fragmetig its legth ito small segmets = + -, ad treatig each fragmet as a rectagular barrier of height (U ) ef = [U( + ) U( )]/ - see Fig. 5. Sice very efficiet umerical algorithms are readily available for fast multiplicatio of matrices (especially as small as ), this approach is broadly used i practice for the computatio of trasparecy of tuel barriers with complicated profiles U(). (This is much more efficiet the the direct umerical solutio of the Schrödiger equatio.) Trasfer matri of a composite scatterer A ( U ) ef U () A N B N B N Fig..5. The trasfer matri approach to a log tuel barrier of a arbitrary profile. I order to use this approach for several coceptually importat systems, let us calculate the trasfer matrices for a few elemetary scatterers, startig from the delta-fuctioal barrier located at =. Takig = =, we ca merely chage the otatio of wave amplitudes i Eq. (7) to get Chapter Page 3 of 76

62 Trasfer matri of a short scatterer Idetity matri Trasfer matri of a space iterval i S, S. (.4a) i i A absolutely similar aalysis of the wave icidece from the left yields i S, S, (.4b) i i ad usig Eqs. (35), we get i i T. (.43) i i The et eample may seem strage at the first glace: what if there is o scatterer at all betwee poits ad? If poits ad coicide, the aswer is ideed trivial ad ca be obtaied, e.g., from Eq. (43) by takig W =, i.e. = : T I (.44) - the so-called idetity matri. However, we are free to choose the referece poits, participatig i Eq. (9) as we wish. For eample, what if = a? Let us first take the forward-propagatig wave aloe: B = (ad hece B = ); the ik ( ) ik( ) ik( ) A e A e e. (.45) Compariso of this epressio with the defiitio (9) for = shows that A = A ep{ik( - )} = A ep{ika}, i.e. T = ep{ika}. Repeatig the calculatio for the back-propagatig wave, we see that T = ep{-ika}, ad sice this o-potetial (space iterval) provides o particle reflectio, we fially get ika e T a ika, (.46) e idepedetly of the mutual positio of poits ad. At a =, we aturally recover the special case (43). Now let us use these results to aalyze the double-barrier system show i Fig. 6. We could of course calculate its properties as before, writig dow eplicit epressios for all 5 travelig waves show by arrows i Fig. 6, ad the usig boudary coditios (4) ad (5) at each of poits, to get a system of 4 liear equatios, ad the solvig it for 4 amplitude ratios. W a W E Fig..6. Double-barrier system. Dashed lies show (schematically) the positio of metastable eergy levels. Chapter Page 3 of 76

63 However, the trasfer matri approach simplifies the calculatios, because we may immediately use Eqs. (4), (43), ad (46) to write i i ika e i i T T TaT. (.47) i i ika e i i Let me hope that the reader remembers the row by colum rule of the multiplicatio of square matrices; 33 usig it for two last matrices, we reduce Eq. (47) to i i ika ika ( i) e ie T. (.48) i i ika ika ie ( i) e Now there is o eed to calculate all elemets of the full product T, because, accordig to Eq. (37), for the calculatio of barrier trasparecy T we eed oly oe its elemet, T : T T ika e ( i) ika e. (.49) Double barrier trasparecy This result is similar to that followig from Eq. (79) for E > U : the trasparecy is a -periodic fuctio of the product ka, reachig the maimum (T = ) at some poit of each period see Fig. 7a. (a) (b).8.3 Im T.6.4. k k ka / Fig..7. Resoat tuelig through a quatum well with delta-fuctioal walls : (a) trasparecy a fuctio of ka, ad (b) calculatig resoace s FWHM at >>. Re However, the ew result is differet i that for >>, the resoace peaks of trasparecy are very arrow, reachig their maima at ka k a, with =,, Physics of this effect is immediately clear from the compariso of this result with our aalysis of the simplest quatum well see Fig..7 ad its discussio. At k k, the icidet wave, which udertakes multiple sequetial reflectios from the semi-trasparet walls of the well, forms a early stadig wave, which at >> virtually coicides with oe of eigefuctios of the well with ifiite walls, with the stadig wave amplitude much larger that that of the icidet wave. As a result, the trasmitted wave amplitude is 33 I the aalytical form: ' N AB A B, where N is the matri rak (i our curret case, N = ). " " "' Chapter Page 3 of 76

64 proportioately icreased. This is the famous effect of resoat tuelig, 34 i mathematical descriptio idetical to the resoat trasmissio of light through a optical Fabry-Perot resoator formed by two parallel semi-trasparet mirrors. 35 Probably, the most surprisig feature of this system is the fact that its maimum trasparecy is perfect (T ma = ) eve at, i.e. i the case of a very low trasparecy of each of two compoet barriers. 36 Ideed, the deomiator i Eq. (49) may be iterpreted as the squared legth of the differece betwee two vectors, oe of legth, ad aother of legth ( - i) = +, with agle = ka + cost betwee them. At the resoace, the vectors are aliged, ad the differece is smallest (equal to ) see Fig. 7b, so that T ma =. We ca use the same vector diagram to calculate the so-called FWHM, the commo acroym for the Full Width [of the resoace curve at] Half-Maimum, i.e. the differece k = k + - k - betwee such two poits o the opposite slopes of the same resoace, at which T = T ma / - see arrows i Fig. 7a. Let the vectors i Fig. 7b be slightly misaliged, by a agle ~ / <<, so that the legth of the differece vector (of the order of ~ ) is still much smaller tha the legth of each vector. I order to double its legth squared, ad hece reduce T by a factor of i compariso with its maimum value, the arc,, betwee the vectors should also become equal, i.e. (k a + cost) =. Subtractig these two equatios from each other, we fially get k ( k k ) k. (.5) a Now let us use the simple potetial show i Fig. 6 to discuss a issue of large coceptual importace. For that, cosider what would happe if at some iitial momet (say, t = ) we have placed a D quatum particle iside the double-barrier well with >>, ad left it there aloe, without ay icidet wave. To simplify the aalysis, let us prepare the iitial state so that it coicides with the groud state of the ifiite-wall well see Eq. (.76): / (,) ( ) sik ( ), where k. (.5) a a At, this is a eigestate of the system, ad from our aalysis i Sec..5 we kow its time evolutio: (, t) ( ) e i t, E with k m ma, (.5) tellig us that the particle remais i the well at all times with costat probability W(t) = W() =. 37 However, if parameter is large but fiite, the de Broglie wave should slowly leak out from the well, so that W(t) would slowly decrease. Let us cosider this effect approimately, assumig that 34 I older literature, it is sometimes called the Ramsauer (or Towsed, or Ramsauer-Towsed ) effect. However, it is curretly more commo to use that ame(s) oly for a similar 3D effect, especially at scatterig of low-eergy electros o rare gas atoms this is how it was first observed, idepedetly, by C. Ramsauer ad J. Towsed i the early 9s. 35 See also, e.g., EM Sec The eact equality T ma = is correct oly if both compoet barriers are eactly equal. 37 Probability W(t) should ot be cofused with the delta-fuctioal barrier s area W, defied by Eq. (). Chapter Page 33 of 76

65 the slow leakage, with a characteristic time >> /, does ot affect the istat wave distributio iside the well, besides the reductio of W. 38 The we ca geeralize Eqs. (5), (5) as follows: / W it (, t) sik ( )] e. (.53) a makig the probability of fidig the particle i the well equal to W. This solutio may be preseted as a sum of two travelig waves: i ) ( ) (, ) ( kt i k t t Ae Be, (.54) with equal magitudes of their amplitudes ad probability currets A W B a /, I A m A k W, m a a I B I A. (.55) But we already kow from Eq. (8) that at >> the delta-fuctioal wall trasparecy T approimately equals /, so that the wave carryig curret I A, icidet o the right wall from iside, iduces a outcomig waves outside of the well (Fig. 8) with the followig probability curret: Absolutely similarly, I R W I A. (.56a) ma I L I B I R. (.56b) I L I R ~ v gr E v gr t v gr t Fig..8. Metastable state s decay i the simple model of a D potetial well with low-trasparet walls schematically. Now we may combie the D versio (6) of the probability coservatio law for well s iterior, with Eqs. (56) to write dw dt dw dt I R I L, (.57) W. (.58) ma 38 This almost evidet assumptio fids its formal ustificatio i the perturbatio theory to be discussed i Chapter 6. Chapter Page 34 of 76

66 Metastable state s decay law Metastable state s lifetime Eergy-time ucertaity relatio This is ust the stadard differetial equatio, dw W, (.59) dt of the epoetial decay, with solutio W(t) = W()ep{-t/}, where costat, i our case equal to ma, (.6) is called the metastable state s lifetime. Usig epressio (.34) for the de Broglie waves group velocity, i our particular wave vector givig v gr = k /m = /ma, Eq. (59) may be rewritte as t A, (.6) T where i our case the attempt time t A is equal to a/v gr, ad T = /. Relatio (6), that is valid for a large class of metastable systems, 39 may be iterpreted i the followig semi-classical way. The cofied particle travels back ad forth betwee the cofiig walls, with time itervals t A betwee the momets of icidece, each time makig a attempt to leak through the wall, with a success probability of T, so the reductio of W per each icidece is W = -WT, immediately leadig to Eq. (6). Aother importat look at Eq. (6) may be take by returig to the resoat tuelig problem ad epressig the resoace width (5) i terms of icidet particle s eergy: k k k E k m. (.6) m m a ma Comparig Eqs. (6) ad (6), we get a remarkably simple formula E. (.63) This so-called eergy-time ucertaity relatio is certaily more geeral tha our simple model; for eample, it is valid for the lifetime ad resoace tuelig width of ay metastable state. This seems very atural, sice because of the eergy idetificatio with frequecy, E =, typical for quatum mechaics, Eq. (63) may be rewritte as = ad seems to follow directly from the Fourier trasform i time, ust as the Heiseberg s ucertaity relatio (.35) follows from the Fourier trasform i space. I some cases, these two relatios are ideed iterchageable; for eample, Eq. (4) for the Gaussia wave packet width may be rewritte as Et =, where E = (d/dk)k = v gr k is the r.m.s. spread of eergies of moochromatic compoets of the packet, while t /v gr is the time scale of the packet passage through a fied observatio poit. However, Eq. (63) it is much less geeral tha Heiseberg s ucertaity relatio (.35). Ideed, i o-relativistic quatum mechaics, Cartesia coordiates (say, ) of a particle, compoets of its mometum (say, p ), ad eergy E are regular observables, preseted by operators. I cotract, time is treated as a c-umber argumet, ad is ot preseted by a operator, so that Eq. (63) caot be derived 39 Essetially the oly requiremet is to have the attempt time t A to be much loger tha the effective time (istato time, see Sec. 5.3 below) of tuelig through the barrier. I the delta-fuctioal approimatio for the barrier, the latter time vaishes, so that this requiremet is always fulfilled. Chapter Page 35 of 76

67 i such geeral assumptios as Eq. (.35). Thus the time-eergy ucertaity relatio should be applied with great cautio. Ufortuately, ot everybody is so careful. Oe ca fid, for eample, wrog claims that due to this relatio, the eergy dissipated by ay system performig a elemetary (sigle-bit) calculatio durig time iterval t has to be larger tha /t. 4 Aother icorrect statemet is that the eergy of a system caot be measured, durig time t, with a accuracy better tha /t. 4 Now let us use our simple model of metastable state s decay for a prelimiary discussio of oe aspect of quatum measuremets. Figure 8 shows (schematically) oe of the travelig wave packets emitted by the quatum well after its iitial state (5) had bee prepared at t =. (A similar packet is emitted to the left.) At t >>, the well becomes essetially empty (W << ), ad the whole probability distributio is localized i two clearly separated wave packets of equal amplitudes, movig from away with speed v gr, each carryig the particle away with a probability of 5%. Now assume a eperimet has detected the particle o the left side of the well. Though the formalisms suitable for a quatitative aalysis of the detectio process will ot be discussed util Sec. 7.7, due to the wide separatio of the packets, we may safely assume that the detectio may be doe without ay actual physical effect o the couterpart wave packet. 4 But if we kow that the particle has bee foud o the left, there is o chace to fid it o the right. If we attributed the wavefuctio to all stages of this particular eperimet, this situatio might be rather cofusig. Ideed, this would mea that the wavefuctio withi the right packet should istatly tur ito zero - the so-called wave packet reductio a process that could ot be described by either Schrödiger equatio or ay other law of physics. However, if (as was already discussed i Sec..3) we attribute the wavefuctio to a statistical esemble of similar eperimets, there is o parado here at all. While the two-packet picture we have calculated (Fig. 8) describes the full iitial esemble (regardless of the particle detectio results), the reduced packet picture (with o wave packet o the right of the well) describes oly a sub-esemble of eperimets with the particle detected o the left side. As was discussed o completely classical eamples i Sec..3, for such sub-esemble the probability distributio, ad hece the wavefuctio, may be dramatically differet..6. Coupled quatum wells Let us ow move o to tuelig through a more comple potetial profile show i Fig. 9: a sequece of (N ) similar quatum wells separated by N similar delta-fuctioal tuel barriers. Accordig to Eq. (4), its trasfer matri is the followig product T T TaT... TaT, (.64) ( N ) N terms with the compoet matrices give by Eqs. (43) ad (46), ad the barrier height parameter defied by the last of Eqs. (7). 4 Here I dare to refer the reader to my ow old work K. Likharev, It. J. Theor. Phys., 3 (98) that preseted a costructive proof that at reversible computatio (itroduced i 973 by C. Beett) the eergy dissipatio may be lower tha this apparet quatum limit. 4 See, e.g., a detailed discussio of this issue i the moograph by V. Bragisky ad F. Khalili, Quatum Measuremet, Cambridge U. Press, This argumet is especially covicig if the particle detectio time is much shorter tha the time t c = v gr t/c, where c is the speed of light i vacuum, i.e. the maimum velocity of ay iformatio trasfer ( sigalig ). Chapter Page 36 of 76

68 a a E I A N TI A Fig..9. Resoat tuelig through a system of N similar, equidistat barriers, i.e. (N ) similar quatum wells. Trasparecy of N equidistat tuel barriers Remarkably, this multiplicatio may be carried out aalytically, 43 givig si ka cos ka T T cos Nqa si Nqa si qa, (.65) where q is a ew parameter, with the wave umber dimesioality, defied by the followig relatio: cosqa coska si ka. (.66) For N =, Eqs. (65) ad (66) immediately yield our old result (8), while for N = they may be reduced to Eq. (49) see Fig. 7a. Figure shows its predictios for two larger umbers N, ad several values of parameter. N 3 N T.8 ) ka / Fig... Trasparecy of the system show i Fig. 9 as a fuctio of product ka. Sice the fuctio T(ka) is -periodic (ust like for N =, see Fig. 7a), oly oe period is show..8 ).6 T ka / 3. Let us start discussio of the plots from case N = 3, i.e. two coupled quatum wells. The compariso of Fig. a ad Fig. 7a shows that the trasmissio patters, ad their depedece o parameter, are very similar, besides that i the coupled wells each resoat tuelig peak splits ito two, with the ka-differece betwee them scalig as /. I order to comprehed the physics of this importat result, let us aalyze a auiliary system show i Fig. : two similar quatum wells 43 This formula will be easier to prove after we have discussed properties of Pauli matrices i Chapter 4. Chapter Page 37 of 76

69 cofied by ifiitely high potetial walls at = a, ad coupled via a trasparet, short tuel barrier at =. U() W () A E A E S a S a Fig... Two lowest eigefuctios ad eigeeergies of a system of two coupled quatum wells schematically. The barrier may be agai, for calculatio simplicity, approimated by a delta-fuctio:, for a, U ( ) (.67) W ( ), for a. We already kow that the stadig-wave eigefuctios of the Schrödiger equatio i regios with U() =, i our curret case, segmets a < < ad < < +a, may be always preseted as liear superpositios of sik ad cosk. I order to immediately satisfy the boudary coditios = at = a, we ca take these solutios i the form C si k( a), for a, ( ) (.68) C si k( a), for a. What remais is to satisfy the boudary coditios at =. Pluggig Eq. (67) ito Eqs. (4) ad (5), we get the followig system of two liear equatios: k C mw C )cos ka C si ka, (.69) C si ka C si ka. (.7) ( The system has two types of solutios, with the two lowest-eergy eigefuctios sketched i Fig. : (i) Atisymmetric solutios (which will be marked with ide A), C C, i.e. C si k, A A A A A (.7) with eigevalues idepedet of W, si kaa, i.e. kaa ka,,,... (.7) Notice that these values of k, ad hece eigeeergies of these atisymmetric states, k A E A, m ma (.73) Chapter Page 38 of 76

70 Characteristic equatio for two coupled quatum wells coicide with those of the simple quatum well of width a see Fig..7 ad its discussio. (ii) Symmetric solutios (ide S): C C, i.e. C si k ( a), S S S S S (.74) with Eq. (69) givig the followig characteristic equatio for costat k S : ta k S a. (.75) Figure shows the graphic solutio of this equatio for three values of parameter, i.e. for various quatum well couplig stregth. For each solutio, k S a is cofied withi iterval k S a, (.76) so that the atisymmetric ad symmetric states alterate o the scale of k (ad hece of the eergy), with the differece k A - k S, for each pair of adacet states, smaller the /a for ay value of. The physics of the splittig betwee eigeeergies correspodig to the symmetric ad atisymmetric states is very simple: it is the chage of kietic eergy of the particle due to differet cofiemet types see Fig.. I each atisymmetric mode, () = (a) =, i.e. the wavefuctio is essetially cofied withi a segmet of legth a; as a result, its eergy (73) does ot deped o the barrier height. O the cotrary, i the symmetric mode, that does egage the potetial barrier, the wavefuctio effectively spreads ito the couterpart well. As a result, it chages slower, ad hece its kietic eergy is also lower that that of the adacet atisymmetric mode. 4 ta ka, k S k A 4 ka / 3.3 Fig... Graphical solutio of the characteristic equatio (75) for the eigevalue of ka i the symmetric mode, for 3 values of parameter, cosiderig it idepedet of ka. The dashed lie shows approimatio (78). By the way, this problem may serve as a toy model of the strogest (ad most importat) type of atom cohesio - the covalet (or chemical ) bodig i molecules, liquids, ad solids. The classical eample of such bodig is that of hydroge atoms i a H molecule. 44 Each of two electros of this system 45 reduces its kietic eergy very substatially by spreadig its wavefuctio aroud both uclei 44 Historically, the developmet of the fully quatum theory of H bodig by W. Heitler ad F. Lodo i 97 was the breakthrough decisive for the acceptace of the-emergig quatum mechaics by chemists. 45 Due to the opposite spis, the Pauli priciple allows them to be i the same orbital groud state see Chapter 8. Chapter Page 39 of 76

71 protos, rather that beig cofied ear oe of them - as it had to be i a sigle atom. As a result, the bodig is very strog: i chemical uits, 49 kj/mol, i.e. 8.6 ev per molecule. 46 Somewhat couterituitive, this eergy is substatially larger tha the strogest classical (ioic) bodig due to electro trasfer betwee atoms, leadig to the Coulomb attractio of the resultig ios. (For eample, the atomic cohesio i the NaCl molecule is ust 3.8 ev.) I the limit (o partitio betwee the wells), k S a ( - /), i.e. the eigestates approach the shape ad eergy of symmetric states of a quatum well of width a. I the opposite limit >>, k S a, ad i the viciity of each such poit we may approimate tak S a with (k S a - ) see the dashed lie i Fig.. As a result, the characteristic equatio (75) is reduced to k S a, (.77) so that the splittig betwee the wave umbers ad eigeeergies of the adacet symmetric ad atisymmetric states is small: k A k S k a, E A E S de dk ma a E A k k. A S (.78) (By costructio, this result is valid oly if >>, i.e. << E A E S.) Let us aalyze properties of the system i this limit i much more detail - first, because the results will help us to develop the importat tight bidig approimatio i the bad theory, ad secod, because the weakly coupled quatum wells will be our first eample of very importat two-level (or spi-½-like ) systems. Let us focus o oe couple of symmetric ad atisymmetric states, correspodig to virtually the same E. Accordig to Eqs. (7) ad (74), i the limit, system s eigefuctios may be approimately represeted as follows: S ( ) R ( ) L ( ), A ( ) R ( ) L ( ), (.79) where R,L are the ormalized groud states of the completely isulated wells: /, for a, / a si k, for a, R ( ) / L ( ) (.8) / a si k, for a,, for a. Let us perform the followig coceptually importat thought eperimet: place the particle, at t =, ito oe of the localized states, say R (), ad leave the system aloe to evolve. Solvig Eqs (8) for R, we may preset the iitial state as a liear superpositio of eigefuctios: (,) R ( ) S ( ) A ( ). (.8) Now, accordig to the geeral solutio (.67) of the time-depedet Schrödiger equatio, time dyamics may be obtaied by ust multiplyig each eigefuctio by the correspodig factor (.6): 46 Uit remider: kj/mol.434 ev. Chapter Page 4 of 76

72 Quatum oscillatios i two coupled wells E S E A (, t) S ( )ep i t A ( )ep i t. (.8) Now, itroducig the followig atural otatio, E A ES E A ES E,. (.83) Ad usig Eqs. (79), this epressio may be rewritte as (, ) ( )ep ( )ep E ep t S i t A i t i t (.84) E ( )cos t i ( )si t ep R L i t. This result implies, i particular, that the probabilities W R ad W L to fid the particle, correspodigly, i the right ad left wells chage with time as WR cos t, WL si t, (.85) mercifully leavig the total probability costat W R + W L =. (If our calculatio had ot passed this saity check, we would be i a big trouble.) This is the famous effect of periodic quatum oscillatios, with frequecy = / = (E A E S )/, of the particle betwee two similar quatum wells, due to their couplig through via tuelig through the tuel barrier. The physics of this effect is straightforward: ust as i the sigle well problem discussed i Sec. 5, the particle iitially placed ito a certai quatum well tries to escape from it via tuelig through the semi-trasparet wall. However, i our curret situatio (Fig. ) the particle ca oly escape ito the adacet well. After the tuelig ito that secod well, the tries to escape from it, ad hece comes back, etc. - ust as a classical D oscillator, iitially deflected from its equilibrium positio. Maybe the most surprisig feature of this effect is its relatively high frequecy: accordig to Eq. (78), the time period of the quatum oscillatios, ma t, for, (.86) E E A S is a factor of / >> shorter tha the lifetime (6) of the metastable state of the particle i a similar but sigle quatum well limited by delta-fuctioal walls with similar parameter. This is a very couterituitive result ideed: the speed of particle tuelig ito a similar adacet well is much higher tha that, through a similar barrier, to the free space! To see whether this result is a artifact of the delta-fuctioal model of the tuel barrier, let us calculate splittig for system of two similar, symmetric, soft quatum wells formed by a smooth potetial profile U() = U(-) see Fig. 3. Chapter Page 4 of 76

73 U() L () R () E a c ' c a Fig..3. Weak couplig betwee two similar, soft quatum wells. If the barrier trasparecy is low, the quasi-localized wavefuctios R () ad L () = R (-) ad their eigeeergies may be foud approimately by solvig the Schrödiger equatios i oe of the wells, eglectig tuelig through the barrier, but fidig requires a little bit more care. Let us write the statioary Schrödiger equatios for the symmetric ad atisymmetric solutios i the form d A d S E A U ( ) A, E U ( ), S S (.87) m d m d the multiply the former equatio by S, the latter oe by A, subtract them from each other, ad itegrate the result from to : d S d A ( E A ES ) S Ad. A S d (.88) m d d If U(), ad hece d A,S /d, are fiite for all, 47 we may itegrate the right-had side by parts to get d d S A ( E A ES ) S Ad A S m d d. (.89) So far, this is a eact equatio. For weakly coupled wells, we ca do more. I this case, the left had side may be approimated as (E A E S )/, because the itegral is domiated by the viciity of poit a, where the secod terms i each of Eqs. (79) are egligible, ad the itegral is equal to ½, due to the proper ormalizatio of fuctio R (). I the right-had side, the substitutio at = vaishes (due to the wavefuctio decay i the classically forbidde regio), ad so does the first term at =, because for the atisymmetric solutio A () =. As a result, we get d A d R d R d L S () () R () () L () () L () (). (.9) m d m d m d m d It is straightforward to show that withi the limits of the WKB approimatio validity, Eq. (9) may be reduced to t A ' c ep ( ' ) d', (.9) c WKB result for couplig eergy 47 Sice it is ot true for potetial (67), oe should ot be surprised that the resultig Eq. (89) is ivalid for our iitial problem, givig twice larger tha the correct epressio (78). Chapter Page 4 of 76

74 where t A is the time period of classical motio of the particle iside oe of the wells, fuctio () is defied by Eq. (97), ad c ad c are the classical turig poits limitig the potetial barrier at the level E of particle s eergy see Fig. 3. Comparig this result with Eq. (7), we ca otice that agai, ust as i the case of the delta-fuctioal barriers, the trasmissio coefficiet T of a tuel barrier (ad hece the reciprocal lifetime of a metastable state i a potetial well separated by such a barrier from a cotiuum) scales as the square of the WKB epoet participatig i Eq. (9), so that the period of quatum oscillatios betwee the well is much smaller tha the lifetime. We will retur to the discussio of this result, i a more geeral form, i Chapter 5. Returig for a secod to Fig. a, we may ow readily iterpret the results for tuelig through the double quatum well: each pair of resoace peaks of trasparecy correspods to the aligmet of icidet particle s eergy with the pair of eergy levels E A, E S of the symmetric ad atisymmetric states of the system..7. D bad theory Let us ow retur to Eqs. (65) ad (66) describig the resoat tuelig, ad discuss their predictios for larger N see, for eample, Fig. b. We see that the icrease of N results i the icrease of the umber of resoat peaks per period to (N - ), ad at N the peaks merge ito the so-called allowed eergy bads (frequetly called ust the eergy bads ) of relatively high trasparecy, separated from similar bads i the adacet periods of fuctio T(ka) by eergy gaps 48 where T. Notice the followig importat features of the patter: (i) at N, the bad/gap edges become sharp for ay, ad ted to fied positios (determied by but idepedet of N); (ii) the larger iterwell couplig ( ), the broader the allowed eergy bads ad arrower the gaps betwee them. Our discussio of resoat tuelig i the previous sectio gives us a evidet clue for a semiquatitative iterpretatio of this patter: if (N - ) quatum wells are weakly coupled by tuelig through the tuel barriers separatig them, system s eergy spectrum cosists of groups (N ) eergy levels. Each level correspods to a eigefuctio that is the set of similar local fuctios i each well, but with certai phase shifts betwee them. It is atural to epect that, ust as for coupled wells (N = ), that at the upper level, = (thus providig the highest cofiemet), with ka at, while at the lowest level all =, providig the most loose cofiemet. 49 However, what about for other levels? Aswers to all these questios are easy to get i the most importat limit N, i.e. for periodic structures - which are, i particular, good D approimatios for solid state crystals, whose samples may feature more tha similar atoms or molecules i each directio of the crystal lattice. It is almost self-evidet that at N, due to the traslatioal ivariace of U(), U ( a) U ( ), (.9) 48 I solid state (especially semicoductor) physics ad electroics, term badgaps is more commo. 49 This epectatio is implicitly cofirmed by Fig. : at >>, the highest resoace peak i each group teds to ka =, ad the lowest peak also ted to a positio idepedet of N (though depedet o ). Chapter Page 43 of 76

75 the phase shift betwee local wavefuctios i all adacet quatum wells should be the same for each period of the system, i.e. i ( a) ( ) e (.93a) for all. (A reasoably fair classical image of is the geometric agle betwee similar obects - e.g., similar paper clips - attached at equal distaces to a log, uiform rubber bad. If the bad s eds are twisted, the twist is equally distributed betwee the structure s periods, represetig the costacy of. 5 ) Equatio (93a) is the (D versio of the) much-celebrated Bloch theorem. 5 Mathematical rigor aside, 5 it is a virtually evidet fact, because the particle s desity w() = *()(), that has to be periodic i this a-periodic system, may be so oly is costat. For what follows, it is more coveiet to preset the real umber i the form qa (there is o loss of geerality here, because parameter q may deped o a as well as other parameters of the system), so that the Bloch theorem takes the form iqa ( a) ( ) e. (.93b) The physical sese of parameter q will be discussed i detail below; for ow ust ote that accordig to Eq. (93b), a additio of (/a) to it yields the same wavefuctio; hece all observables have to be (/a)-periodic fuctios of q. 53 Now let us use the Bloch theorem to fid eigefuctios ad eigeeergies for a particular, ad probably the simplest periodic fuctio U(): a ifiite set of similar quatum wells separated by deltafuctioal tuel barriers (Fig. 4). D Bloch theorem a a a E Fig..4. The simplest periodic potetial: a ifiite set of similar, equidistat, delta-fuctioal tuel barriers. 5 I am ashamed to cofess that, due to the lack of time, this was virtually the oly lecture demostratio i my QM courses. 5 Named after F. Bloch who applied this cocept to wave mechaics i 99, i.e. very soo after its formulatio. Admittedly, i mathematics, a equivalet statemet, usually called the Floquet theorem, has bee kow sice at least I will address this rigor i two steps. Later i this sectio, we will see that the fuctio obeyig Eq. (93) is ideed a solutio of the Schrödiger equatio. However, to save time/space, it will be better for us to postpoe the proof that ay eigefuctio of the equatio, with periodic boudary coditios, obeys the Bloch theorem, util Chapter 4. As a partial reward for the delay, that proof will be valid for a arbitrary spatial dimesioality. 53 Product q, which has the dimesioality of mometum, is called either the quasi-mometum or (especially i the solid state physics) the crystal mometum of the particle. Chapter Page 44 of 76

76 To start, cosider two poits separated by distace a: oe of them,, ust left of positio of oe of the barriers, ad aother oe, +, ust left of the followig barrier. Eigefuctios i each of the poits may be preseted as liear superpositios of two simple waves ep{ik}, ad amplitudes of their compoets should be related by a trasfer matri T of the potetial fragmet separatig them. Accordig to Eq. (4), this matri may be foud as the product of the matri (46) of oe iterval a ad the matri (43) of oe barrier: A ika A e i i A Ta T. (.94) B ika B e i i B However, accordig to the Bloch theorem (93b), the compoet amplitudes should be also related as A B iqa e A B iqa e iqa e A B. (.95) The coditio of self-cosistecy of these two equatios leads to the followig characteristic equatio: e ika i e ika i i iqa e i iqa e. (.96) I Sec. 5, we have already calculated the matri product participatig i this equatio see Eq. (48). Usig it, we see that Eq. (96) is reduced to the same simple Eq. (66) that has already umped at us from the solutio of the differet (resoat tuelig) problem. Let us eplore that simple result i detail. First of all, the right had part of Eq. (66) is a siusoidal fuctio of ka, with amplitude ( + ) / see Fig. 5, while its left had part is a siusoidal fuctio of qa with amplitude. gap gap bad bad cos qa 3 4 ka / Fig..5. Graphical solutio of the characteristic equatio (66) for a fied value of parameter. The rages of ka that yield with cos qa <, correspod to the allowed eergy bads, while those with cos qa >, to gaps betwee them. As a result, withi each period (ka) =, the characteristic equatio does ot have a real solutio for q iside two itervals of ka - ad hece iside two itervals of eergy E = k /m. (These itervals are eactly the eergy gaps metioed above, while the complemetary itervals of ka ad E, where a real q eists, are the allowed eergy bads.) I cotrast, parameter q ca take ay real values, so it is more coveiet to plot the eigeeergy E = k /m as the fuctio of q (or, eve more Chapter Page 45 of 76

77 coveietly, qa) rather tha ka. 54 While doig that, we eed to recall that parameter, defied by the last of Eqs. (7), depeds o wave vector k as well, so that if we vary q (ad hece k), it is better to characterize the structure by a differet, k-idepedet dimesioless parameter, for eample maw ( ka), (.97) so that Eq. (66) becomes si ka cos qa cos ka. (.98) ka Figure 6 shows the plots of E ad k, followig from Eq. (98), for a particular, moderate value of parameter. The bad structure of the eergy spectrum is apparet. Aother evidet feature is the -periodicity of the patter, that we have already predicted from the geeral Bloch theorem argumets. (Due to this periodicity, the complete bad/gap patter may be studied o ust oe iterval - qa +, called the st Brilloui zoe the so-called reduced zoe picture. For some applicatios, however, it is more coveiet to use the eteded zoe picture with - qa + - see, e.g., the et sectio.) Characteristic equatio for system i Fig. 4 st Brilloui zoe (a) st Brilloui zoe (b) ka E E qa / qa / Fig..6. (a) Real mometum k of a particle i the periodic delta-fuctioal potetial profile show i Fig. 4, ad (b) its eergy E = k /m (i uits of E /ma ), as fuctios of the quasi-mometum q, for a particular value ( = 3) of the dimesioless potetial parameter (ka) = mwa/. Arrows i the lower right corer of pael (b) illustrate the defiitio of the eergy bad (E ) ad eergy gap ( ) widths. 54 Perhaps a more importat reaso for takig q as the argumet is that for motio i a geeral potetial U(), particle s mometum k is ot a costat of motio, while (accordig to the Bloch theorem), the quasi-mometum q is. E Chapter Page 46 of 76

78 However, maybe the most surprisig fact, clearly visible i Fig. 6, is that there is a ifiite umber of eergy bads, with differet eergies E (q) for the same value of q. Mathematically, it is evidet from Eq. (98) see also Fig. 5. Ideed, for each value of qa there are two solutios ka to this equatio o each period (ka) = - see also pael (a) i Fig. 6. Each of such solutios gives a differet value of particle eergy E = k /m. A cotiuous set of similar solutios for various qa forms a particular eergy bad. Sice the bad theory is oe of the most vital results of quatum mechaics, it is importat to uderstad the physics of these differet solutios - ad hece of the whole bad picture. For that, let us eplore aalytically two differet potetial stregth limits. A importat advatage of this approach is that both aalyses may be carried out for a arbitrary periodic potetial U(), rather tha for the simplest model show i Fig. 4. (i) Tight-bidig approimatio. This approimatio is soud whe eigeeergy E is much lower tha the height of the potetial barriers separatig the potetial miima (servig as quatum wells) see Fig. 7. As should be clear from our discussio i Sec. 6, the wavefuctio is mostly localized i the classically allowed regios at poits of the potetial eergy miima - see the dashed lies i Fig. 7. Essetially the oly role of couplig betwee these quatum well states (via tuelig through the separatig barriers) is to establish certai phase shifts = qa betwee the pairs of adacet quasi-localized wavefuctio lumps u( - ) ad u( + ). a a U() u ( ) u ( ) E a Fig.. 7. Tight bidig approimatio (schematically). To describe this effect quatitatively, let us first retur to the problem of two coupled wells cosidered i Sec. 6, ad recast result (84) as E (, ) ( ) ( ) ( ) ( ) ep t ar t R al t L i t, (.99) where fuctios a R ad a L oscillate siusoidally i time: ar ( t) cos t, al ( t) i si t. (.) This evolutio satisfies the followig system of two equatios whose structure remids Eq. (.59): i a a, ia ar. (.) R L Later i the course (i Chapter 6) we will prove that such equatios are ideed valid, i the tightbidig approimatio, for ay system of two coupled quatum wells. These equatios may be readily geeralized to the case of may similar coupled wells. Here, i this case, istead of Eq. (99), we evidetly should write L Chapter Page 47 of 76

79 E (, t) ep i t a ( t) u ( ), (.) where E are the eigeeergies, ad u the eigefuctios of each isolated well. I the tight bidig limit, oly the adacet wells are coupled, so that istead of Eq. () we should write a ifiite system of similar equatios i a a a, (.3) for each well umber, where parameters describe the couplig betwee two adacet quatum wells. Repeatig the calculatio outlied i the ed of Sec. 6 for our ew situatio, we get the result essetially similar to the last form of Eq. (9): du u ( ) ( a ), (.4) m d where is the distace betwee the well bottom ad the middle of the tuel barrier o the right of it see Fig. 7. The oly substatial ew feature of this epressio i compariso with Eq. (9) is that the sig of alterates with the level umber : >, <, 3 >, etc. Ideed, the umber of wiggles (formally, zeros) of eigefuctios u () of ay potetial well icreases as see, e.g., Fig..7, 55 so that the differece of the epoetial tails of the fuctios, seakig uder the left ad right barriers limitig the well also alterates with. The ifiite system of ordiary differetial equatios (3) allows oe to eplore a large rage of importat problems (such as the spread of the wavefuctio that was iitially localized i oe well, etc.), but our mai task ow is to fid its statioary states, i.e. the solutios proportioal to ep{- i( /)t}, where is a still ukow, q-depedet additio to the backgroud eergy E of -th level. I order to satisfy the Bloch theorem (93) as well, such solutio should have the form a ( t) a epiq i t cost, (.5) where a is a costat. Pluggig this solutio ito Eq. (3) ad cacelig the commo epoet, we get E E E iqa iqa e e E cos qa, (.6) so that i this approimatio, the eergy bad width E (see Fig. 6b) equals 4. Relatio (6), whose validity is restricted to << E, describes the particular lowest eergy bads plotted i Fig. 6b reasoably well. (For larger, the agreemet would be eve better.) So, this calculatio eplais what the eergy bads really are i the tight bidig limit they are best iterpreted as isolated well s eergy levels E, broadeed ito bads by the iterwell iteractio. Also, this result gives a clear proof that the eergy bad etremes correspod to qa = l ad qa = (l + ½), with iteger l. Fially, the sig alteratio of the couplig coefficiet (4) with umber eplais why the eergy maima of oe bad are aliged, o the qa ais, with eergy miima of the adacet bads. Tight bidig limit: couplig eergy Tight bidig limit: eergy bads 55 Below, we will see several other eamples of this behavior. This alteratio rule is also i accordace with the Bohr-Sommerfeld quatizatio coditio Chapter Page 48 of 76

80 Bloch theorem: alterative form (ii) Weak-potetial limit. Surprisigly, the eergy bad structure is also compatible with a completely differet physical picture that ca be developed i the opposite limit. Let eergy E be so high that the periodic potetial U() may be treated as a small perturbatio. Naively, we would have the parabolic dispersio relatio betwee particle s eergy ad mometum. However, if we are plottig eergy as a fuctio of q rather tha k, we eed to add l/a, with arbitrary iteger l, to the argumet. Let us show this by epadig all variables ito the spatial Fourier series. For a periodic potetial eergy U() such a epasio is straightforward: 56 U ( ) U l" ep i l", (.7) l" a where the summatio is over all itegers l, from - to +. However, for the wavefuctio we should show due respect to the Bloch theorem (93). To uderstad how to proceed, let us defie aother fuctio ad study its periodicity: iq u( ) ( ) e, (.8) iq( a) iq u( a) ( a) e ( ) e u( ). (.9) We see that the ew fuctio is a-periodic, ad hece we ca use Eqs. (8)-(9) to rewrite the Bloch theorem as iq ( ) u( ) e, with u( a) u( ). (.) Now it is safe to epad the periodic fuctio u() eactly as U(): u( ) ul' ep i l', (.) l ' a so that, accordig to the Bloch theorem, ( ) iq e ul' ep i l' ul' epi q l'. (.) l' a l' a The oly otrivial part of pluggig this epressio ito the statioary Schrödiger equatio (6) is the calculatio of the product term, usig epasios (7) ad (): U ( ) U l" ul' epi q ( l' l" ). (.3) l', l" a At fied l, we may chage summatio over l to that over l l + l (so that l = l l ), ad write: U ( ) epi q l u l' U l l'. (.4) l a l' 56 The beefits of my uusual choice of the summatio ide (l istead of, say, l) will be clear i a few lies. Chapter Page 49 of 76

81 Now pluggig Eqs. () (with ide l ow replaced by l) ad (4) ito the statioary Schrödiger equatio (6), ad requirig the coefficiets of each spatial epoet to match, we get a ifiite system of liear equatios for u l : 57 l' U ll' u l' E q l m a ul. (.5) So far, this system is a equivalet alterative to the iitial Schrödiger equatio ad, by the way, is very efficiet for fast umerical calculatios, for virtually ay potetial stregth, though i systems with tight bidig it may require takig ito accout a large umber of harmoics u l. I the weak potetial limit, i.e. if all the Fourier coefficiets U are small, 58 we ca complete all the calculatio aalytically. 59 Ideed, i the so-called th approimatio we ca igore all U, so that i order to have at least oe u l differet from, Eq. (5) requires that l E El q. (.6) m a (u l itself should be obtaied from the ormalizatio coditio). This result meas that the dispersio relatio E(q) has a ifiite umber of similar quadratic braches umbered by iteger l see Fig. 8. () E () E l l l l qa / Fig..8. D bad picture i the weak potetial case ( << E () ). Shadig shows the st Brilloui zoe. O ay brach, the eigefuctio has ust oe Fourier coefficiet, i.e. presets a moochromatic travelig wave l u e l ik u l l ep i q. (.7) a 57 Note that we have essetially proved that the Bloch wavefuctio () is ideed a solutio of Eq. (6), provided that the quasi-mometum q is selected i a way to make the system of liear equatio (5) compatible, i.e. is a solutio of its characteristic equatio see, e.g., Eq. (3) below. 58 Besides the costat potetial U that, as we kow from Sec., may be icluded ito eergy i a trivial way, so that we may take U =. 59 This method is so powerful that its multi-dimesioal versio is ot much more comple tha the D versio described here see, e.g., Sec. 3. i the classical tetbook by J. M. Zima, Priciples of the Theory of Solids, d ed., Cambridge U. Press, 979. Chapter Page 5 of 76

82 Weak potetial limit: eergy gap positios Weak potetial limit: eergies ear badgap This fact allows us to rewrite Eq. (5) i a more trasparet form that may be formally solved for u l : l' l u l U u E E ) u l' l l ' ( l l, (.8) E E l l' l U l' l u l'. (.9) If the Fourier coefficiets U are ovaishig but small, this formula shows that wavefuctios do acquire other Fourier compoets (besides the mai oe, with the ide correspodig to the brach umber), but these additios are all small, besides arrow regios ear the poits E l = E l where two braches (6) of the dispersio relatio E(q), with some specific umbers l ad l, cross. This happes whe q l q l', (.) a a i.e. at q q m m/a (with iteger m l + l ) 6 correspodig to ( ) El El' ( l l') l E, (.) ma ma with iteger l l. (Equatio () shows that ide is ust the umber of the brach crossig o the eergy scale see Fig. 8.) I such a regio, E has to be close to both E l ad E l, so that the deomiator i ust oe of the ifiite umber of terms i Eq. (9) is very small, makig the term substatial despite the smalless of U.. Hece we ca take ito accout oly oe term i each of the sums (writte for l ad l ): U ul' ( E El ) ul, (.) U u ( E E ) u. l Takig ito accout that for ay real fuctio U() the Fourier coefficiets i series (7) have to be related as U - = U *, Eq. () yields the followig simple characteristic equatio with solutio E E U l U l' * E E l' l', (.3) / E l El' * E Eave U U. (.4) Accordig to Eq. (6), close to the brach crossig poit q m = (l + l )/a, the fractio participatig i this result may be approimated as 6 6 Let me hope that the differece betwee this ew iteger ad particle s mass, both called m, is absolutely clear from the cotet. 6 Physically, / = (/a)m = k () /m is ust the velocity of a free classical particle with eergy E (). Chapter Page 5 of 76

83 E l E l' q~, del with dq qq m E ma ( ), ad q~ q q, (.5) while parameters E ave (E l + E l )/ = E () ad U U * = U do ot deped o q ~, i.e. the distace from the cetral poit q m. This is why Eq. (4) may be plotted as the famous level aticrossig (also called avoided crossig, or iteded crossig, or o-crossig ) diagram (Fig. 9), with the eergy gap width equal to U, i.e. ust double the magitude of the -th Fourier harmoic of the periodic potetial U(). Such aticrossigs are also clearly visible i Fig. 8 that shows the results of the eact solutio of Eq. (98) for =.5. 6 m () E E E U E l E l ' q q m E Fig..9. Level aticrossig diagram. We will ru ito the aticrossig diagram agai ad agai i the course, otably at the discussio of spi. Such diagram characterizes ay quatum systems with two weakly-iteractig eigestates with close eergies. It is also repeatedly met i classical mechaics, for eample at the calculatio of eigefrequecies of coupled oscillators. 63, 64 I our curret case of the weak potetial limit, the diagram describes the weak iteractio of two siusoidal de Broglie waves (6), with oppositely directed wave vectors, l ad l, via the (l - l ) th (i.e. th ) Fourier harmoic of the potetial profile U(). This effect eists also for the classical wave theory, ad is kow as the Bragg reflectio, describig, for eample, the D case of the wave reflectio by a crystal lattice (Fig..5) i the limit of weak iteractio betwee the icidet particles ad the lattice. Returig for the last time to our iitial result the bad structure for the delta-fuctioal U() (Fig. 4), show i Fig. 6, we may woder how geeral it is, takig ito accout the peculiar properties of the delta-fuctio approimatio. A partial aswer may be obtaied from the bad structure for two more realistic ad relatively simple periodic fuctios U(): the siusoidal potetial (Fig. 3a) ad the rectagular Kroig-Peey potetial show i Fig. 3b. For the siusoidal potetial (Fig. 3a), with U() = U cos(/a), the statioary Schrödiger equatio (6) takes the form 6 From that figure, it is also clear that i the weak potetial limit, width E of the -th eergy bad is ust E () E ( - ) see Eq. (). Note that this is eactly the distace betwee adacet eergy levels of the simplest D quatum well of ifiite depth cf. Eq. (.77). 63 See, e.g., CM Sec. 5. ad i particular Fig Actually, we could obtai this diagram earlier i this sectio, for the system of two weakly coupled quatum wells (Fig. 3), if we assumed the wells to be slightly dissimilar. Chapter Page 5 of 76

84 Mathieu equatio d U cos E. (.6) m d a By the itroductio of dimesioless variables E U,,, (.7) () () a E E where E () is defied by Eq. (), Eq. (6) may be reduced to the caoical form of the well-kow Mathieu equatio 65 d ( cos ). d (.8) U() a (a) U U U() d (b) U a Fig..3. Two simple periodic potetial profiles: (a) the siusoidal ( Mathieu ) potetial ad (b) the Kroig-Peey potetial. Figure 3 shows the so-called characteristic curves of the Mathieu equatio, i.e. the relatios betwee parameters ad, correspodig to the eergy bad edges separatig them from the adacet bads. (Such curves may be readily calculated umerically, for eample, usig Eqs. (5) with the badedge values qa = ad qa = ). I such phase plae plots, the detailed iformatio about the eergy depedece o the quasi-mometum is lost, but we already kow from Fig. 6 that the depedece is ot too evetful. The most remarkable feature of these plots is the fast (epoetial) disappearace of the allowed eergy bads at > (i Fig. 3, above the red dashed lie), i.e. at E < U. This may be readily eplaied by our tight-bidig approimatio result (6): as soo as the eigeeergy drops sigificatly below the potetial maimum U ma = U (see Fig. 3a), quatum states i the adacet potetial wells are oly coected by tuelig through the separatig potetial barriers, with epoetially small amplitudes see Eq. (4). O the other had, the characteristic curves below the dashed lie, i.e. at <, correspod to virtually free motio of the particle with eergy E above U ma = U. Naturally, i this regio the eergy bads rapidly epad while gaps virtually disappear. This could be epected from the weak potetial limit aalysis (see Fig. 8 ad its discussio); however, based o that aalysis oe could epect that the 65 This equatio, first studied i the 86s by É. Mathieu i the cotet of a rather practical problem of vibratig elliptical drumheads (!), has may other importat applicatios i physics ad egieerig, otably icludig the parametric ecitatio of oscillatios see, e.g., CM Sec Chapter Page 53 of 76

85 eergy gaps U would disappear more gradually. The fast declie of the gaps at U (i.e. ) i the Mathieu equatio is a artifact of the siusoidal potetial U(), with o Fourier harmoics U above the first oe. (I order to calculate the correct asymptotic behavior at, oe eeds to go beyod the first approimatio we have used i the weak potetial limit aalysis.) Fig..3. Characteristic curves of the Mathieu equatio. I applicatio to the bad theory, dotted regios correspod to the eergy gaps, while regios betwee them, to eergy bads. The red dashed lie correspods to coditio =, i.e. E = U U ma, separatig the regios of tuelig ad over-barrier motio. Figure adapted from / 4 If oe wats to study the details of trasitio betwee the two limits i the D bad theory without the artifacts of the delta-fuctioal model show i Fig. 4 (with ifiite umber of harmoics U idepedet of ) ad of the Mathieu equatio (with all U = for ), the stadard way is to eamie the Kroig-Peey potetial show i Fig. 3b. For this potetial, the characteristic equatio may be readily derived usig our rectagular barrier aalysis i Sec. 3. For the case E < U, the result is the followig atural geeralizatio of Eq. (66): k cos qa cosh d cos k( a d) sihd si k( a d), (.9) k where parameters k ad are defied, as fuctios of E ad U, by Eqs. (6) ad (65). I the opposite case E > U, oe ca use the same formula with the replacemet (73). Plots E(q), described by these formulas, 66 are very similar to those show i Figs. 6b ad 8 above. I order to see some differece, oe eeds to plot the characteristic curves U (E). This may be doe by takig qa = ad qa = (i.e. cosqa = ) i Eq. (9), ad solvig the resultig trascedetal equatio for U umerically. The curves are geerally similar to those show i Fig. 3, but, i accordace with Eq. (4), ehibit a more gradual decrease of eergy gaps: U ( ) U, at E ~ E U. (.3) To coclude this sectio, let me address the effect of periodic potetial o the umber of eigestates i D systems of large but fiite legth l >> a, k -. Surprisigly, the Bloch theorem makes the aalysis of this problem elemetary, for arbitrary U(). Ideed, let us assume that l is comprised of 66 Such plots, for several particular values of parameters, may be foud, for eample, i Figs of E. Merzbacher s tetbook cited above. Chapter Page 54 of 76

86 D desity of states ad summatio rule a iteger umber of periods a, ad its eds are described by the similar boudary coditios both assumptios evidetly icosequetial for l >> a (such as a -cm-scale crystal with ~ 8 atoms alog each directio). The, accordig to Eq. (), the boudary coditios impose, o the quasi-mometum q, eactly the same quatizatio coditio as we had for k for a free D motio. Hece, istead of Eq. (.94) we ca write l dn dq, (.3) with the correspodig chage of the summatio rule: l f ( q) f ( q) dk. (.3) q Hece, the desity of states i D q-space, dn/dq = l/, does ot deped o the potetial profile at all! Note, however, that the profile does affect the desity of states o the eergy ais, dn/de. As a etreme eample, o the bottom ad at the top of each eergy bad we have de/dq, ad hece dn de dn dq / de dq l / de dq. (.33) This divergece (which survives i higher spatial dimesioalities as well) of the state desity has importat implicatios for the operatio of several electro ad optical devices, i particular semicoductor lasers..8. Effective mass ad the Bloch oscillatios The bad structure of the eergy spectrum has profoud implicatios ot oly o the desity of states, but also o the dyamics of particles i periodic potetials. I order to see that, let us cosider the simplest case: motio of a wave packet cosistig of Bloch fuctios (), all i the same (say, th ) eergy bad. Similarly to Eq. (7) for the a free particle, we ca describe such a packet as i q qt (, t) a u ( ) e dq, (.34) q where the a-periodic fuctios u(), defied by Eq. (8), are ow ideed to emphasize their depedece o the quasi-mometum, ad (q) E (q)/ is the fuctio of q describig the shape of the correspodig eergy bad see, e.g., Fig. 6b or Fig. 8. If the packet is arrow, i.e. the width q of the distributio a q is much smaller tha all the characteristic scales of the dispersio relatio (q), i particular /a, we may simplify Eq. (34) eactly as we have doe i Sec. for a free particle, despite the presece of factors u q () uder the itegral. I the liear approimatio of the Taylor epasio, we agai get Eq. (3), but ow with 67 v gr d dq qq q v q, ad ph qq, (.35) 67 A geeralizatio of this epressio to the case of essetial iterbad trasitios is ot difficult usig the Heiseberg picture of quatum mechaics (which will be discussed i Chapter 4 of this course) - see, e.g., Sec. 55 i E. M. Lifshitz ad L. P. Pitaevskii, Statistical Physics, Part, Pergamo,98. Chapter Page 55 of 76

87 where q is the cetral poit of the quasi-mometum distributio. Despite the formal similarity with Eq. (33) for the free particle, this result is much more evetful; for eample, as evidet from the dispersio relatio s topology (see Figs. 6b, 8), the group velocity vaishes ot oly at q =, but at all values of q that are multiples of (/a), at the bottom ad o the top of each eergy bad. At these poits, packet s evelope does ot move i either directio - though may keep spreadig. 68 Eve more fasciatig pheomea take place if a particle i the periodic potetial is the subect of a additioal eteral force F(t). (For electros i a crystal lattice, this may be, for eample, the Loretz force of the applied electric ad/or magetic field.) Let the force be relatively weak, so that product Fa (i.e. the scale of eergy icremet from the additioal force per oe lattice period) is much smaller tha the relevat eergy scales the dispersio relatio E(q) see Fig. 6b: Fa,. (.36) This relatio allows oe to eglect the force-iduced iterbad trasitios, so that the wave packet (34) icludes the Bloch eigefuctios belogig to oly oe (iitial) eergy bad at all times. For the time evolutio of its ceter q, theory yields 69 a etremely simple equatio of motio q F( t). (.37) This equatio is physically very trasparet: it is essetially the d Newto law for the time evolutio of the quasi-mometum q uder the effect of the additioal force F(t) oly, ecludig the periodic force -U()/ of the backgroud potetial U(). This is very atural, because q is essetially the particle s mometum averaged over potetial s period, ad the periodic force effect drops out at such a averagig. Despite the simplicity of Eq. (37), the results of its solutio may be highly otrivial. First, let us use Eqs. (35) ad (37) fid the istat group acceleratio of the particle (i.e. the acceleratio of its wave packet s evelope): a gr dv dt gr d dt d dq q d dq dq dq E dq dt d ( q ) dq dq dt d dq qq F( t). (.38) This meas that the secod derivative of the dispersio relatio plays the role of the effective reciprocal mass of the particle: m ef d / dq d E / dq. (.39) For the particular case of a free particle, described by Eq. (6), this epressio is reduced to the origial (ad costat) mass m, but geerally the effective mass depeds o the wave packet s mometum. Accordig to Eq. (39), at the bottom of ay eergy bad, m ef is always positive, but depeds o the stregth of particle s iteractio with the periodic potetial. I particular, accordig to Eq. (6), i the tight bidig limit, the effective mass is very large: Time evolutio of quasimometum Effective mass 68 For a Gaussia packet, the spreadig is described by Eq. (39), with the replacemet k q; it is curious that at the iflectio poits with d /dq = (which are preset i each eergy bad) the packet does ot spread. 69 The proof of Eq. (37) is ot difficult, but becomes more compact i the bra-ket formalism, to be discussed i Chapters 4 ad 5. This is why I recommed the proof to the reader as a eercise after readig those two chapters. Chapter Page 56 of 76

88 m ef q( a () E m / a) m. (.4) O the cotrary, i the weak potetial limit, the effective mass is close to m at most poits of each eergy bad, but at the edges of the (arrow) badgaps it is much smaller. Ideed, epadig Eq. (4) i the Taylor series ear poit q = q m, we get E where ad q ~ are defied by Eq. (5), so that del ~ ~ ( ) U q U q, (.4) EE U dq qq U m m ef q U m E q U ( ) m m. (.4) The effective mass effects i real solids may be very sigificat. For eample, the charge carriers i the ubiquitous field-effect trasistors of silico itegrated circuits have m ef.9 m e i the lowest ormally-empty eergy bad (traditioally called the coductio bad), ad m ef.98 m e i the lower, ormally-filled valece bad. I some semicoductig compouds the coductio-bad electro mass may be eve smaller - dow to.45 m e i ISb! However, the absolute value of the effective mass i ot the most surprisig effect. The more shockig corollary of Eq. (39) is that o the top of each eergy bad the effective mass is egative please revisit Figs. 6, 8, ad 9 agai. This meas that the particle (or more strictly its wave packet s evelope) is accelerated i the directio opposite to the force. This is eactly what electroic egieers, workig with electros i semicoductors, call holes, characterizig them by positive mass ad positive charge. If the particle does ot leave a close viciity of the eergy bad s top (say, due to scatterig effects), such flip of sigs does ot lead to a error, because the Loretz force is proportioal to electro s charge (q = -e), so that particle s acceleratio a gr is proportioal to ratio (q/m ef ). 7 However, at some pheomea the usual image of a hole as a particle with q > ad m ef > is uacceptable. For eample, let us form a arrow wave packet at the bottom of the lowest eergy bad, 7 ad the eert o it a costat force F > say, due to a costat eteral electric field directed alog ais. Accordig to Eq. (37), this would lead to a liear growth of q i time, so that i the quasimometum space, the packet s ceter would slide, with costat speed, alog the q ais see Fig. 3a. Close to the eergy bad bottom, this motio would correspod to a positive effective mass (possibly, somewhat larger tha the geuie particle s mass m), ad hece be close to free particle s acceleratio. However, as soo as q has reached the iflectio poit, where d E /dq =, the effective mass, ad hece acceleratio (38) chage sigs to egative, i.e. the packet starts to slow dow (i the direct space 7 The laguage is which the hole has a positive charge ad mass has a additioal coveiece for states o the top of the valece bad whose sigle-particle states are ormally filled. The the simplest, sigle-particle ecitatio of this multi-particle groud state may be created by givig oe electro eough eergy to lift it to a referece (e.g., Fermi-eergy) level E F that is, by defiitio of the valece bad, is higher tha all values E - (q). The it is atural to prescribe to the ecitatio a positive mass m ef, because the eergy E = E F E - (q) ecessary for the ecitatio grows with the deviatio of q from q m. 7 Ituitio tells us (ad statistical physics duly cofirms :-) that this may be readily doe, for eample, by weakly couplig the system to a low-temperature eviromet, ad lettig it to rela to the lowest possible eergy. Chapter Page 57 of 76

89 ) while still movig ahead i the quasi-mometum space. Fially, at the eergy bad s top the particle stops at certai ma, while cotiuig to move i the q-space. E(q) (a) E (b) E ( q ) / F E E E ( q ) qa a E / F Fig..3. The Bloch oscillatios (red lies) ad the Ladau-Zeer tuelig (blue arrows) withi: (a) the time-domai picture, ad (b) the eergy-domai picture. O pael (b), the tilted gray strips show the allowed eergy bads, ad the bold red lies, the Waier-Stark ladder. ma Now we have two alterative ways to look at the further time evolutio of the wave packet. From the eteded zoe picture (which is the simplest for this aalysis, see Fig. 3a), 7 we may say that the particle crosses the st Brilloui zoe boudary ad starts goig forward i q, i.e. dow the lowest eergy bad. Accordig to Eq. (35), this regio (up to the et iflectio poit) correspods to a egative group velocity. After q has reached the et miimum of the eergy bad at qa =, the whole process repeats agai (ad agai, ad agai). These are the famous Bloch oscillatios the effect that was predicted (by the same F. Bloch) as early as i 99, but evaded eperimetal observatio util the 98s - see below. Their time period may be readily foud from Eq. (37): q / a tb, (.43) dq / dt F / Fa so that the Bloch oscillatio frequecy Fa B. (.44) t The direct-space motio of the wave packet s ceter (t) durig the Bloch oscillatio process may be aalyzed by itegratig Eq. (35) over some time iterval t: B Bloch oscillatios: frequecy 7 This pheomeo may be also discussed from the poit of view of the reduced zoe picture, but the it requires the itroductio of istat umps betwee the Brilloui zoe boudary poits (see the dashed red lie i Fig. 3) that correspod to physically equivalet states of the particle. Evidetly, this laguage is more artificial. Chapter Page 58 of 76

90 Bloch oscillatios: spatial swig Waier- Stark ladder d( q ) d( q t t t tt ( t) vgrdt dt d q dq dq / dt F F t ). (.45) If iterval t is equal to the Bloch oscillatio period t B (34), the iitial ad fial momets of E(q ) = (q ) are equal, givig = : i the ed of the period, the wave packet returs to its iitial positio. However, if we carry this itegratio oly from the smallest to the largest values of (q ), i.e. the poits where the group velocity vaishes, we get the oscillatio swig E ma ma mi. (.46) F F This simple result may iterpreted usig a alterative eergy diagram (Fig. 3b) that results from the followig argumets. The additioal force F may be described ot oly via the d Newto law versio (37), but, alteratively, by its cotributio U F = - F to the total ( Gibbs 73 ) potetial eergy U ( ) U ( F (.47) ) of the system. The direct solutio of the Schrödiger equatio (6) with such potetial may be hard to fid, but if the force is weak i the sese of Eq. (36), as we are assumig ow, oe ca argue that our quatum-mechaical treatmet icludig the periodic potetial U() should be still correct, if the secod term i Eq. (47) is cosidered as a costat at the wave packet width scale, but depedet o positio of the packet s ceter. I this approimatio, the total eergy of the wave packet may be foud as E E( q F. (.48) I a plot of such eergy as a fuctio of (Fig. 3b), the iformatio o eergy depedece o q is lost, but we already kow it is rather uevetful, ad well characterized by the positio of bad-gap edges o the eergy ais. 74 I this represetatio, the Bloch oscillatios of a relatively wide ( >> a) wave packet should keep the full eergy E costat, i.e. follow a horizotal lie i Fig. 3b, limited by the classical turig poits correspodig to the bottom ad the top of the allowed eergy bad. The distace ma betwee these poit is evidetly give by Eq. (46). Besides this secod look at the oscillatio swig result, the total eergy diagram show i Fig. 3b eables oe more remarkable result. Let a wave packet be so arrow i the mometum space (q ) that /q >> ma ; the the horizotal lie segmet i Fig. 3b presets the spatial etesio of the eigefuctio of the Schrödiger equatio with potetial (47). But this equatio is evidetly ivariat with respect to the followig simultaeous traslatio i coordiate ad eergy: ) a, E E Fa. (.49) This meas that it is satisfied with a ifiite set of similar solutios, each correspodig to oe of the horizotal red lies show i Fig. 3b. This is the famous Waier-Stark ladder, with the step height Fa. (.5) E S 73 See, e.g., CM Sec I semicoductor device physics ad egieerig, such plots are called the bad edge diagrams, ad are the virtually uavoidable compoets of ay discussio or publicatio. Chapter Page 59 of 76

91 The importace of this alterative represetatio of the Bloch oscillatios is due to the followig fact. I most eperimetal realizatios, the power of radiatio at frequecy (44), that may be etracted from the oscillatios by their electromagetic couplig to a eteral detector, is very low, so that their direct detectio presets a hard problem. 75 However, let us apply to a Bloch oscillator a additioal rf field at frequecy ~ B. As these frequecies are brought close together, the eteral sigal should sychroize ( phase lock ) Bloch oscillatios, 76 resultig i certai observable chages for eample, a resoat absorptio of the eteral radiatio. Now let us otice that Eqs. (44) ad (5) yield the followig remarkable relatio: E B. (.5) S This meas that the resoat pheomea at B allow for a alterative (but equivalet) iterpretatio as the result of rf-iduced trasitios 77 betwee the steps of the Waier-Stark ladder! (Such occasios whe two very differet laguages may be used for the iterpretatio of the same pheomeo is oe of the most beautiful features of physics.) This effect has bee used for the first eperimetal cofirmatio of the Bloch oscillatio theory. For this purpose, the atural periodic structures, solid state crystals, are icoveiet due to their very small period a ~ - m. Ideed, accordig to Eq. (44), such structures require very high forces F (ad hece high electric fields E = F/e) to brig B to a eperimetally coveiet rage. This problem has bee overcome by fabricatig artificial periodic structures (superlattices) of certai semicoductor compouds, such as Ga - Al As with various degrees of gallium to alumium atom replacemet, whose layers may be grow over each other epitaially, i.e., without very few crystal structure violatios. These superlattices, with periods a ~ m, has allowed a clear observatio of resoat effects at B, ad hece the measuremet of the Bloch oscillatio frequecy, i particular its proportioality to the applied dc electric field, predicted by Eq. (44). 78 Very soo after this observatio, the Bloch oscillatios have bee observed i small Josephso uctios. 79 Sice this eperimet ivolved two importat coceptual issues, let me discuss it i a little bit more detail. As was discussed i Sec..3, the Josephso uctio dyamics may be reasoably well described by two simple equatios (54) ad (55). They may be combied to calculate the work of a eteral voltage source at Josephso phase chage betwee arbitrary iitial ( ii ) ad fial ( fi ) values, as the itegral of its power IV over the time iterval t of the chage: work t IVdt t d e dt I e fi c c I si dt sid cos cos c ii I e fi ii. (.5) We see that the work depeds oly o the iitial ad fial values of (but ot o the law phase evolutio i time), i.e. may be preseted as the differece U( fi ) U( ii ), where fuctio 75 I systems with may idepedet particles (such as semicoductors), the detectio problem is eacerbated by phase icoherece of the Bloch oscillatios performed by each particle. This drawback is abset i atomic Bose- Eistei codesates whose Bloch oscillatios (i a periodic potetial created by stadig optical waves) were evetually observed by M. Be Daha et al., Phys. Rev. Lett. 76, 458 (996). 76 A simple aalysis of phase lockig of a classical oscillator may be foud, e.g., i CM Sec A qualitative theory of such trasitios will be discussed i Sec. 6.6 ad the i Chapter E. Medez et al., Phys. Lev. Lett. 6, 46 (988). 79 L. Kuzmi ad D. Havilad, Phys. Rev. Lett. 67, 89 (99). Chapter Page 6 of 76

92 Josephso couplig eergy SFQ pulse I C U EJ cos cost, with EJ, (.5) e may be iterpreted as the potetial eergy of the uctio (if we cosider the Josephso phase as a geeralized coordiate). This eergy apart, the Josephso uctio, as a system of two close, early isolated (super)coductors, has a certai capacitace C ad the associated electrostatic eergy E C = CV /. Usig Eq. (54) agai, we may preset it as E C C V C e d. (.5) dt This meas that from the poit of view at phase as a geeralized coordiate, E C should be cosidered the kietic eergy of the system, whose depedece o the geeralized velocity d/dt is similar to that of a D mechaical particle, with a effective mass 8 J m C. (.5) e Hece the total eergy of the uctio, E C + U(), is formally similar to that of a D o-relativistic particle i the siusoidal potetial with the -ais period a J =. However, before usig the results of the D bad theory to this system, we have to resolve oe parado (that was the subect of a lively discussio ust about 3 years ago). Whe we develop the bad theory, we imply that its traslatio by period a is (i priciple) measurable, i.e. particle positios ad ( + a) are distiguishable otherwise Eq. (93) with q would ot have much sese. For a mechaical particle this assumptio is very plausible, but less so for a Josephso uctio. Ideed, for eample, if we chage by a J = via chagig the phase of oe of supercoductors, say (Fig. 3) by, the its wavefuctio becomes ep{i( + )} = ep{i }, ad it is ot immediately clear whether these two states may be distiguished. I order to resolve this cotradictio, it is sufficiet to have a look at Eq. (54). It shows that if chages i time by (say, by a fast ramp-up), voltage V across the uctio ehibits a pulse with area V ( t) dt e d dt dt e d e h e 5 V s. (.53) Such sigle-flu-quatum (SFQ) pulses 8 ot oly may be measured eperimetally, but eve have bee used for sigalig ad ultrafast (sub-thz) computatio, to the best of my kowledge still keepig the absolute records for the highest speed ad smallest eergy cosumptio at computatio. 8 Hece, the -shifts of phase are measurable, ad i the absece of dissipatio the Josephso uctio dyamics is ideed similar to that of a D particle i a periodic (siusoidal) potetial, ad its eergy spectrum forms eergy bads ad gaps described by the Mathieu equatio see Fig. 3. Eperimetally, the easiest way to verify this picture is to measure the correspodig Bloch oscillatios 8 Of course, the dimesioality of m ef so defied is differet from kg. 8 This term has origiated from the fact that the right-had part of Eq. (53) equals to the sigle quatum uit ( ) of the magetic flu i supercoductors see Sec. 3. below. 8 See, e.g., P. Buyk et al., It. J. o High Speed Electroics ad Systems, 57 (). Chapter Page 6 of 76

93 iduced by a eteral curret I e (t). I order to fid the frequecy of these oscillatios, it is sufficiet to replace Eq. (37), which epresses the d Newto law averaged over period a of potetial U(), with the charge balace equatio dq I e, t (.54) dt where Q is the quasi-charge 83, i.e. the electric charge of the capacitor averaged over the period of the periodic potetial U(). (Notice that at such averagig, curret (55) is averaged out from the equatio, so that is affects the pheomea oly via its cotributio to the eergy bad structure.) Sice the Josephso-uctio aalog of the geuie wave umber k = m(d/dt)/ of a particle is mj d mj e CV kj V, (.55) dt e ad CV is the geuie charge o the capacitor, the aalog of q (the quasi-mometum divided by ) may be obtaied ust by the replacemet of that product with quasi-charge Q: Q qj. (.56) e Comparig this epressio with Eq. (54), we see that q J obeys the followig equatio of motio: dqj I e t. (.57) dt e so that the role of force F is ow played by F J = I/e. Hece if I e (t) = cost = I, we ca use Eq. (44) with that replacemet, ad also a a J =, to get f B B FJ aj I e. (.58) This very simple result has the followig physical sese. 84 I the quatum operatio mode, the uctio is recharged by the eteral curret, followig Eq. (56), util its electric charge reaches e (i.e. q J a J = (Q/e) reaches - see Fig. 3a); the oe Cooper pair passes through the uctio chagig its charge to e (e) = -e, with the same chargig eergy (5) the process aalogous to crossig the border of the st Brilloui zoe; the the process repeats agai ad agai. It is remarkable that Eq. (58), describig the frequecy of such a quatum property of the Josephso phase as its Bloch oscillatios, does ot iclude the Plack costat, while Eq. (56), describig the classical motio of, does. 85 Bloch oscillatios i supercoductivity 83 Eq. (54) tells us that quasi-charge Q has the simple physical sese of the eteral electric charge beig iserted ito the uctio by the eteral curret I e - ust like the physical sese of quasi-mometum q of a mechaical particle, accordig to Eq. (37), is the cotributio to particle s mometum by the eteral force F. 84 D. Averi et al., Sov. Phys. JETP 6, 47 (985). 85 Phase lockig of the Bloch oscillatios, with frequecy (58), as well as that of very similar SET oscillatios of frequecy f SET = I/e, by a sigal of well characterize frequecy, eable fudametal stadards of dc curret. The eperimetally achieved accuracy of such stadards is close to -8, a few times worse tha that of a less direct way - usig the Josephso voltage stadard ad the resistace stadard based o the quatum Hall effect. Chapter Page 6 of 76

94 Quatum uit of resistace I this cotet, oe may woder which of these two types of oscillatios would a dc-biased Josephso uctio geerate. For the dissipatio-free uctio, the aswer is: the Bloch oscillatios (58) with frequecy proportioal to dc curret. However, ay practical uctio has some eergy losses that may be (approimately) described by a certai Ohmic coductace G coected i parallel to the uctio. Very luckily for Dr. Josephso ad his Nobel Prize, it is much easier to fabricate ad test uctios with G >> / R Q, where R Q is the so-called quatum uit of resistace R Q e 6.45k, (.59) the fudametal costat that umps out at aalysis of several other effects as well see, e.g., Sec. 3.. As will be discussed i Chapter 7, such high eergy losses provide what is called dephasig the suppressio of the quatum coherece betwee differet quatum states of the system i our curret case, betwee the wavefuctios u( - ) localized at differet miima of the periodic potetial U(), ad thus make the dyamics of the Josephso phase virtually classical, obeyig equatios (54) ad (55). As we have see i Sec., dc biasig of such a uctio leads to Josephso oscillatios with frequecy (56) proportioal to the applied dc voltage..9. Ladau-Zeer tuelig All the Bloch oscillatio discussio i the last sectio was based o the premise that the particle stays withi oe (say, the lowest) eergy bad. However, ust a sigle look at Fig. 3 shows that this assumptio becomes urealistic if the eergy gap separatig this bad from the et oe becomes very small,. Ideed, i the weak potetial approimatio, that is adequate i this limit, at U, the two dispersio curve braches (6) cross without ay iteractio, so that if our particle (the wave packet) is drive to approach that poit, it should cotiue to move up i eergy - see the dashed blue arrow i Fig. 3a. Similarly, i the eergy-domai presetatio show i Fig. 3b, it is ituitively clear that at, the particle residig at oe of the steps of the Waier-Stark ladder should able to somehow overcome the vaishig spatial gap = /F ad to leak ito the et bad see the horizotal dashed blue arrow o that pael. This process, called the Ladau-Zeer (or iterbad, or bad-to-bad ) tuelig 86 is ideed possible. I order to aalyze it, let us first take F =, ad cosider what happes if a quatum particle described by a -log (i.e. E-arrow) wave packet is icidet from the free space upo a periodic structure of a large but fiite legth l >> a. If packet s eergy E is withi oe of the eergy bads, it may evidetly propagate through the structure (though may be partly reflected from its frot ed). The correspodig quasi-mometum may be foud by solvig the dispersio relatio for q; for eample, i the weak-potetial limit, Eq. (4), which is valid ear the gap, yields q q m q, ~ q ad is give by the secod of Eqs. (5). ~ ( ) / ~ E U, where E E E ~, (.6) 86 It was predicted idepedetly by L. D. Ladau, Phys. Z. Sowetuio, 46 (93) ad C. Zeer, Proc. R. Soc. Lodo A 37, 696 (93). Chapter Page 63 of 76

95 Now, if eergy E correspods to oe of the eergy gaps, the propagatio is impossible, so that the packet is completely reflected back. However, our aalysis of the potetial step problem i Sec. 3 implies that the wavefuctio would still have a epoetial tail protrudig ito the periodic structure ad decayig o some legth - see Eq. (67). Ideed, a review of the calculatio leadig to Eq. (6) shows that they remai valid withi the gap as well, if the quasi-mometum is uderstood as a purely imagiary umber: ~ / ~ U E, for E U q~ i, where. (.6) With such cotributio, the Bloch solutio (93b) ideed describes a epoetial decay of the wavefuctio at legth = /. Now returig to the effects of weak force F i the eergy-domai approach, preseted by Eq. (48) ad illustrated i Fig. 3b, we may recast Eq. (6) as U ( F~ ) / ( ), (.6) where ~ is particle s (i.e. wave packet ceter s) deviatio from the mid-gap poit. Thus the gap has created a potetial barrier of a fiite width = F/U, through which the wave packet may tuel with a fiite probability. As we already kow, i the WKB approimatio (i our case requirig >> ) this probability is ust the tuel barrier s trasparecy T, which may be calculated from Eq. (7): lt ( ( ) d ) c c ~ / ~ U U ( F ) d c / d. (.63) where c / = F/U are the classical turig poits. Workig out this simple itegral (which may be viewed upo as the quarter of the uit circle s area, ad hece equal to /4), we get T U ep. (.64) F Ladau- Zeer tuelig probability This famous result was obtaied by Ladau ad Zeer i a more comple way, whose advatage is a costructive proof that Eq. (64) is valid for arbitrary relatio betwee F ad U, i.e. arbitrary T, while our simple derivatio was limited to the WKB approimatio, i.e. to T <<. 87 Returig to Eq. (5) ad (37), we ca rewrite the product F participatig i Eq. (64) as E E dq de E d l l' l l' u F, (.65) dq dt dt E E E l l ' E E E l l ' where u has the meaig of the speed of the eergy level crossig i the absece of the gap. Hece, Eq. (64) may be preseted i a form 87 Note that Eq. (64) is still limited to the hyperbolic dispersio relatio, i.e. (i the bad theory) to the weak potetial limit. I the opposite, tight-bidig limit, the iterbad tuelig may be treated as a ecitatio of the upper bad states by siusoidal Bloch oscillatios, ad is completely suppressed at B <. Chapter Page 64 of 76

96 T U ep u, (.66) that is more physically trasparet. 88 Ideed, the fractio U /u = u gives the time scale t of eergy s crossig the gap regio, ad accordig to the Fourier trasform, its reciprocal, ma ~ /t gives the upper cutoff of frequecies ivolved i the Bloch oscillatio process. Hece Eq. (66) meas that lt. (.67) This formula allows us to iterpret the Ladau-Zeer tuelig as for system s ecitatio across the eergy gap, by the maimum eergy quatum ma available from the Bloch oscillatio process. The iterbad tuelig is a importat igrediet of several physical pheomea ad eve some practical devices, for eample the tuelig (or Esaki ) diodes. This simple device is ust a uctio of two semicoductor electrodes, oe of them is so strogly -doped by electro doors that the additioal electros form a degeerate Fermi gas at the bottom of the coductio bad. Similarly, the opposite electrode is p-doped so strogly that the Fermi level of electros i the valece bad is lowered below the bad edge (Fig. 33). ma (a) (b) I (c) -doped p-doped ev ev V Fig..33. Tuelig diode: (a) the bad edge diagram of the device at zero bias; (b) the same diagram at modest positive bias ev ~ /, ad (c) the I-V curve (schematically). Dashed lies show the Fermi level positios. At thermal equilibrium, ad i the absece of eteral voltage bias, the Fermi levels self-alig, 89 leadig to the build-up of the cotact potetial differece /e, with somewhat larger tha the eergy badgap - see Fig. 33a. This potetial differece creates a iteral electric field that tilts the eergy bads (ust as the eteral field did i Fig. 3b), ad leads to the formatio of the so-called deletio layer i which the Fermi level located is withi the eergy gap ad hece there are o charge carriers ready to move. I usual p- uctios, this layer is broad ad prevets ay curret at applied voltages V lower tha ~/e. I cotrast, i a tuelig diode the depletio layer is so thi (below ~ m) that the 88 I Chapter 6, Eq. (66) will be derived usig a differet method based o the Golde Rule of quatum mechaics. 89 See, e.g., SM Secs..5 ad 6.4. Chapter Page 65 of 76

97 iterbad tuelig is possible ad provides a substatial Ohmic curret at small applied voltages see Fig. 33c. However, at substatial positive bias, ev ~ /, the coductio bad become aliged with the middle of the gap i the p-doped electrode, ad electros caot tuel there. Similarly, these are o electros i the -doped semicoductor to tuel ito the available states ust above the Fermi level i the p-doped electrode see Fig. 33b. As a result, curret drops sigificatly, to grow agai oly whe ev eceeds ~ ad allows the electro motio through the withi each eergy bad. Thus the tuel uctio s I-V curve has a part with egative differetial resistace (dv/di < ). This effect may be used for the amplificatio of aalog sigals, icludig self-ecitatio of electrical oscillators (i.e. rf sigal geeratio), 9 ad sigal swig restoratio i digital electroics... Harmoic oscillator: A brute force approach To complete our review of D systems, we have to cosider the famous harmoic oscillator, i.e. a D particle movig i the quadratic-parabolic potetial (). This is ust a smooth quatum well providig soft cofiemet, whose discrete spectrum we have already foud i the WKB approimatio see Eq. (4). Let us try to solve the same problem eactly ot because there is aythig coceptually iterestig i it (there is ot :-), but because of its eormous importace for applicatios. For that, let us write the statioary Schrödiger equatio for potetial (): d m E. (.68) m d From the solutio of Eercise Problem.5, the reader already kows 9 oe of the eigefuctios of this equatio, m C ep, (.69) ad the correspodig eigeeergy E. (.7) Epressio (69) shows that the characteristic scale of wavefuctio s spatial spread 9 is equal to Groud state wavefuctio Groud state eergy m /. (.7) Due to the importace of this scale, let us give its crude estimates for several typical systems: Wavefuctio spread scale 9 See, e.g., CM Sec If ot yet, I am ivitig him or her to check this fact ow by the direct substitutio of solutio (69) ito the differetial equatio (68), simultaeously provig Eq. (7). 9 Quatitatively, as was already metioed i Sec.., = = /. Chapter Page 66 of 76

98 Hermite polyomials (i) Electros i solids ad fluids: m -3 kg, ~ 5 s -, givig ~.3 m, comparable with iter-atomic distaces a. As a result, classical mechaics is ot valid at all for the aalysis of their motio. (ii) Atoms i solids: m kg, ~ 3 s -, givig ~. -. m, i.e. from ~a few percet to a few tes percet of a. Because of that, methods based classical mechaics (e.g., molecular dyamics) are approimately valid for the aalysis of atomic motio, though may miss some fie effects of motio of lighter atoms e.g., quatum tuelig of hydroge atoms through eergy barriers of the potetial profile created by its eighbors. (iii) Probe masses i moder gravity-wave detectors (Advaced LIGO, VIRGO, KAGRA, etc.): 93 m ~ kg, ~ s -, givig ~ -9 m. After several decades of developmet, the sesitivity of these istrumets is still limited by various oise sources at the level of the order of -8 m. 94 Thus the prospects of observig quatum-mechaical effects i such istallatios do ot look very realistic. Returig to the Schrödiger equatio (68), let us recast it ito a dimesioless form by itroducig dimesioless variable /. This gives d, (.7) d where E/ = E/E. I this otatio, the groud state wavefuctio is proportioal to ep{- /}, so that let us look for the solutios to Eq. (7) i the form C ep H ( ), (.73) where H() is a ew fuctio. With this substitutio, Eq. (7) yields d H d dh ( ) H d. (.74) It is evidet that H = cost ad = is oe of its solutios, describig the eigestate (69) with eergy (7), but what are the other eigestates ad eigevalues? This equatio has bee studied i detail i the mid-8s by C. Hermite who has show that all eigevalues are give by equatio, with =,,,, (.75) so that our WKB result (4) is ideed eact for ay, ad Eqs. (69) ad (7) describe the groudstate of the oscillator. The eigefuctio correspodig to eigevalue is a polyomial (ow called the Hermite polyomial) of degree, that may be most coveietly calculated usig the followig eplicit formula: d ep ep H. (.76) d 93 See, e.g., ad a recet update by T. Feder, Phys. Today 68, No. 9, (5). 94 Accordig to the recet aoucemet by B. Abbott et al., Phys. Rev. Lett. 6, 6 (6), this sesitivity was sufficiet for the first direct detectio of gravitatioal waves emitted at a merger of two black holes. Chapter Page 67 of 76

99 It is easy to use this formula to calculate several lowest-degree polyomials see Fig. 34a: 3 4 H, H, H 4, H 8, H 6-48,... (.77) 3 4 (a) ( ) H (b) E 3 E E E U () / Fig..34. (a) A few lowest Hermite polyomials ad (b) the correspodig eigeeergies (dashed lies) ad eigefuctios (solid lies) of the harmoic oscillator. The black dashed lie shows the potetial profile U(), draw o the same scale as eergies E, so that the lie crossigs with the eergy levels correspod to the classical turig poits. The most importat properties of the polyomials are as follows: (i) their parity (symmetry-atisymmetry) alterates with umber, (ii) H () crosses the -ais eactly times (has zeros), ad (iii) the polyomials are mutually orthoormal i the followig sese: H / d!. ( ) H ' ( )ep, ' (.78) Chapter Page 68 of 76

100 Harmoic oscillator s eigefuctios Usig Eq. (73) to traslate this result to fuctios (), we get the followig orthoormal eigefuctios of the harmoic oscillator (Fig.34b): 95 ( ) ep H. (.79) / / 4 /! Besides its ow importace, this is a typical eample of eigestates of particle cofied i a softwall quatum well. It is very istructive to compare them with eigestates of a the rectagular quatum well, with its ultimately-hard walls see Eq. (.76) ad Fig..7. Let us list their similar features: (i) Wavefuctios oscillate i the classically-allowed regios with E > U(), while droppig epoetially beyod the boudaries of that regio. (ii) Each step up the eergy level ladder icreases the umber of the oscillatio halfwaves (ad hece the umber of its zeros), by oe. 96 Here are the maor features specific for the soft cofiemet: (i) The spatial spread of the wavefuctio grows with, followig the gradual icrease of the classically allowed regio. (ii) Correspodigly, E ehibits a slower growth tha the E law give by Eq. (.77), because of the gradual reductio of cofiemet, which moderates the growth of kietic eergy. Ufortuately, the brute-force approach to the harmoic oscillator problem, discussed above, is ot too appealig itellectually. First, the proof of Eq. (76) is rather logish. More importatly, it is hard to use Eq. (79) for calculatio of the so-called matri elemets of the system as we will see i Chapter 4, virtually the oly umbers importat for applicatios. Fially, it is also almost evidet that there should be some straightforward math leadig to ay formula as simple as Eq. (4) for E. Ideed, there is a much more efficiet, operator-based approach to this problem; it will be described i Sec Eercise problems.. The iitial wave packet of a free D particle is described by Eq. (.) of the lecture otes: ik, a e dk. (i) Obtai a compact epressio for the epectatio value p of particle's mometum. Does p deped o time? (ii) Calculate p for the case whe fuctio a k is symmetric with respect to some value k... Calculate the fuctio a k, defied by Eq. (.), for the wave packet with a rectagular evelope: k 95 These statioary states of the harmoic oscillator are sometimes called its Fock states, to distiguish them from other fudametal solutios (such as Glauber states) which will be discussed i Sec. 5.5 ad beyod.. 96 I mathematics, a slightly more geeral statemet, valid for a broader class of ordiary liear differetial equatios, is frequetly called the Sturm oscillatio theorem, ad is a part of the Sturm-Liouville theory of such equatios see, e.g., Chapter i the hadbook by G. Arfke et al. recommeded i MA Sec. 6. Chapter Page 69 of 76

101 C ep ik for a / a /, (,), otherwise. Aalyze the result i the limit k a.,.3. Prove Eq. (49) for the D propagator of a free quatum particle, startig from Eq. (48)..4. Epress the D propagator, defied by Eq. (44), via eigefuctios ad eigeeergies of a particle movig i a arbitrary statioary potetial U(). (For the otatio simplicity, assume that the eergy spectrum of the system is discrete.).5. Calculate the chage of the wavefuctio of a D particle, resultig from a short pulse of a eteral force, which may be approimated by the delta-fuctio: 97 F t t P..6. * Aalyze the effect of phase lockig of Josephso oscillatios o the dc curret flowig through the uctio, assumig that eteral microwave source applies a fied siusoidal ac voltage, V ( t) V Acost, to a uctio with siusoidal curret-phase relatio (55), usig Eq. (54) for time evolutio of phase..7. Calculate the trasmissio coefficiet T as a fuctio of particle s eergy E for the rectagular potetial barrier,, for d /, U ( ) U, for d / d /,, for d /, for the case E > U. Aalyze ad iterpret the result, takig ito accout that U may be either positive or egative. (I the last case, we are speakig about particle s passage over a rectagular potetial well of fiite depth.).8. Lookig at the lower (red) lie i Fig..7, it seems plausible that the D groud-state fuctio X() si(/a) of the simple quatum well (.69) may be well approimated by a iverted parabola: X C a trial, where C is the ormalizatio costat, ad a a for brevity. Eplore how good this approimatio is Spell out the statioary wavefuctios of a harmoic oscillator i the WKB approimatio, ad use them to calculate the epectatio values ad 4 for arbitrary state umber. 97 The costat P is called the force s impulse. (I higher dimesioalities, it is a vector - ust as the force is.) 98 Solvig this problem is a good preparatio to the use of the full variatioal method i the et two problems (ad beyod). Chapter Page 7 of 76

102 .. * A D particle of mass m is placed ito the followig triagular quatum well: 99, for, U with F. F, for, (i) Calculate its eergy spectrum usig the WKB approimatio. (ii) Estimate the groud state eergy usig the variatioal method. (iii) Calculate the three lowest eergy levels, ad also for the th level, with at least.% accuracy, from the eact solutio of the problem. (iv) Compare ad discuss the results. Hits: - I Task (ii), try to icorporate a certai parameter ito your trial wavefuctio, ad the use its adustmet to miimize the epectatio value of system s Hamiltoia (metioed i Chapter ): H trial * trialh triald, where the trial fuctio is assumed to be properly ormalized. The variatioal method is based o the easily provable fact that this epectatio value caot be less tha the geuie E g, coicidig with it oly if the trial fuctio eactly coicides with the geuie wavefuctio g of the groud state. Hece, the lower H trial you reach, the better is your result. - The values of the first zeros of the Airy fuctio, ecessary for Task (iii), may be foud i may math hadbooks, for eample, i Table.3 of the collectio edited by Abramowitz ad Stegu see MA Sec. 6(i)... For a D particle of mass m placed ito a potetial well with the followig profile, U s a, with a ad s (i) calculate its eergy spectrum usig the WKB approimatio, ad (ii) estimate the groud state eergy usig the variatioal method. Compare the groud state eergy results for parameter s equal to,, 3, ad... Prove Eq. (7) for the case T WKB <<, usig the coectio formulas (4)..3. Use the WKB approimatio to epress the epectatio value of the kietic eergy of a D particle, cofied i a soft potetial well, i its th statioary state, via the derivative de /d, for >>..4. * Use the WKB approimatio to calculate the trasparecy T as a fuctio of particle eergy E, for the followig triagular potetial barrier:, U ( ) U F, for, for,, 99 With F = mg, this is ust the well-kow boucig ball problem. See, e.g., Sec. 8. below. Chapter Page 7 of 76

103 with F, U >. Hit: Be careful treatig the sharp potetial step at =..5. * Prove that the symmetry of the scatterig matri elemets describig a arbitrary timeidepedet scatterer allows its represetatio i the form (36a), with the additioal restrictio (36b)..6. Prove the uiversal relatios betwee elemets of the trasfer matri T of a statioary (but otherwise arbitrary) D scatterer, which were metioed i Sec For a deep ad arrow D quatum well, modeled by a delta-fuctio, U ( ) W ( ), with W, (*) fid the localized eigefuctio(s) (with ( ) at ), ad the correspodig value(s) E..8. A D particle was localized i the delta-fuctioal well, with U() = -W(), such as the oe aalyzed i the previous problem. The (say, at t = ) the well s bottom is suddely lifted, so that the particle becomes free to move. Calculate the probability desity, w(k) to fid the particle i a state with wave umber k at t >, ad the fial total eergy of the system..9. Calculate the lifetime of the metastable localized state of a D particle i the potetial W F, with W U, usig the WKB approimatio. Formulate the coditio of validity of the result... Aalyze the localized eigefuctio(s) ad the characteristic equatio(s) for eigeeergies of a D particle i the followig two-well potetial a a U ( ) W, with W. Eplore asymptotic behaviors of the eigeeergies i the limits of very strog ad very weak potetial, ad fid the umber of localized states as a fuctio of distace a... * Cosider a symmetric system of two quatum wells of the type show i Fig. 3, but with U() = U() = see Fig. o the right. What is the sig of well iteractio force due to a quatum particle of mass m, shared by U them, for the cases whe the particle is i: (i) a symmetric eigestate, with s (-) = s ()? (ii) a asymmetric eigestate, with a (-) = - a ()? Use a differet approach to cofirm your result for the particular case of delta-fuctioal wells, cosidered i the previous problem... Derive ad aalyze the characteristic equatio for eigevalues for a particle i a rectagular well of a fiite depth: Chapter Page 7 of 76

104 U, for a/, U ( ), otherwise. I particular, calculate the umber of localized states as a fuctio of well s width a, ad eplore the limit U << /ma..3. Calculate eergy E of the localized state i a quatum well of a arbitrary shape U(), provided that its width a is fiite, ad the average depth is very small: U ma, where U a well U d..4. * A particle of mass m is movig i a field with the followig potetial: U U W, where U () describes a smooth, symmetric fuctio with U () =, growig mootoically at. (i) Use the WKB approimatio to derive the characteristic equatio for the eergy spectrum; (ii) semi-quatitatively describe the spectrum structure evolutio at the icrease of W, for both sigs of this parameter, ad make the results more specific for the quadratic potetial U m..5. Prove Eq. (9), startig from Eq. (9)..6. For the problem eplored i the begiig of Sec. 7, i.e. D particle s motio i a deltafuctioal periodic potetial show i Fig. 4, W a, with W U, (where are itegers), write eplicit epressios for its eigefuctios: (i) at the bottom, ad (ii) at the top of the lowest eergy bad. Sketch both eigefuctios..7. * A D particle of mass m moves i a ifiite periodic system of very arrow ad deep quatum wells that may be described by delta-fuctios: W a, with W U. (i) Sketch the eergy bad structure of the system for relatively small ad relatively large values of the quatum well s area W, ad Chapter Page 73 of 76

105 (ii) calculate eplicitly the groud state eergy of the system i the limits of very small ad very large W..8. * For the system discussed i the previous problem, write eplicit epressios for the eigefuctios of the system, correspodig to: (i) the bottom poits of the lowest eergy bad, ad (ii) the top poits of that bad, ad (iii) the lowest poits of each higher eergy bad, ad sketch the fuctios..9. * The D crystal, aalyzed i the last two problems, ow eteds alog oly to >, while borderig a flat potetial step at = : U W U, a, with W, for, for. Prove that the system ca have a set of so-called Tamm states, localized ear the surface =, ad calculate their eergies i the limit whe U is very large but fiite. (Quatify this coditio.).3. Calculate the whole trasfer matri of the rectagular tuel barrier, specified by Eq. (76), for particle eergies both below ad above U..3. Use results of the previous problem to calculate the trasfer matri of oe period of the periodic Kroig-Peey potetial show i Fig. 3b (reproduced i Fig. o the right). U() U d a.3. Usig results of the previous problem, derive the characteristic equatios for particle s motio i the periodic Kroig-Peey potetial, for both E < U ad E > U. Try to brig the equatios to a form similar to that obtaied i Sec. 5 for the delta-fuctioal barriers see Eq. (66). Use the equatios to formulate the coditios of applicability of the tight-bidig ad weak-potetial approimatios, i terms of parameters U, d, ad a of the potetial profile, ad particle s mass m ad eergy E..33. * For the Kroig-Peey potetial, use the tight bidig approimatio to calculate the widths of the allowed eergy bads. Compare the results with those of the previous problem (i the correspodig limit). I applicatios to electros i solid-state crystals, the delta-fuctioal quatum wells model the attractive potetial of atomic uclei, while U represets the workfuctio, i.e. the eergy ecessary for the etractio of a electro from the crystal to the free space see, e.g., EM Sec..6 ad SM Sec Chapter Page 74 of 76

106 .34. * For the same Kroig-Peey potetial, use the weak potetial limit formulas to calculate the eergy gap widths. Agai, compare the results with those of Problem 3, i the correspodig limit..35. D periodic chais of atoms may ehibit what is called the so-called Peierls istability, leadig to the Peierls trasitio to phase i which atoms are slightly displaced by = (-), with << a. These displacemets lead to the alteratio of couplig amplitudes (see Eq. (4)) betwee some values + ad -. Use the tight-bidig approimatio to calculate the resultig chage of the th eergy bad, ad discuss the result..36. Assumig the quatum effects to be small, calculate the lower part of the eergy spectrum of the followig system: a small bead of mass m, free to move without frictio alog a rig of radius R that is rotated about its vertical diameter with a costat agular velocity - see Fig. o the right. Formulate a quatitative coditio of validity of your results. R mg.37. A D harmoic oscillator (with mass m ad frequecy ) had bee i its groud state; the a additioal force F was suddely applied (ad retaied costat i time). Fid the probability of the oscillator stayig i its groud state..38. A D particle of mass m has bee placed ito a quadratic potetial well (), m U ( ), ad allowed to rela ito the groud state. harmoic oscillator had bee i its groud state. At t =, the well starts to be moved with velocity v, without chagig its profile, so that at t the above formula for U is valid with the replacemet vt. Calculate the probability for the system to still be i the groud state at t >..39. A D particle is placed ito the followig potetial well:, for, U ( ) m /, for. (i) Fid its eigestates ad eigeeergies. (ii) This system had bee let to rela ito its groud state, ad the the potetial wall at < was rapidly removed, so that the system was istatly tured ito the usual harmoic oscillator (with the same m ad ). Fid the probability for the oscillator to be i its groud state..4. Prove the followig formula for the propagator of the D harmoic oscillator: This system was used as the aalytical mechaics testbed problem i the CM part of this series, ad the reader is welcome to use ay relatios derived there - but remember that they pertai to the classical mechaics domai! Chapter Page 75 of 76

107 / cos[ ( t t )]. m im G(, t;, t ) ep si[ ( )] i t t si[ ( t t )] Discuss the relatio betwee this formula ad the propagator of a free D particle..4. Use the variatioal method to estimate the groud state eergy E g of the followig cofied D systems: (i) a harmoic oscillator, with U() = m /, ad (ii) a particle i the followig potetial well: U() = -U ep{- }, ad U >. I the latter case, get eplicit results i the limits of small ad large U, ad give their iterpretatio..4. * Use the WKB approimatio to calculate the lifetime of the metastable groud state of a D particle of mass m i the pocket of the potetial profile m U ( ) Cotemplate the sigificace of this problem..43. I the cotet of the Sturm oscillatio theorem metioed i Sec., prove that the umber of zeros of statioary wavefuctios of a particle, cofied i a arbitrary potetial well, always icreases with eergy. Hit: You may like to use the suitably modified Eq. (89). 3. Chapter Page 76 of 76

108 Chapter 3. Higher Dimesioality Effects The coverage of multi-dimesioal problems of wave mechaics i this course is miimal: it is limited to a few pheomea (such as the AB effect ad Ladau levels) that caot take place i oe dimesio due to topological reasos, ad a few key 3D problems (such as the Bor approimatio i scatterig theory ad the Bohr atom) whose solutios are ecessary for umerous applicatios. 3.. Quatum iterferece ad the AB effect I the past two chapters, we have already discussed some effects of the de Broglie wave iterferece. For eample, stadig waves iside a quatum well, or eve o the top of a tuel barrier, may be cosidered as a result of the icidet ad reflected waves. However, there are some remarkable ew effects made possible by the spatial separatio of such travelig waves, ad such separatio requires a higher (either D or 3D) dimesioality. A good eample of such separatio is provided by the Youg-type eperimet (Fig. ) i which particles are passed through two arrow holes (or slits) is a otherwise opaque partitio. particle source l ' l z'' ' l C partitio with slits '' l particle detector W w(r) Fig. 3.. Scheme of the two-slit (Youg-type) iterferece eperimet. If the particles emitted by the source do ot iteract (which is always true if the emissio rate is sufficietly low), the average rate of particle coutig by the detector is proportioal to the probability desity w(r, t) = (r, t) *(r, t) to fid a sigle particle at the detector s locatio r, where (r, t) is the solutio of the sigle-particle Schrödiger equatio (.5). Let us describe this eperimet for the case whe the particles may be represeted by moochromatic waves of eergy E (e.g., very r-log wave packets), so that the wavefuctio may be take i the form give by Eqs. (.56) ad (.6): (r, t) = (r) ep{-iet/}. I this case, i the free-space parts of the system, (r) satisfies the statioary Schrödiger equatio (.6) with Hamiltoia (.7a): 3D Helmholtz equatio E. (3.a) m With the stadard defiitio k (me) / /, it may be rewritte as the 3D Helmholtz equatio k (3.b) K. Likharev

109 a evidet 3D geeralizatio of Eqs. (.75) or (.8). The opaque parts of the partitio may be well described as classically forbidde regios, so if their size scale a is much larger tha the wavefuctio peetratio depth (.67), we ca use o their surface S the same boudary coditios as for the quatum barrier of ifiite height:. (3.) S Equatios () ad () formulate the stadard boudary problem of the theory of propagatio of scalar waves of ay ature. For a arbitrary geometry, such problem does ot have a simple aalytical solutio. However, for a coceptual discussio of iterferece we use certai atural assumptios that will allow us to fid its particular, approimate solutio. First, let us discuss wave emissio, ito free space, by a small-size source located at the origi. Naturally, the emitted wave should be spherically-symmetric: (r) = (r). Usig the well-kow epressio for the Laplace operator i spherical coordiates, we the reduce Eq. () to a ordiary differetial equatio d dr d dr r k r. (3.3) Let us itroduce a ew fuctio, f(r) = r(r). Pluggig the reciprocal relatio = f/r ito Eq. (3), we see that it is reduced to the D wave equatio, d f k f, (3.4) dr whose solutios were discussed i detail i Sec... For a fied k, the geeral solutio of Eq. (4) is so that the full wavefuctio ikr ikr f f e f e (3.5) f ikr f ikr f i kr t f i kr t ( ) e e i.e., t e e with r ( ) ( ),, r r r r r If the source is located at poit r, the obvious geeralizatio of Eq. (6) f i( krt) f i( krt) ( r, t) e e, R R with R R, E k. (3.6) m R r r'. (3.7) The first term of this solutio describes a spherically-symmetric wave propagatig from the source outward, while the secod oe, a wave covergig oto the source poit r from large distaces. Though the latter solutio is possible at some very special circumstaces (say, whe the outgoig wave is reflected back from a spherical shell), for our problem, oly the outgoig waves are relevat, so that we may keep oly the first term (proportioal to f + ) i Eq. (7). Note that factor R is the deomiator (that was abset i D geometry) has a simple physical sese: it provides the idepedece of the full probability curret I = 4R (R), with (R) k* /R, of the distace R betwee the observatio poit ad the source. See, e.g., MA Eq. (.9). Chapter 3 Page of 56

110 Kirchhoff itegral Wavefuctio superpositio Now let us assume that the partitio s geometry is ot too complicated for eample, it is plaar as show i Fig., ad cosider the regio of the particle detector locatio far behid the partitio (at z >> /k), ad at a relatively small agle to it: << z. The it should be physically clear that the spherical waves (7) emitted by each poit iside the slit caot be perturbed too much by the opaque parts of the partitio, ad their oly role is the restrictio of the set of such emittig poits by the area of the slits. Hece, a approimate solutio of the boudary problem is give by the followig Huyges priciple: the wave behid the partitio looks as if it was the sum of cotributios (7) of poit sources located i the slits, with each source s stregth f + proportioal to the amplitude of the wave arrivig at this pseudo-source from the real source see Fig.. This priciple fids its cofirmatio i strict wave theory, which shows that with our assumptios, the solutio of the boudary problem ()-() may be preseted as the followig Kirchhoff itegral: ( r' ) ikr k ( r ) c e d r', with c. (3.8) R i slits If the source is also far from the partitio, its wave frot is almost parallel to the slit plae, ad the slits are ot too broad, we ca take (r ) costat (, ) at each slit, so that Eq. (8) is reduced to ca, ( r ) a" epikl" a" epikl", with a",,, (3.9) l" where A, are the slit areas. The wavefuctios o the slits be calculated approimately 3 by applyig the same Eq. (7) to the space before the slits:, (f + /l, )ep{ikl, }. As a result, Eq. (9) may be rewritte as ( r) a ikl a epikl, ep with l, l', l'',, a,, c f A l' l",,,. (3.) (As Fig. shows, each of l, is the legth of the full classical path of the particle from the source, through the correspodig slit, ad further to the observatio poit r see Fig. ). Accordig to Eq. (), the resultig rate of particle coutig is proportioal to Quatum iterferece: the patter ad phase shift where * a aa cos w( r ) ( r) ( r) a, (3.) k( l ) (3.) l is the differece of the total wave phase accumulatios alog each of two alterative traectories. The last epressio may be evidetly geeralized as For a proof of Eq. (8), see, e.g., EM Sec A possible (ad reasoable) cocer about the applicatio of Eq. (7) to the field i the slits is that it igores the effect of opaque parts of the partitio. However, as we kow from Chapter, the mai role of the classically forbidde regio is providig the reflectio of the icidet wave towards its source (i.e. to the left i Fig. ). As a result, the cotributio of this reflectio to the field iside the slits is isigificat is A, >>, ad eve i the opposite case provides ust some rescalig of the amplitudes a,, which is uimportat for our coceptual discussio. Chapter 3 Page 3 of 56

111 k dr, (3.3) C with itegratio alog the virtually closed cotour C (see the dashed lie i Fig. ), i.e. from poit, i the positive (i.e. couterclockwise) directio to poit. (From our eperiece with the D WKB approimatio we may epect such geeralizatio to be valid eve if k chages, sufficietly slowly, alog the paths.) Our result () shows that the coutig rate oscillates as a fuctio of the differece (l l ) that i tur chages with detector s positio, givig the famous iterferece patter, with the amplitude proportioal to the product a a, ad hece vaishig if ay of the slits is closed. For a wave theory, this is a well-kow result, 4 but for particle physics, is was (ad still is :-) rather shockig. Ideed, our aalysis pertais to a very low particle emissio/detectio rate, so that there is o other way to iterpret it rather tha resultig from particle s iterferece with itself, or rather the iterferece of its wavefuctio parts passig through each of two slits. Let us ow discuss a very iterestig effect of magetic field o the quatum iterferece. I order to make the discussio simpler, let us cosider a alterative versio of the two-slit eperimet, i which each of alterative path is fied to a arrow chael usig a partial cofiemet see Fig.. (I this arragemet, movig the particle detector without chagig chaels geometry, ad hece local values of k may be more problematic i eperimetal practice, so let us thik about its positio r fied.) regio with B chael C w w(b) chael Fig. 3.. The AB effect. I this case, because of the effect of the walls providig the path cofiemet, we caot use epressios () for amplitudes a,. However, from the discussios i Sec..6 ad Sec.., it should be clear that the first of epressios () remais valid, though may be with a value of k specific for each chael. The beefit of this geometry is that we ca ow apply magetic field B, perpedicular to the plae of particle motio, that would pierce cotour C, but would ot touch the particle propagatio chaels. I classical physics, magetic field s effect o a particle with electric charge q is described by the Loretz force 5 FB qv B, (3.4) 4 See, e.g., a detailed discussio i EM Sec See, e.g., Sec. 5.. Note that Eq. (4), as well as all other formulas of this course, are i the SI uits; i Gaussia uits, all terms which iclude either B or A should be divided by c, the speed of light i free space. Chapter 3 Page 4 of 56

112 where B is the field value at the poit of its particle s locatio, so that for the eperimet show i Fig., F B =, ad the field would ot affect the particle motio at all. I quatum mechaics, this is ot so, ad the field does affect the probability desity w, eve if B = i all poits where the wavefuctio (r) is ot equal to zero. I order to describe this surprisig effect, let us first develop a geeral framework for accout of effects of electromagetic fields o a quatum particle, which will also give us some importat byproduct results. I order to do that, we eed to calculate the Hamiltoia operator of a charged particle i the field. For a electrostatic field, this hardly preset ay problem. Ideed, from classical electrodyamics we kow that such field may be preseted as a gradiet of its electrostatic potetial, E r, (3.5) so that the force eerted by the field o a particle with electric charge q, F E qe, (3.6) may be described by addig the potetial eergy of the field, U r qr, (3.7) to other (possible) compoets of the full potetial eergy of the particle. As we have already discussed, such a fuctio of coordiates may be icluded to the Hamiltoia operator ust by addig it to the kietic eergy operator (.7). However, magetic field s effect is peculiar: sice its Loretz force (4) caot do ay work o the particle: dwb FB dr FB vdt q( v B) vdt, (3.8) the field caot be preseted by ay potetial eergy, so it may ot be immediately clear how to accout for it i the Hamiltoia. Help comes from the aalytical-mechaics approach to classical electrodyamics: 6 i the o-relativistic limit, the Hamiltoia fuctio of a particle i electromagetic field looks superficially like that i electrostatic field oly: mv p H U q ; (3.9) m however, the mometum p mv that participates i this epressio is ow the differece p P qa. (3.) Here A is the vector-potetial, defied by the well-kow relatios for the electric ad magetic field: 7 A E, B A, (3.) t while P is the caoical mometum whose Cartesia compoets may be calculated (i classics) from the Lagragia fuctio, 8 usig the stadard formula of aalytical mechaics, 6 See, e.g., EM Sec See, e.g., EM Sec. 6.7, i particular Eqs. (6.6). 8 Just for reader s referece, the classical Lagragia correspodig to Hamiltoia (9) is Chapter 3 Page 5 of 56

113 P L. (3.) v To emphasize the differece betwee the two mometa, p = mv is frequetly called the kiematic mometum (or mv-mometum ). The distictio betwee p ad P = p + qa becomes eve more clear if we otice that vector-potetial is ot gauge-ivariat: accordig to the secod of Eqs. (), at the so-called gauge trasformatio A A, (3.3) with a arbitrary sigle-valued scalar gauge fuctio = (r, t), the magetic field does ot chage. Moreover, accordig to the first of Eqs. (), if we make the simultaeous replacemet, (3.4) t the gauge trasformatio does ot affect the electric field either. With that, the gauge fuctio does ot chage the classical particle s equatio of motio, ad hece the velocity v ad mometum p. Hece, the kiematic mometum is gauge-ivariat, while P is ot, because it chages by q. Now the stadard way of trasfer to quatum mechaics is to treat the caoical rather tha kiematic mometum accordig to correspodece postulate discussed i Sec... This meas that i the coordiate represetatio, the operator of this variable is give by Eq. (.6): 9 P i. (3.5) Hece the Hamiltoia operator correspodig to the classical fuctio (9) is iq H i qa q A q, (3.6) m m so that the Schrödiger equatio of a particle movig i electromagetic field (but otherwise free) is Caoical mometum operator iq A q E, (3.7) m We may ow repeat all the calculatios of Sec..4 for the case A, ad readily get the followig geeralized epressio for the probability curret desity: q p c.c A. iq c.c A (3.8) im m m Charged particle i EM field mv L qv A q - see EM Sec Note that this fuctio icludes A withi a term that caot be iterpreted as either the purely kietic eergy (as the first term) or the purely potetial eergy (as the last term with the mius sig). 9 The validity of this choice is clear from the fact that if the kietic mometum was described by this differetial operator, the Hamiltoia operator correspodig to the classical Hamiltoia fuctio (9) would ot iclude the magetic field at all, ad hece solutios of the correspodig Schrödiger equatio could ot satisfy the correspodece priciple. Chapter 3 Page 6 of 56

114 AB effect We see that the curret desity is gauge-ivariat (as required for ay observable) oly if the wavefuctio s phase chages as q. (3.9) This may be a poit of cocer: sice the quatum iterferece is described by the spatial depedece of phase, ca the observed iterferece patter deped o the gauge fuctio choice (which would ot make sese)? Fortuately, this is ot true, because the spatial phase differece betwee two iterferig paths, participatig i Eq. (), is gauge-trasformed as q. (3.3) But has to be a sigle-valued fuctio of coordiates, hece i the limit whe poits ad coicide, =, so that (ad hece the iterferece patter) is gauge-ivariat. However, the differece may be affected by the magetic field, eve if it is localized outside the chaels i which the particle propagates. Ideed, i this case the field caot ot affect particle s velocity ad curret desity : r r, (3.3) ( ) B ( ) B so that the last form of Eq. (8) yields q ( r) B ( r) B A. (3.3) Itegratig this equatio alog cotour C (Fig. ), for the phase differece betwee poits ad we get q B B A dr, (3.33) where the itegral should be take alog the same virtually closed cotour C as before (i Fig., from poit, couterclockwise alog the dashed lie to poit ). But from the classical electrodyamics we kow that as poits ad are overlapped, i.e. cotour C becomes closed, the last itegral is ust the magetic flu B d r through ay smooth surface limited by cotour C, so that Eq. (33) may be preseted as q B B Φ. (3.34a) I terms of the iterferece patter, this meas a shift of iterferece friges, proportioal to the magetic flu (Fig. 3). This pheomeo is usually called the Aharoov-Bohm (or ust the AB) effect. For particles with a sigle elemetary charge, q = e, this result is frequetly preseted as C See, e.g., EM Sec I persoally prefer the latter, less persoable ame, because the effect had bee actually predicted by W. Ehreberg ad R. Siday i 949, before it was rediscovered by Y Aharoov ad D. Bohm i 959. To be fair to Aharoov ad Bohm, it was their work that triggered a wave of iterest to the pheomeo, resultig i its first Chapter 3 Page 7 of 56

115 B B, (3.34b) ' where the fudametal costat /e = h/e Wb has the meaig of flu ecessary to chage by, i.e. shift the iterferece patter () by oe period, ad is called the ormal magetic flu quatum, because of the reasos we will soo discuss. (a) (b) Fig Typical results of a two-paths iterferece eperimet by A. Toomura et al., Phys. Rev. Lett. 56, 79 (986), showig the AB effect for electros well shielded from the applied magetic field. I this particular eperimetal geometry, the AB effect produces a relative shift of the iterferece patters iside ad outside the dark rig. (a) = /, (b) =. 986 APS. The AB effect may be almost eplaied classically, i terms of Faraday s electromagetic iductio. Ideed, a chage of magetic flu i time causes a vorte-like electric field E aroud it. That field is ot restricted to the magetic field s locatio, i.e. may reach particle s traectories. The field s magitude (or rather of its itegral alog cotour C) may be readily calculated by itegratio of the first of Eqs. (): dφ ΔV ΔE dr, (3.35) dt C I hope that i this epressio the reader readily recogizes the itegral ( udergraduate ) form of Faraday s iductio law. Now let us assume that the variable separatio described i Sec..5 may be applied to the ed poits ad of particle s alterative traectories as two idepedet systems, ad that the magetic flu chage by certai amout does ot chage the spatial parts of wavefuctios of these systems. The chage (35) leads to the chage of potetial eergy differece U = qv betwee the two poits, ad repeatig the argumets that were used i Sec..3 at the discussio of the Josephso effect, we may rewrite Eq. (.53) as d U q q d V. (3.36) dt dt Itegratig this relatio over the time of magetic field s chage, we get eperimetal observatio by R. Chambers i 96 ad several other groups soo after that. Later, the eperimets were improved, usig ferromagetic cores ad/or supercoductig shieldig to provide better separatio betwee the electro traectories ad the applied magetic field, see i the work whose results are show i Fig. 3. This assumptio may seem a bit of a stretch, but the resultig relatio (37) may be ideed prove for a rather realistic model, though that would take more time ad space that I ca afford. Chapter 3 Page 8 of 56

116 Supercoductig flu quatum q, (3.37) - superficially, the same result as give by Eq. (34). However, this iterpretatio of the AB effect is limited. Ideed, it requires the particle to be i the system (o the way from the source to the detector) durig the flu chage, i.e. whe the iduced electric field E may affect its dyamics. O the cotrary, Eq. (34) predicts that the iterferece patter would shift eve if the field chage has bee made whe the there is o particle i the system, ad hece field E could ot be felt by it. Eperimet cofirms the latter coclusio. Hece, there is somethig i the space where a particle propagates (i.e., outside of the magetic field regio), which trasfers iformatio about eve the static magetic field to the particle. The stadard iterpretatio of this surprisig fact is as follows: the vector-potetial A is ot ust a coveiet mathematical tool, but a physical reality (ust as its electric couterpart ), despite the large freedom of choice we have i prescribig specific spatial ad temporal depedeces of these potetials without affectig ay observable see Eqs. (3)-(4). Let me briefly discuss the very iterestig form the AB effect takes i supercoductivity. I this case, our results require two chages. The first oe is simple: sice supercoductivity may be iterpreted as the Bose-Eistei codesate of Cooper pairs with electric charge q = e, has to be replaced by the so-called supercoductig flu quatum Φ.7 Wb.7 Gs cm. (3.38) e Secod, sice the pairs are Bose particles ad are all codesed i the same quatum state, described by the same wavefuctio, the total electric curret desity, proportioal to the probability curret desity, may be etremely large i real supercoductig materials, up to ~ A/m. I these coditios, oe caot eglect the cotributio of that curret ito the magetic field ad hece its flu, which (accordig to the Lez rule of the Faraday iductio law) tries to compesate chages i eteral flu. I order to see possible results of this cotributio, let us cosider a closed supercoductig loop (Fig. 4). B C Fig Flu quatizatio i a supercoductig loop. Due to the Meisser effect (which is ust aother versio of the flu self-compesatio), curret ad magetic field peetrate iside the supercoductor by oly a small distace (called the Lodo 3 Oe more bad, though commo, term a wire ca (super)coduct, but a quatum hardly ca! Chapter 3 Page 9 of 56

117 peetratio depth) L ~ -7 m. 4 If the loop is made of a supercoductig wire that is cosiderably thicker tha L, we ca draw a cotour deep iside the wire, at that the curret desity is egligible. Accordig to Eq. (8), everywhere at the cotour, q A. (3.39) Itegratig this equatio alog the cotour as before (from poit to the virtually coicidig poit ), we eed to have the phase differece =, because the wavefuctio ep{i} i the iitial ad fial poits ad should be essetially the same, i.e. produce the same observables. As a result, we get Φ A dr. (3.4) q e C This is the famous flu quatizatio effect, 5 which ustifies the term magetic flu quatum for the costat give by Eq. (38). Here I have to metio i passig very iterestig effects of partial flu quatizatio, that arise whe a supercoductor loop is closed by a Josephso uctio, formig the so-called Supercoductor QUatum Iterferece Device - SQUID. Such devices are used, i particular, for supersesitive magetometry ad ultrafast, low-power computig. 6 Flu quatizatio 3.. Ladau levels ad quatum Hall effect I the last sectio, we have used the Schrödiger equatio (7) for aalysis of static magetic field effects i almost-d, circular geometries show i Figs.,, ad 4. However, this equatio describes very iterestig effects i higher dimesios as well, especially i the D case. Let us cosider a uiform D quatum well (say, parallel to the [, y] plae), with strog cofiemet i the perpedicular directio z. Accordig to the discussio i Sec..6, eergy-relaed particles will always reside i the lowest eergy subbad, with costat quatizatio eergy (E z ). Addig this shift to well s flat floor U(,y) = cost, ad takig the resultig costat eergy as the referece, for the D motio of the particle i the well, we reduce Eq. (7) to the similar equatio, but with the Laplace operator actig oly i directios ad y: m y q i A E. (3.4) y Let us fid its solutios for the simplest case whe the applied static magetic field is uiform ad perpedicular to the plae: B B z. (3.4) 4 For more detail, see EM Sec It was predicted i 949 by F. Lodo ad eperimetally discovered (idepedetly ad virtually simultaeously) i 96 by two eperimetal groups: B. Deaver ad W. Fairbak, ad R. Doll ad M. Näbauer. 6 A brief review of these effects, ad recommedatios for further readig may be foud i EM Sec Chapter 3 Page of 56

118 Ladau levels Accordig to the secod of Eqs. (), this imposes the followig restrictio o the choice of vectorpotetial: Ay A B, (3.43) y but the gauge trasformatios still give us a lot of freedom i its choice. The atural aiallysymmetric form, A = B/, where = ( + y ) / is the distace from some z-ais, leads to a cumbersome math. I 93, L. Ladau realized that the eergy spectrum of Eq. (4) may be obtaied by makig a very simple choice A, A B, (3.44) y (with arbitrary ), which evidetly satisfies Eq. (43), though it igores the physical equivalece of the ad y directios. Now, epadig the eigefuctio ito the Fourier itegral i directio y: iky y y X e (, ) ( ) dk, (3.45) we see that for each compoet of this itegral, Eq. (4) yields a specific equatio k d q i y k X k EX k m B( ) d. (3.46) Sice the vectors iside the square brackets are mutually perpedicular, its square has o crossterms, so that Eq. (46) may be rewritte as d q k X k B ' X k EX k ' m d m, where. (3.47) qb But this D Schrödiger equatio is idetical to Eq. (.68) for the D harmoic oscillator, but with the ceter at poit, ad frequecy equal to qb c. (3.48) m I this epressio, it is easy to recogize the classical cyclotro frequecy of particle s motio i the magetic field. (It may be readily obtaied usig the d Newto law for a circular orbit of radius r, v m F B qvb, (3.49) r ad otig that the resultig ratio v/r = qb/m is ust the radius-idepedet agular velocity c of particle s rotatio.) Hece, the eergy spectrum for each Fourier compoet of itegral (45) is the same: E c, (3.5) ad does ot deped o either, or y, or k. This is a eample of a highly degeerate system: for each eigevalue E, there are may differet eigefuctios that differ by the positios {, y } of their ceter o ais, ad the rate k of Chapter 3 Page of 56

119 their phase chage alog ais y. They may be used to assemble a large variety of liear combiatios, icludig D wave packets whose ceters move alog classical circular orbits with some radius r determied by iitial coditios. Note, however, that such radius caot be smaller tha the so-called Ladau radius, rl, (3.5) qb which characterizes the miimum radius of the wave packet itself, ad results from Eq. (.7) after replacemet (48). This radius is remarkably idepedet o particle s mass, ad may be iterpreted i the followig way: the scale BA mi of the applied magetic field s flu through the effective area A mi = r L of the smallest wave packet is ust oe ormal flu quatum = /q. A detailed aalysis of such wave packets (for which we would ot have time i this course) shows that magetic field does ot chage the average desity dn /de of differet D states o the eergy scale, but ust assembles them o the Ladau levels (see Fig. 5a), so that the umber of states o each Ladau area (per uit area) is N dn dn m qb qb L B ΔE B ΔE. c (3.5) A A de A de h This epressio may agai be iterpreted i terms of magetic flu quata: L = B, i.e. there is oe particular state o each Ladau level per each flu quatum. / Ladau radius E B = (a) B c c (b) electrodes E F Fig (a) Codesatio of D states o Ladau levels, ad (b) fillig the levels by eteral electros at the quatum Hall effect. The most famous applicatio of the Ladau levels cocept is the eplaatio of the quatum Hall effect 7. Geerally, the Hall effect 8 is observed i the geometry sketched i Fig. 6, where electric curret I is passed through a thi rectagular coductig sample (frequetly called the Hall bar) placed ito a magetic field B perpedicular to the sample plae. The classical aalysis of the effect is based o the otio of the Loretz force (4). This force the deviates charge carriers (say, electros) from their straight motio from oe eteral electrode to aother, bedig them to the isolated edges of the bar (i Fig. 6, parallel to ais ). Here the carriers accumulate, geeratig a gradually icreasig electric field E, util its force (6) eactly balaces the Loretz force (4): 7 It was first observed i 98 by K. vo Klitzig ad coworkers. 8 Discovered i 879 by E. Hall. Chapter 3 Page of 56

120 qe qvb, (3.53) y where v is the drift velocity of the electros alog the bar (Fig. 6), providig the sustaied balace coditio E y /v = B z at each poit of the D sample. W y E B v, I Fig Hall bar geometry. Darker rectagles show eteral (3D) electrodes. Classical Hall effect Quatum Hall effect With carriers per uit area, i a sample of width W, this coditio yields the followig classical epressio for the so-called Hall resistace R H : R H Vy E yw B. (3.54) I q v W q This formula is broadly used i practice for the measuremet of the carrier desity, ad (i semicoductors) the carrier type egative electros or positive holes. However, i eperimets with high-quality (low-defect) D quatum wells at very low, subkelvi temperatures 9 ad high magetic fields, the liear growth of R H with B, described by Eq. (54), is iterrupted by virtually horizotal plateaus (Fig. 7) with costat values RH R K, (3.55) i where i (oly i this cotet, followig traditio!) is a iteger, ad value R K k (3.56) is reproduced with etremely high accuracy (~ -9 ) from eperimet to eperimet ad from sample to sample. Such stability is a very rare eceptio i solid state physics were most results are oticeably depedet o the particular material ad particular sample uder study. Let us apply the Ladau level picture. The D sample is typically i a weak cotact with 3D electrodes whose coductivity electros form a Fermi sea with certai Fermi eergy E F, so that at low temperatures all states with E < E F are filled with electros see Fig. 5b. As B is icreased, spacig c betwee the Ladau levels icreases, so that fewer ad fewer of these levels are below E F ad are filled, ad withi broad rages of field variatio, the umber i of filled levels is costat. (I Fig. 5b, i =.) So, pluggig = i L ad q = e ito Eq. (54), we get R H B i q L h, i e i e (3.57) 9 Recetly, the quatum Hall effect was observed at room temperature i the graphee (a virtually perfect D sheet of carbo atoms, see Sec. 4 below) see K. Novoselov et al., Sciece 35, 379 (7). Chapter 3 Page 3 of 56

121 i.e. eactly the eperimetal result (55), with h R K 4. (3.58) e e This costat, eactly 4 times the quatum uit of resistace R Q give by Eq. (.59), is i a ecellet agreemet with eperimetal value (56), ad is sometimes called the Klitzig costat. R H B Fig Typical record of the quatum Hall effect. The lower trace (with sharp peaks) shows the logitudial compoet, V /I, of the resistace tesor. (Adapted from However, this oversimplified eplaatio of the quatum Hall effect does ot take ito accout several importat factors, icludig: (i) the role of ouiformity of the quatum well bottom potetial U(, y), ad of the localized states this ouiformity produces, ad the surprisigly small effect of these factors o the etraordiary accuracy of Eq. (55); ad (ii) the mutual Coulomb iteractio of the electros, i high-quality samples leadig to the formatio of R H plateaus with ot oly iteger, but also fractioal values of i (/3, /5, 3/7, etc.). Ufortuately, a thorough discussio of these iterestig features is well beyod the framework of this course Scatterig ad diffractio The secod class of quatum effects that become more rich i multi-dimesioal space is typically referred to as either diffractio or scatterig - depedig o the cotet. (Diffractio is essetially the iterferece, but of waves emitted by several may coheret sources.) Just as i the two The eplaatio of this parado may be obtaied i terms of the so-called quatum edge chaels the quasi- D regios of width (5), alog the lies were the Ladau levels cross the Fermi surface. Particle motio alog these chaels, which is resposible for electro trasfer, is effectively oe-dimesioal ad thus caot be affected by modest uiformities of the potetial distributio U(, y). This fractioal quatum Hall effect was discovered i 98 by D. Tsui, H. Stormer, ad A. Gossard. I cotrast, the effect described by Eq. (55) with iteger i (Fig. 7) is ow called the iteger quatum Hall effect. For a comprehesive discussio of these effects I ca recommed, e.g., either the moograph by D. Yoshioka, The Quatum Hall Effect, Spriger, 998, or the review by D. Yeie, Rev. Mod. Phys. 59, 78 (987). Chapter 3 Page 4 of 56

122 slits i the Youg-type eperimet (Fig. ), these sources are most frequetly the elemetary re-emitters of some iitial wave from a sigle source. More geerally, such re-emittig is called scatterig; this term is also applied to particles eve if their quatum properties may be igored. 3 Figure 8 shows the geeral scatterig situatio. Most commoly, the detector of scattered particles (i the quatum case, read de Broglie waves) is located at a large distace r >> a from the scatterer. 4 I this case, the mai observable idepedet of r is the flu (umber of particles per uit time) of particles scattered i a certai directio, i.e. their flu per uit solid agle. Sice such flu is proportioal to the icidet flu of particles per uit area, the ability of the scatterer to re-emit i a particular directio may be characterized by the ratio of these two flues. This ratio has the dimesioality of area per uit agle, ad is called the differetial cross-sectio of the scatterer: Differetial crosssectio d flu of scatterd particles per uit solid agle. (3.58) d flu of icidet particles per uit area k a r a, k k detector scattered particles icidet particles scatterer Fig D scatterig (schematically). Full crosssectio Such ame ad otatio stem from the fact that the itegral of d/d over all scatterig agles, d total flu of scattered particles dω, (3.59) dω icidet flu per per uit area (also with the dimesioality of area), has a simple iterpretatio as the full cross-sectio of scatterig. For the simplest case whe a macroscopic solid obect scatters all classical particles hittig its surface, but does ot affect the particles flyig by it, is ust the geometrical cross-sectio of the obect, as visible from the directio of icomig particles. I classical mechaics, 5 we first calculate the particle scatterig agle as a fuctio of the impact parameter b, ad the average the result over all values of b, cosidered radom. I this sese the calculatios i wave mechaics are simpler, because a parallel beam of icidet particles of fied eergy E may be fairly preseted by a plae de Broglie wave ik r a e, (3.6) 3 See, e.g., CM Sec I optics, this limit is called the Frauhofer diffractio see, e.g., EM Secs. (8.6) ad (8.8). 5 For eample, i the simplest task of derivatio of the so-called Rutherford formula for scatterig of a charged o-relativistic particle by a poit fied charge, due to their Coulomb iteractio see, e.g., CM Sec Chapter 3 Page 5 of 56

123 with the free-space wave umber k = (me) / / ad costat probability curret desity (.49): a k. (3.6) m This curret desity is eactly the flu of icidet particles per uit area that is used i the deomiator of defiitio (58), so the oly remaiig thig to do is to calculate the omiator of that fractio. To do this, let us write the Schrödiger equatio for the scatterig problem (ow i the whole space icludig the scatterer) i the form where E H U ( r), (3.6) k H k, ad E. (3.63) m m m the potetial eergy U(r) describes the effect of scatterer. Lookig for the solutio of Eq. (6) i the atural form, (3.64) where is the icidet wave (6), ad s has the sese of the scattered wave, ad takig ito accout that former wave satisfies the free-space equatio we may reduce Eq. (6) to s H E, (3.65) E H U s r. (3.66) The most straightforward (ad commo) simplificatio of this problem is possible if the scatterig potetial U(r) is i some sese weak. (We will derive the eact coditio of this smalless below.) The sice at U(r) = the scatterig wave s disappears, we may epect that at small but ovaishig U(r), the mai part of s is proportioal to its scale U. The all terms i Eq. (66) are proportioal to U, besides the product U s, which is proportioal to U. Hece, i the first approimatio i U, that term may be igored, ad Eq. (66) reduces to the famous equatio of the Bor approimatio: 6 E H s U r s. (3.67a) This simplificatio chages the situatio drastically, because the liear superpositio priciple allows fidig a eplicit solutio of this equatio (i the form of a itegral) for a arbitrary fuctio U(r). Ideed, after rewritig Eq. (67a) as Bor approimatio 6 Named after M. Bor, who was the first oe to apply this approimatio i quatum mechaics. However, the basic idea of this approach had bee developed much earlier (i 88) by Lord Rayleigh i the cotet of electromagetic wave scatterig see, e.g., EM Sec Note that the cotets of that sectio repeats much of our curret discussio regrettably but uavoidably so, because the Bor approimatio is a ceterpiece of scatterig theory for both electromagetic ad de Broglie waves. Chapter 3 Page 6 of 56

124 Spatial Gree s fuctio Gree s fuctio for free space m k U ( r) ( r) s, (3.67b) we may otice that s is ust a respose of a liear system to a certai ecitatio (represeted by the right-had part) that is fied, i.e. does ot deped o the solutio. Hece we ca preset s as a sum of resposes to elemetary ecitatios from elemetary volumes d 3 r : m 3 s ( r) U ( r' ) ( r' ) G( r, r' ) d r'. (3.68) Here G(r, r ) is the spatial Gree s fuctio, defied as such a elemetary respose of the free-space Schrödiger equatio to a poit ecitatio, i.e. the solutio of the followig equatio 7 k G ( r r' ). (3.69) But we already kow the physically-relevat spherically-symmetric solutio of this equatio see Eq. (7) ad its discussio: f ikr G( r, r' ) e, (3.7) R so that we eed ust to calculate the coefficiet f + for Eq. (67). This ca be doe i several ways, for eample by oticig that at r << k -, the secod term i Eq. (7) is egligible, ad it is reduced to the well-kow Poisso equatio with delta-fuctioal right-had part, which describes, for eample, the electrostatic potetial geerated by a poit electric charge. Either recallig the Coulomb law, or applyig the Gauss theorem, 8 we readily get the asymptote G, at kr, (3.7) 4R which is compatible with Eq. (7) oly if f + = -/4, i.e. if ikr G( r, r' ) e. (3.7) 4R Pluggig this result ito Eq. (68), we get the fial solutio of Eq. (67): m ( r' ) ikr 3 s ( r) U ( ') e d r' r. (3.73) R Note that if fuctio U(r) is smooth, the sigularity i the deomiator is itegrable (i.e. ot dagerous); ideed, the cotributio of a sphere of radius R, with the ceter i poit r =, scales as RR 3 d R R 4 R R dr R R 4 RdR R. (3.74) 7 Please otice both the similarity ad differece betwee this Gree s fuctio ad the propagator discussed i Sec... I both cases, we use the liear superpositio priciple to solve wave equatios, but while Eq. (68) gives the solutio of the ihomogeeous equatio (67), Eq. (.44) does that for a homogeeous Schrödiger equatio i which the wave sources are preseted by iitial coditios rather tha by equatio s right-had part. 8 See, e.g., EM Sec... Chapter 3 Page 7 of 56

125 Actually, Eq. (73) gives us more tha we wated: it evaluates the scattered wave at ay poit, icludig those withi of the scatterig obect, while our goal was to fid the wave far from the scatterer please revisit Fig. 8 if you eed. However, before goig to that limit, we ca use the geeral formula to fid the quatitative criterio of the Bor approimatio s validity. Ideed, let us estimate the magitude of the right had part of this equatio, for a scatterer of liear size ~a, ad the potetial magitude scale U, i two limits: (i) If ka <<, the iside the scatterer (i.e., at distaces r ~ a), both ~ ep{ikr} ad the secod epoet uder the itegral chage slowly, so that a crude estimate of the solutio is ~ m U a s. (3.75) (ii) I the opposite limit ka >>, the itegratio alog oe of the dimesios (that of the wave propagatio) is cut out o distaces of the order of the de Broglie wavelegth k -, so that the itegral is correspodigly smaller: m a ~ s U. (3.76) ka Sice the reductio of Eq. (66) to Eq. (67) requires s << everywhere withi the scatterer, we may ow formulate the coditios of this requiremet as U ma[ ka, ]. (3.77) ma I the first factor of the right-had part, we may readily recogize the scale of the kietic (quatumcofiemet) eergy E a of the particle iside a quatum well of size ~ a, so that the Bor approimatio is valid essetially if the potetial eergy of particle s iteractio with the scatterer is smaller tha E a. Note, however, that estimates (75) ad (76) are ot valid i special situatios whe the effects of scatterig accumulate i some directio. This is frequetly the case for small scatterig agles i eteded obects (whe ka >> but ka < ), ad especially i D (or quasi-d) scatterers orieted alog the icidet particle beam. Now let us proceed to large distaces r >> r ~ a, ad simplify Eq. (73) usig a approimatio similar to the dipole epasio i electrodyamics. 9 I deomiator s R, we ca merely igore r i compariso with r, but the epoet requires more care, because eve if r ~a << r, the product kr ~ ka may still be larger tha. I the first approimatio i r, we ca take (Fig. 9a): a r' R r (a) detector k k (b) q Fig (a) Dipole epasio i the Bor approimatio ad (b) defiitios of vector q ad agles ad. 9 See, e.g., EM Sec. 8.. Chapter 3 Page 8 of 56

126 Scatterig fuctio ad sice the directios of vectors k ad r coicide, i.e. k = k r, R r r' r r r', (3.78) ikr ikr ikr' kr kr k r', ad e e e, (3.79) With this replacemet, ad the icidet wave i form (6), the Bor approimatio yields m a ikr i( k k ) r' 3 s ( r) e U ( ') e d r' r r. (3.8) This relatio may be preseted i a geeral form 3 f ( k, k ) a e ikr s, (3.8) r where f(k, k ) is called the scatterig fuctio. 3 Its physical sese becomes clear from the calculatio of the correspodig probability curret desity s. For that, geerally we eed to use Eq. (.47) with the gradiet operator havig all spherical-coordiate compoets. 3 However, at kr >> the mai cotributio to s, proportioal to k >> /r, is provided by the term ep{ikr} which chages fast i the commo directio of vectors r ad k, so that s r s k s, at kr. (3.8) r so that Eq. (.47) yields f ( k, k ) s ( ) a k. (3.83) m r Sice this vector is parallel to k ad hece to r, the flu i the omiator of Eq. (58), i.e. the probability curret per uit solid agle, is ust r s. Hece, the differetial cross-sectio is simply ad the total cross-sectio is d d s r f ( k, k d ), (3.84) f ( k, k ) Ω, (3.85) so that the scatterig fuctio f(k, k ) gives us everythig we eed (ad i fact more, because the fuctio also cotais iformatio about the phase of the scattered wave). 3 It is easy to prove that this form is a asymptotic form of ay solutio s of the scatterig problem (eve that beyod the Bor approimatio) at sufficietly large distaces r >> a, k -. 3 Note that fuctio f has the dimesio of legth, ad does ot accout for the icidet wave. This is why sometimes a dimesioless fuctio, S = + ikf, is used istead. This fuctio S is called the scatterig matri, because it may be cosidered as a atural geeralizatio of the D matri S, defied by Eq. (.33), to higher dimesioality. 3 See, e.g., MA Eq. (.8). Chapter 3 Page 9 of 56

127 Accordig to Eq. (8), i the Bor approimatio the scatterig fuctio may be preseted as the Bor itegral m iq r 3 f ( k, k ) U ( ) e d r r, (3.86) where for the otatio simplicity I have replaced r with r, ad also itroduced the scatterig vector q k, (3.87) k with legth q = k si(/), where is the scatterig agle betwee vectors k ad k see Fig. 9b. For the differetial cross-sectio, Eq. (86) yields ( ) U r d m iq r 3 e d r, (3.88) d ad the total cross-sectio may be ow readily calculated from the first of Eqs. (59). 33 This is the mai result of this sectio; it may be further simplified for spherically-symmetric scatterers, with U ( r ) U ( r). (3.89) Differetial crosssectio i the Bor approimatio Here, it is coveiet to preset the epoet i the Bor itegral as ep{-iqr cos}, where is the agle betwee vectors k (i.e. the directio r toward the detector) ad q (rather tha the icidet wave vector k!) see Fig. 9b. Now, for fied q, we ca take this vector s directio as the polar ais of a spherical coordiate system, ad reduce Eq. (86) to a D itegral: f ( k, k m ) m r dru ( r) d si d ep{ iqr'cos } si qr m r dru ( r) qr q U ( r)si( qr) rdr. As a simple eample, let us use the Bor approimatio to aalyze scatterig o the followig spherically-symmetric potetial: (3.9) r U r U ep. (3.9) a I this particular case, it is better to avoid the temptatio to eploit the spherical symmetry by usig Eq. (9), ad istead use the geeric Eq. (88), because it falls apart ito a product of three similar Cartesia factors: with mu ( k, k ) I I y I z, (3.9) f 33 Note that accordig to Eq. (88), i the Bor approimatio the scatterig itesity does ot deped o the sig of potetial U, ad also that scatterig i a certai directio is completely determied by a specific Fourier harmoic of fuctio U(r), amely by the harmoic with the wave vector equal to the scatterig vector q. Chapter 3 Page of 56

128 Chapter 3 Page of 56, ep d iq a I (3.93) ad similar itegrals for I y ad I z. From Chapter, we already kow that Gaussia itegrals like I may be readily worked out by complemetig the epoet to the full square, i our curret case givig etc., ep / a q a I, (3.94) so that, fially, ), ( a q e a mu a I I I mu f d d z y k k. (3.95) Now, the total cross-sectio is a itegral of d/d over all directios of vector k. Sice i our case the scatterig itesity does ot deped o the azimuthal agle, the itegratio is reduced to that over the scatterig agle (Fig. 9b):. 4 ) cos ( cos ep 4 si ep si 4 si a k e a mu k π d a k a mu a π a k d a mu a π d d d d d d (3.96) Let us aalyze these formulas. I the low-eergy limit, ka << (ad hece qa << for ay scatterig agle), the scattered wave is virtually isotropic: d/d cost a very typical feature of scatterig by small obects, i ay approimatio. Notice that i this limit, the Bor epressio for,, 8 a mu a (3.97) is oly valid if is much smaller tha the scale a of the physical cross-sectio of the scatterer. I the opposite, high-eergy limit ka >>, the scatterig is domiated by small agles q/k ~ /ka ~ /a: ep a k a mu a d d. (3.98) This is, agai, very typical for diffractio. Notice, however, that due to the smooth character of the Gaussia potetial (9), the diffractio patter ehibits o oscillatios; such oscillatios of d/d as fuctio of agle aturally appear for potetials with sharp borders see, e.g., Problems ad 3. The Bor approimatio, while beig very simple ad used more ofte tha ay other scatterig theory, is ot without substatial shortcomigs, as is clear from the followig eample. It is ot too difficult to prove the followig geeral optical theorem, valid for a arbitrary scatterer:

129 k Im f k, k. (3.99) 4 However, Eq. (86) shows that i the Bor approimatio, fuctio f is purely real at q = (i.e. k = k ), ad hece caot satisfy the optical theorem. Eve more evidetly, it caot describe such a simple effect as a dark shadow ( ) cast by a opaque obect (say, with U >> E). There are several ways to improve the Bor approimatio, while still holdig the geeral idea of approimate treatmet of U. (i) Istead of the mai assumptio s U, we ca use a complete perturbatio series: s... (3.) Optical theorem with U, ad fid successive approimatios oe by oe. I the st approimatio we of course retur to the Bor formula, but already the d approimatio yields Im f k k, (3.) 4, k where is the full cross-sectio calculated i the st approimatio, so that the optical theorem (99) is almost satisfied. 34 (ii) As was metioed above, the Bor approimatio does ot work very well for small-agle scatterig by eteded obects. This deficiecy may be corrected by the so-called eikoal approimatio (from Greek word, meaig ico ) that replaces the plae wave epoet ep{ik } represetatio of the icidet wave by a WKB-like epoet, though still i the first ovaishig approimatio i U : / ik m E U ( ' ) m e epi k( ' ) d' epi d' e ik U ( ' ) d'. (3.) k This approimatio s results satisfy the optical theorem (99) already i the st approimatio i U. Eikoal approimatio 3.4. Eergy bads i higher dimesios I Sec..5, we have discussed the D bad theory for potetial profiles U() that obey the periodicity coditio (.9). For what follows, let us otice that that coditio may be rewritte as U ( X ) U ( ), (3.3) 34 The costructio of such series may be facilitated by the followig observatio. If we retai s i the righthad part of Eq. (66), we may write a relatio formally similar to Eq. (68) for the full wavefuctio = + s : ( r) m U ' ' G ' d 3 ( r) ( r ) ( r ) ( r, r ) r'. This is oe of forms of the Lipma-Schwiger equatio that is eactly equivalet to the differetial Schrödiger equatio (66) but is more coveiet for some applicatios, i particular for the calculatio of higher approimatios. Ufortuately, I will have ot time to discuss this approach i detail ad have to refer the reader, for eample, to either Chapter 9 of the tetbook by L. Schiff, Quatum Mechaics, 3 rd ed., McGraw-Hill, 968, or (for eve more details) to moograph by J. Taylor, Scatterig Theory, Dover, 6. Chapter 3 Page of 56

130 where X = a, with beig a arbitrary iteger. Oe ca say that the set of poits X forms a periodic D lattice i the direct (-) space. We have also see that each Bloch state (i.e., each eigestate of the Schrödiger equatio for such periodic potetial) is characterized by the quasi-mometum q ad its eergy does ot chage if q is chaged by a multiple of /a. Hece if we form, i the reciprocal (k-) space, a D lattice of poits Q = lb, with b = /a ad iteger l, ay pair of poits from these two mutually reciprocal lattices satisfies the followig rule: ep a il iqx ep il a e. (3.4) I this form, the results of Sec..5 may be readily eteded to d-dimesioal periodic potetials whose traslatioal symmetry obeys the followig geeralizatio of Eq. (3): Bravais lattice ad its potetial U ( r R) U ( r), (3.5) where poits R, which may be umbered by d itegers, form the so-called Bravais lattice 35 of poits d R a, (3.6) with d primitive vectors a. The simplest eample of a 3D Bravais lattice are give by the simple cubic lattice (Fig. a), which may be described by the system of mutually perpedicular primitive vectors a of equal legth. However, ot i ay lattice these vectors are perpedicular; for eample Figs. b ad c show possible sets of the primitive vectors describig the face-cetered cubic lattice (fcc) ad bodycetered cubic lattice (bcc). I 3D, the sciece of crystallography, based o the group theory, distiguishes, by their symmetry properties, 4 Bravais lattices grouped ito 7 differet lattice systems. 36 (a) (b) (c) a 3 a 3 a a 3 a a a a a Fig. 3.. The simplest (ad most commo) 3D Bravais lattices: (a) simple cubic, (b) face-cetered cubic (fcc), ad (c) body-cetered cubic (bcc), ad possible choices of their primitive vector sets (blue arrows). Note, however, ot all highly symmetric sets of poits form Bravais lattices. As probably the most strikig eample, odes of the very simple D hoeycomb lattice (Fig. a) caot be described by 35 Named after A. Bravais, the crystallographer who itroduced this otio i The strogest motivatio for the bad theory is provided by properties of solid crystals. Thus it is ot surprisig that perhaps the most clear, well illustrated itroductio to the Bravais lattices may be foud i Chapters 4 ad 7 of the famous tetbook by N. Ashcroft ad N. Mermi, Solid State Physics, Sauders College, 976. Chapter 3 Page 3 of 56

131 a Bravais lattice - while the D heagoal lattice, show i Fig. b, ca. The most promiet 3D case of such a lattice is the diamod structure (Fig. c), which describes, i particular, atoms of world s most importat crystal silico. 37 I cases like these, the bad theory is much facilitated by the fact that the Bravais lattices usig some poit assemblies (called primitive uit cells) may describe these poit systems. For eample, Fig. a shows the possible choice of primitive vectors for the hoeycomb structure, 38 with the primitive uit cell formed by ay two adacet poits of the origial lattice (say, withi the dashed ellipses i Fig. a). Similarly, the diamod lattice may be described as the fcc Bravais lattice with two-poit primitive uit cell. 39 Now we are ready for the followig geeralizatio of the D Bloch theorem, give by Eqs. (.93) ad (.), to higher dimesios. Ay eigefuctio of the Schrödiger equatio describig particle s motio i the periodic potetial (5) may be preseted either as or as iq R ( r R) ( r)e, (3.7) iq r ( r) u( r) e, with u( r R) u( r), (3.8) Two forms of the 3D Bloch theorem where the quasi-mometum q is agai a costat of motio, but ow is a vector. (a) (b) (c) a a a a a 3 a a Fig. 3.. Some importat periodic structures that require two-poit primitive cells for their Bravais lattice presetatio: (a) D hoeycomb lattice ad their primitive vectors ad (c) 3D diamod lattice. For a cotrast, pael (b) shows the D heagoal structure which forms a Bravais lattice with a sigle-poit primitive cell. The key otio of the bad theory is the reciprocal lattice i the wavevector space, formed as d Q l b, (3.9) Reciprocal lattice i q-space 37 It may be best uderstood as the sum of two fcc lattices of side a, mutually shifted by vector {,, }a/4, so that the distaces betwee each poit of the combied lattice ad its 4 earest eighbors (see the thick gray lies i Fig. c) are all equal. 38 This structure is presetly very popular due to the recet discovery of graphee isolated moolayer sheets of carbo atoms arraged i a hoeycomb lattice with the iteratomic distace of.4 m. 39 A harder case is preseted by quasicrystals (whose idea may be traced dow to medieval Islamic tiligs, but was discovered i atural crystals, by D. Shechtma et al., oly i 984), which obey high (say, 5-fold) rotatioal symmetry, but caot be described by a Bravais lattice with ay fiite primitive uit cell. For a popular review of quasicrystals see, for eample, P. Stephes ad A. Goldma, Sci. Amer. 64, #4, 4 (99). Chapter 3 Page 4 of 56

132 Primitive vectors of the reciprocal lattice with iteger l, ad vectors b selected i such way that the followig geeralizatio of Eq. (4) is valid for ay pair of poits of the direct ad reciprocal lattices: iqr e. (3.) The importace of lattice Q is immediately clear from the first formulatio of the Bloch theorem, give by Eq. (7): if we add to q ay vector Q of the reciprocal lattice, the wavefuctio does ot chage. This meas that all iformatio about the system is cotaied i ust oe elemetary cell of the reciprocal space q. Its most frequet choice, called the st Brilloui zoe, is the set of all poits q that are closer to the origi tha to ay other poit of lattice Q. It is easy to see that primitive vectors b of the reciprocal 3D lattice 4 may be costructed from those of the iitial, direct lattice as b a a a a a a 3 3, b, b 3. (3.) a a a3 a a a3 a a a3 Ideed, from the operad rotatio rule of the vector algebra 4 it is evidet that a b =. Hece, the epoet i the left-had part of Eq. () is reduced to e i Q R ep i l l l 3 3. (3.) Sice all l ad are itegers, the epressio i the paretheses is also a iteger, so the epoet ideed equals, thus satisfyig the defiitio of the reciprocal lattice give by Eq. (). As the simplest eample, let us retur to the simple cubic lattice of period a (Fig. a), orieted i space so that a a a a, a a, (3.3), y 3 z Accordig to Eq. (), its reciprocal lattice is (of course) also cubic: Q ( l l y y l z z ), (3.4) a so that the st Brilloui zoe is a cube with side b = /a. Almost similarly simple calculatios show that the reciprocal lattice of fcc is bcc, ad vice versa. Figure shows the resultig st Brilloui zoe of the fcc lattice. The otio of the reciprocal lattice 4 makes the multi-dimesioal bad theory ot much more comple tha that i D, especially for umerical calculatios, at least for the sigle-poit Bravais lattices. Ideed, repeatig all the steps that have led to Eq. (.8), but ow with a d-dimesioal Fourier epasio of fuctios U(r) ad u l (r), we readily get its geeralizatio: l' l U u, (3.5) l' lul' ( E El ) l 4 For the D case ( =, ), oe may use, for eample, the first two formulas of Eq. () with a 3 = a a. 4 See, e.g., MA Eq. (7.6). 4 This otio is also the mai startig poit of X-ray diffractio studies of crystals, because it allows rewritig the well-kow Bragg coditio for diffractio peaks i a etremely simple form of the mometum coservatio law: k = k + Q, where k ad k are the wave vectors of the, respectively, icidet ad diffracted photo. Chapter 3 Page 5 of 56

133 where l is ow a d-dimesioal vector of iteger idices l. The summatio i Eq. (5) should be carried over all (essetial) compoets of this vector (i.e. over all relevat odes of the reciprocal lattice), so writig a correspodig computer code requires a bit more care tha i D; however, this is ust a homogeeous system of liear equatios, ad umerous routies of fidig its eigevalues E are readily available from both public sources ad commercial software packages. 43 q z q y q Fig. 3.. st Brilloui zoe of the fcc lattice, ad the traditioal otatio of its mai directios. Adapted from What is ideed more comple tha i D is the presetatio (ad hece the comprehesio :-), of the calculatio results ad eperimetal data. Typically, the presetatio is limited to plottig the Bloch state eigeeergy as a fuctio of compoets of vector q alog certai special directios the reciprocal space of quasi-mometum (see, e.g., the lies show i Fig. ), typically plotted o sigle pael. Figure shows perhaps the most famous (ad certaily the most practically importat) of such plots, the bad structure of silico. The dashed horizotal lies mark the idirect gap of width. ev betwee the valece ad coductio eergy bads, which is the playgroud of virtually all silicobased electroics. Fig Bad structure of silico, alog the special directios show i Fig.. (Adapted from 43 See, e.g., MA Sec. 6 (iv). Chapter 3 Page 6 of 56

134 I order to uderstad the reaso of this bad structure presetatio compleity, let us see how we would start to develop the weak-potetial approimatio for the simplest case of a D square lattice (which is a subset of the cubic lattice, with 3 = ). Its st Brilloui zoe is of course also a square, of area (/a). Let us draw the lies of costat eergy of a free particle (U = ) i this zoe. Repeatig the argumets of Sec..7 (see especially Fig..8 ad its discussio), we should coclude that Eq. (.6) should ow be geeralized as follows, k l l y E E l q q y, (3.6) m m a a with all possible itegers l ad l y. Cosiderig the result oly withi the st Brilloui zoe, we see that as eergy E grows, the lies of equal eergy evolve as show i Fig. 4. Just like i D, the weakpotetial effects are oly importat at the Brilloui zoe boudaries, ad may be crudely cosidered as the appearace of arrow eergy gaps, but oe ca see that the bad structure i q-space is comple eough eve without these effects. q y (a) (b) (c) a / a q Fig Lies of costat eergy E of a free particle, withi the st Brilloui zoe of a square Bravais lattice, for: (a) E/E.95, (b) E/E.5; ad (c) E/E.5, where E /ma. The tight-bidig approimatio is usually easier to follow. For eample, for the same square D lattice, we may repeat the argumets that have led us to Eq. (.3), to write 44 da, i a, a, a, a,, (3.7) dt where idices correspod to the deviatios of itegers ad y from a arbitrarily selected miimum of the potetial eergy - ad hece wavefuctio s hump quasi-localized at this miimum. Now, lookig for the statioary solutio of these equatios, that correspods to the Bloch theorem (7), istead of Eq. (.6) we get iqa iqa iqya iqya E E E e e e e E cos qa cos q ya. (3.8) Figure 5 shows this result, withi the st Brilloui zoe, i two forms: as the color-coded lies of equal eergy ad as a 3D plot (also ehaced by color). 44 Actually, usig the same values of i both directios implies some sort of symmetry of the quasi-localized states. For eample, s-states of aially-symmetric potetials (see the et sectio) always have such a symmetry. Chapter 3 Page 7 of 56

135 q y 4 a q q y / a 4 q Fig Allowed bad eergy = E E for a square D lattice, i the tight-bidig approimatio. It is evidet that the plots of this fuctio alog differet lies o the q-plae, for eample alog oe of aes (say, q ) ad alog a diagoal of the st Brilloui zoe (say, q = q y ) give differet curves, qualitatively similar to those of silico (Fig. 3). The latter structure is complicated by the fact that the primitive cell of their Bravais lattices cotais more tha atoms see Fig. c ad its discussio. I this case, eve the tight-bidig picture becomes more comple. Ideed, eve if the atoms i the differet positios of the primitive uit cell are similar (as they are, for eample, i both graphee ad silico), ad hece the potetial well shape ear those poits ad the correspodig local wavefuctios u(r) are similar as well, the Bloch theorem (which oly pertais to Bravais lattices!) does ot forbid them to have differet comple amplitudes a(t) whose time evolutio should be described by a specific differetial equatio. For eample, i order to describe the hoeycomb lattice show i Fig. a, we have to prescribe differet amplitudes to the top ad bottom poits of its primitive cell - say, ad, correspodigly. Sice each of these poits is surrouded (ad hece weakly iteracts) with 3 eighbors of the opposite type, istead of Eq. (7) we have to write two equatios 3 3 d d i, i ', (3.9) dt dt where each summatio is over 3 et-eighbor poits. (I am usig differet summatio idices ust to emphasize that these directios are differet for the top ad bottom poits of the primitive cell see Fig. a.) Now usig the Bloch theorem (7) i the form similar to Eq. (.5), we get two coupled systems of liear algebraic equatios: 3 ' E E e, E E e, iqr 3 ' iqr (3.) where r ad r are the et-eighbor positios, as see from the top ad bottom poits, respectively. Writig the coditio of cosistecy of this system, we get two equal ad opposite values for eergy correctio for each value of q: E E 3, ' iq r /, where e. (3.) r ' ' ' ' Chapter 3 Page 8 of 56

136 Accordig to Eq. (), these two eergy bads correspod to the phase shifts (o the top of the regular Bloch shift qr) of either or betwee the adacet quasi-localized wavefuctios u(r ). The most iterestig corollary of such eergy symmetry, augmeted by the hoeycomb lattice symmetry, is that for certai values q D of vector q (that tur out to be i each of 6 corers of the hoeycomb-shaped st Brilloui zoe), the double sum vaishes, i.e. the two bad surfaces E (q) touch each other. As a result, i viciities of these Dirac poits 45 the dispersio relatio is liear: E E q ~, where q ~ v q q, (3.) qq D with v beig a costat with the dimesio of velocity (for graphee, close to 6 m/s). Such a liear dispersio relatio esures several iterestig trasport properties of graphee. For their discussio, I have to refer the reader to special literature. 46 D 3.5. Aially-symmetric systems I caot coclude this chapter (ad hece our review of wave mechaics) without addressig the issue of eigestates ad eigevalues at full cofiemet i multi-dimesioal potetials U(r). For a arbitrary potetial, the statioary Schrödiger equatio does ot have a aalytical solutio, but a substatial symmetry of fuctio U(r) may make such solutio possible. This pertais, i particular, to the aial symmetry i D problems ad the spherical symmetry i 3D problems, which are typical for several importat situatios (or their reasoable models), especially i atomic ad uclear physics. I rare cases such symmetry may be eploited by the separatio of variables i Cartesia coordiates. The most famous eample is the d-dimesioal harmoic oscillator, i.e. a particle movig iside the potetial m U. (3.3) d r Separatig the variables eactly as we did for the rectagular quatum well (see Sec..5), for each degree of freedom we get the Schrödiger equatio (.68) of a D oscillator, whose eigefuctios are give by Eq. (.78), ad the eergy spectrum is described by Eq. (.4). As a result, the total eergy spectrum may be ideed by a vector = {,,, d } of d idepedet itegers (quatum umbers): d d E, (3.4) 45 This term is based o a (pretty loose) aalogy with the Dirac theory of relativistic quatum mechaics, to be discussed i Chapter 9 below. Namely, i the viciity of a Dirac poit (), Schrödiger equatios (9), ad hece the dispersio relatio (), may be obtaied from the effective Hamiltoia H σ q ~ v. (Sice vector q ~ is two-dimesioal, this Hamiltoia employs oly two of three Pauli matrices.) This epressio remids the first term of Dirac s Hamiltoia (9.97), which is defied, however, i a differet Hilbert space. 46 See, e.g., a recet review by A. Castro Neto et al., Rev. Mod. Phys. 8, 9 (9). Note that trasport properties of graphee are determied by couplig of p z electro states of carbo atoms, whose wavefuctios are proportioal to ep{i} rather tha are aially-symmetric as implied by Eqs. (). However, due to the lattice symmetry this fact does ot affect the dispersio relatio E(q). Chapter 3 Page 9 of 56

137 all of them ragig from to. Note that every eergy level of this system, with the oly eceptio of the groud state, d d g ( r ) ep d d r, (3.5) / 4 / is degeerate: several differet wavefuctios, each with its ow differet set of quatum umbers, but the same value of their sum, have the same eergy. However, the harmoic oscillator problem is a eceptio: for other cetral- ad sphericallysymmetric problems the solutio is made easier by usig more appropriate coordiates. Let us start with the simplest aially-symmetric problem: the so-called plaar rigid rotator (or rotor ), i.e. a particle costraied (cofied) to move alog a plae, roud circle of radius R (Fig. 5). 47 R m l R Fig Plaar rigid rotator. The plaar rotator has ust oe degree of freedom, say the displacemet arc l = R. So, its classical eergy (ad Hamiltoia fuctio) is H = p l /m, p l mv = m(dl/dt). This fuctio is similar to that of a free D particle (with the replacemet l), ad hece rotator s quatum properties may be described by a similar Hamiltoia operator: p H, with p i, (3.6) m l ad its eigefuctios have a similar structure: ikl Ce. (3.7) The oly ew feature is that i the rotator, all observables should be R-periodic fuctios of l, ad hece, as we have already discussed i the cotet of the magetic flu quatizatio (see Fig. 4 ad its discussio), as the particle makes oe tur about the ceter, its wavefuctio s phase kl may oly chage by, with a arbitrary iteger (from - to +),: i ( l R) ( l) e. (3.8) With eigefuctios (7), this immediately gives coditio gives k R =. Thus, waveumber k ca take oly quatized values k = /R, so that the eigefuctios should be ideed by : l C epi, (3.9) R Plaar rotator: eigefuctios 47 This is a reasoable model for the cofiemet of light atoms, otably hydroge, i some orgaic compouds. Chapter 3 Page 3 of 56

138 Plaar rotator: eigeeergies Plaar rotator i magetic field ad the eergy spectrum is discrete: p k E. (3.3) m m mr So, while the free traslatio motio of a quatum particle is cotiuous, i the sese that its mometum has a cotiuous spectrum, its rotatio is quatized the most importat fact, which has so may implicatios (icludig the eistece of atoms, molecules, ad hece us humas, ad hece sciece icludig this course :-). This simple model allows a eact aalysis of eteral magetic field effects o a quatumcofied motio of a electrically charged particle. Ideed, if this field is uiform ad directed perpedicular to rotator s plae, it does ot violate the aial symmetry of the system. Accordig to Eq. (6), i this case we have to geeralize Eq. (6) as H i qa. (3.3) m l Here, i cotrast to the gauge choice (44), which was so istrumetal i the Ladau level problem, it is ow clearly beeficial to take the vector-potetial i a maifestly aially-symmetric form A = A(), where {, y} is the D radius-vector. Usig the well-kow epressio for curl i cylidrical coordiates, 48 we ca readily check that the requiremet A = B z, with B = cost, is satisfied by the followig fuctio: B A. (3.3) For the plaar rotator, ρ = R = cost, so that the statioary Schrödiger equatio becomes BR i q E. (3.33) m l A little bit surprisigly, this equatio is still satisfied with the sie-wave eigefuctios (7). Moreover, sice the periodicity coditio (8) is also uaffected by the applied magetic field, we retur to field-idepedet eigefuctios (9). However, the field does affect the system s eergy: BR Φ E q Φ, (3.34) m R mr ' where R B is the magetic flu through the area limited by the particle s traectory, ad /q is the ormal magetic flu quatum we have already met i the AB effect cotet see Eq. (34) ad its discussio. The field also chages the electric curret of the particle i -th state: iqr I q * B c.c. q C. (3.35) im l mr ' Normalizig wavefuctio (9) to have W =, we get C = /R, so that Eq. (35) becomes 48 See, e.g., MA Eq. (.5). Chapter 3 Page 3 of 56

139 Φ q I I, with I. Φ ' (3.36) mr Fuctios E () ad I () are show i Fig. 7. Note that sice /q, for ay sig of the particle s charge, di /d <. It is easy to check that this meas that the curret is diamagetic, 49 i.e. correspods to the Lez rule of the Faraday s electromagetic iductio: the field-iduced curret flows i such directio that its ow magetic field tries to compesate the eteral magetic flu applied to the loop. E / ' I / ' Fig Effect of magetic field o a charged plaar rotator. Dashed lies show possible ielastic trasitios betwee metastable ad groud states, due to weak iteractio with eviromet, as the magetic field is beig icreased. This result may be iterpreted as a differet implemetatio of the AB effect. 5 I cotrast to the two-slit iterferece eperimet that was discussed i Sec., i the situatio show i Fig. 7 the particle is ot absorbed by the detector, but travels aroud the rig cotiuously. As a result, its wavefuctio is rigid: due to the boudary coditio (8), the topological quatum umber is discrete, ad magetic field caot chage the wavefuctio gradually. I this sese, the system is similar to a supercoductig loop - see Fig. 4 ad its discussio. The differece betwee these systems is two-fold: (i) For a sigle charged particle, i a macroscopic systems with practicable values of q, R, ad m, the curret scale I is very small. For eample, for m = m e, q = -e, ad R = m, Eq. (36) yields I 3 pa. 5 The cotributio LI ~ RI ~ -4 Wb of the curret so small ito the et magetic flu is 49 This effect, whose qualitative features remai the same for all D or 3D localized states (see Chapter 6 below), is frequetly referred to as the orbital diamagetism. I magetic materials cosistig of particles with ucompesated spis, this effect competes with aother effect, spi paramagetism - see, e.g., EM Sec It is straightforward to check that Eqs. (33) ad hece (35) remai valid eve if the magetic field lies do ot touch the particle s traectory, ad the field is localized well iside rotator s rig. 5 Such persistet, macroscopic diamagetic currets i o-supercoductig systems may be eperimetally observed, for eample, by measurig the weak magetic field geerated by electros i a system of a large umber (~ 7 ) of similar coductig rigs see, e.g., L. Lévy et al., Phys. Rev. Lett. 64, 74 (99). Due to the Chapter 3 Page 3 of 56

140 egligible i compariso with ~ -5 Wb, so that the quatizatio of does ot lead to the magetic flu quatizatio. (ii) As soo as the magetic field raises the eigestate eergy E above that of aother eigestate E, the former state becomes metastable, ad weak iteractios of the system with its eviromet (which are eglected i our simple model) may iduce a quatum trasitio of the system to the lowereergy state, thus reducig the diamagetic curret s magitude see the dashed lies i Fig. 7. The flu quatizatio i supercoductors is much more robust to such perturbatios. 5 Now let us retur, for oe more time, to Eq. (9), ad see what do they give for oe more observable, particle s agular mometum L r p, (3.37) I our curret problem, vector L has ust oe compoet perpedicular to the rotator plae, L z Rp. (3.38) I classical mechaics, L z of the rotator should be coserved (due to the absece of eteral torque), but ca take arbitrary values. I quatum mechaics the situatio chages: with p = k, our result k = /R may be rewritte as L ( L ) Rk. (3.39) z z Thus, the agular mometum is quatized: it may be oly a multiple of the Plack costat - cofirmig Bohr s guess see Eq. (.). As we will see i Chapter 5, this result is very geeral (though may be modified by spi effects) ad that wavefuctios (9) may be iterpreted as eigefuctios of the agular mometum operator. I order to implemet the plaar rotator i our 3D world, we eeded to provide rigid cofiemet of the particle both i the motio plae, ad alog radius. Let us proceed to the more geeral problem whe oly the former cofiemet is strict, i.e. to a D particle movig i a arbitrary cetrally-symmetric potetial U ( ρ ) U ( ). (3.4) Usig the well-kow epressio for the D Laplace operator i polar coordiates, 53 we may preset the D statioary Schrödiger equatio i the form U ( ) E. (3.4) m Separatig the radial ad agular variables as 54 dephasig effects of electro scatterig by phoos ad other electros, the effect s observatio requires submicro samples ad millikelvi temperatures. 5 Iterruptig a supercoductig rig with a weak lik (Josephso uctio), i.e. formig a SQUID, we may get the switchig behavior similar to that show with dashed arrows i Fig. 7 see, e.g., EM Sec See, e.g., MA Eq. (.3) with /z =. 54 At this stage, I do ot wat to mark the particular solutio (eigefuctio) ad correspodig eigeeergy E by ay ide, because we already may suspect that i a D problem the role of this ide will be played by two itegers two quatum umbers. Chapter 3 Page 33 of 56

141 Chapter 3 Page 34 of 56 ) ( ) ( F R, (3.4) we get, after the divisio by ad multiplicatio by ρ, the followig equatio: E U d F d F d d d d m ) ( R R. (3.43) It is clear that the fractio (d F/d )/F should be a costat (because all other terms of the equatio may be oly fuctios of ρ aloe), so that we get for fuctio F() a ordiary differetial equatio, F d F d, (3.44) where is the variable separatio costat. The fudametal solutio of Eq. (44) is evidetly F ep{i}. Now requirig, as we did for the plaar rotator, the periodicity of ay observable, i.e. i e F F ) ( ) (, (3.45) so that costat has to be iteger (say, ), ad we ca write: 55, i e C F (3.46) Pluggig the resultig relatio (d F/d )/F = - ito Eq. (43), we may rewrite is as E U d d d d m ) ( R R. (3.47) The physical iterpretatio of this equatio is that the full eergy is a sum, E E, E (3.48) of the radial-motio part ) ( U d d d d m E R. (3.49) ad the agular-motio part m E. (3.5) Now let us otice that a similar separatio eists i classical mechaics, 56 because the total eergy of a particle movig i a cetral field may be preseted, withi the plae of motio, as, ) ( ) ( E E U m U v m E (3.5) where 55 Notig that for the plaar rotator (Fig. 6) l/r =, we ca preset Eq. (9) i a similar form. This is atural, because the rotator is ust a particular case of our curret problem - with a rigid cofiemet alog ais. 56 See, e.g., CM Sec. 3.5.

142 p Lz E U ( ), E. (3.5) m m The compariso of the latter relatio with Eqs. (39) ad (5) gives us grouds to suspect that the quatizatio rule L z = may be valid for this problem as well, ad may be i other cases as well. I Sec. 5.6, we will see that this is ideed the case. Returig to Eq. (47), o the basis of our eperiece with D wave mechaics we may epect that this ordiary, liear, secod-order differetial equatio should have (for a motio cofied to a certai fial regio of its argumet ρ), for ay fied, a discrete eergy spectrum described by some other iteger quatum umber (say, l). This meas that eigefuctios (4), ad correspodig eigeeergies (48) should be ideed by two quatum umbers. Note, however, that sice the radial fuctio obeys equatio (47), which already depeds o, fuctio R(ρ) should carry both idices, so the variable separatio is ot so clea as it was for the rectagular quatum well. Normalizig the agular fuctio to the full circle, =, we may rewrite Eq. (4) as i, l R,l ( ) F ( ) R,l ( ) e. (3.53) / A good (ad importat) eample of a solvable problem of this type is a free D particle whose motio is rigidly cofied to a disk of radius R:, for R, U ( ) (3.54), for R. I this case, the solutios R,l () of Eq. (47) are proportioal to the first-order Bessel fuctios J (k l ρ), 57 ad the spectrum of possible values of parameter k l should foud the boudary coditio R,l (R) =. Let me leave the detailed solutio ad aalysis of this problem for reader s eercise Spherically-symmetric systems: Brute force approach Now let us address the (mathematically more ivolved) case of 3D motio, with sphericallysymmetric potetial U ( r ) U ( r). (3.55) Let me start, agai, with a rigid rotator - ow a spherical rotator, i.e. a particle cofied to move o the surface of a sphere of radius R. It has degrees of freedom, because ay positio o the spherical surface is completely described by two coordiates say, the polar agle ad the azimuthal agle. I this case, the kietic eergy we eed to cosider is limited to its agular part, so that i the Laplace operator i spherical coordiates 58 we may keep oly those parts, with fied r = R. The the statioary Schrödiger equatio becomes 57 A short summary of properties of these fuctio, plus a few plots ad a useful table of values, may be foud i EM Sec..4. For more o of Bessel fuctios, see the literature recommeded i MA Sec. 6(ii). 58 See, e.g., MA Eq. (.9). Chapter 3 Page 35 of 56

143 si E. (3.56) mr si si (Agai, I abstai from attachig ay idices to ad E for the time beig.) With the usual variable separatio assumptio, ( ) F( ), (3.57) Eq. (56), with all terms multiplied by si /F, yields si d d d F si E si. (3.58) mr d d F d Just as i Eq. (43), fractio (d F/d )/F may be a fuctio of oly, ad hece has to be costat, givig for it a equatio similar to Eq. (44). So, the azimuthal fuctios are ust the sie waves (46) agai, ad we ca use the same periodicity coditio (45) to write them i the ormalized form 59 im F m ( ) e. (3.59) / With that, fractio (d F/d )/F equals (-m ), ad Eq. (58), after multiplicatio by /si, is reduced to the followig ordiary, liear differetial equatio for fuctio (): d d m si, with E /, si si d d mr It is coveiet to recast it ito a equatio for a ew variable P() (), with cos : (3.6) d dp ( ) m l( l ) P, (3.6) d d where a ew otatio for the ormalized eergy is itroduced: l(l+). The motivatio for such otatio is that, accordig to a mathematical aalysis, 6 Eq. (6) with iteger m, has solutios oly if parameter l is iteger: l =,,,, ad oly if that iteger is ot smaller tha m, i.e. if l m l. (3.6) This immediately gives the followig eergy spectrum of the spherical rotator: l( l ) E l, (3.63) mr Eergy spectrum of spherical rotator 59 Here, rather regrettably, I had to replace the otatio of the iteger from to m, i order to comply with the geerally accepted covetio for this so-called magetic quatum umber. Let me hope that the differece betwee this iteger ad particle s mass is absolutely clear from the cotet. 6 It was carried out by A.-M. Legedre (75-833). Just as a historic ote: besides may origial mathematical results, Dr. Legedre has authored the famous tetbook Élémets de Géométrie which domiated teachig geometry through the 9th cetury. Chapter 3 Page 36 of 56

144 Legedre equatio Rodrigues formula for Legedre polyomials so that the oly effect of the magetic quatum umber m here is imposig the restrictio (6) o the orbital quatum umber l. This meas, i particular, that each of eergy level (63) correspods to (l + ) differet values of m, i.e. is (l + ) degeerate. To uderstad the physics of this degeeracy, we eed to eplore the correspodig eigefuctios of Eq. (6). They are aturally umbered by two itegers, m ad l, ad are called the associated Legedre fuctios P m l. For the particular, simplest case m =, these fuctios are ust (Legedre) polyomials P l () P l (), which may be either defied as the solutios of the Legedre equatio followig from Eq. (6) at m = : d d ( ) P l( l ) P, (3.64) d d or calculated eplicitly from the followig Rodrigues formula: 6 l d l Pl ( ) ( ), l,,,.... (3.65) l l l! d Usig this formula, it easy to spell out a few lowest Legedre polyomials: 3 3, P ( ) 5 3,... P ( ), P ( ), P ( ) 3, (3.66) though such epressios become more ad more bulky as l is icreased. As Fig. 8 shows, as argumet is decreased, all these fuctios start i oe poit, P l (+) = +, ad ed up either i the same poit or i the opposite poit: P l (-) = (-) l. O the way betwee these two ed poits, the l th polyomial crosses the horizotal ais eactly l times, i.e. has l roots. 6 It may be show that o the segmet [-,+], the Lagrage polyomials form a full orthogoal set of fuctios, with the followig ormalizatio rule: P l l ( ) Pl ' ( ) d ll '. (3.67).5 ( ) P l.5 l cos 4 Fig A few lowest Legedre polyomials. 6 Derived idepedetly by B. O. Rodrigues i 86, J. Ivory i 84, ad C. Jacobi i I this behavior, we readily recogize the stadig wave patter typical for all D eigeproblems cf. Fig..7. The quatitative deviatio from the siusoidal waveform is due to the differet metric of the sphere. Chapter 3 Page 37 of 56

145 For m >, the associated Legedre fuctios may be epressed via the Legedre polyomials (65) usig the followig formula, which remids Eq. (65): m m m m / d Pl ( ) ( ) ( ) Pl ( ), (3.68) m d while if the ide m is egative, the followig simple relatio may be used: m m ( l m)! m Pl ( ) ( ) Pl ( ). (3.69) ( l m)! O the segmet = [-, +], each set of the associated Legedre fuctios with fied ide m forms a full orthogoal set, with the ormalizatio relatio, Associated Legedre fuctios P ( l m)!, (3.7) l ( l m)! m m l ( ) Pl ' ( ) d ll' which is evidetly a geeralizatio of Eq. (67) for arbitrary m. Sice the differece betwee agles ad is to some etet artificial (caused by the arbitrary directio of the polar ais), physicists prefer to use ot the fuctios () P m l (cos) ad F m () ep{im} separately, but their products (57), which are called spherical harmoics: Y l ( l m)! / m im (, ) Pl (cos e 4 ( l m)!. (3.7) m l ) The specific coefficiet i Eq. (7) is chose i a way to simplify the followig two relatios: the equatio for egative m, ad the ormalizatio relatio m m m Y (, ) ( ) Y (, ), (3.7) l m' Y l' (, ) d ll' ' l m Yl (, ) mm, (3.73) with itegratio over the whole solid agle 4. The last relatio shows that the spherical harmoics form a orthoormal set of fuctios. This set is also full, so that ay fuctio defied o a sphere may be uiquely preseted as a liear combiatio of Y m l. Despite a somewhat itimidatig formulas give above, they yield rather simple epressios for the lowest spherical harmoics: / / l : Y, (3.74) 4 / i Y 3/ 8 si e, / l : Y 3/ 4 cos, (3.75) / i Y 3/ 8 si e, Spherical harmoics Chapter 3 Page 38 of 56

146 / i Y 5/ 3 si e, / i Y 5/8 si cos e, / l : Y 3/6 (3cos ), (3.76) / i Y 5/8 si cos e, / i Y 5/ 3 si e. It is importat to uderstad the symmetry of these fuctios. Sice spherical fuctios with m are comple, the most popular way of their graphical represetatio is first to form their real combiatios correspodig to two opposite values of m, 63 ` m m m cos m, for m, Ylm Y l sg( m)( ) Yl (3.77) si m, for m, (for m =, Y l Y l ), ad the plot the magitude of these combiatios i spherical coordiates as the distace from the origi, while usig two colors to show their sig see Fig. 9. m = Fig Several lowest real spherical harmoics Y lm. (Adapted from Web site l = (p states): m = - m = m = + l = (d states): m = - m = - m = m = + m = + 63 Such real fuctios Y lm, which also form the full set of orthoormal eigefuctios ad are frequetly called the real spherical harmoics, are more coveiet tha the comple fuctios Y l m for several applicatios, especially whe the variables of iterest are real by defiitio. Chapter 3 Page 39 of 56

147 Let us startig from the simplest case l =. Accordig to Eq. (6), there could be oly oe such s state, 64 with m =. The spherical harmoic correspodig to that state is ust a costat, so that the wavefuctio is uiformly distributed over the sphere. Sice the fuctios does ot have gradiet i ay directio, the kietic eergy (63) of the particle equals is zero. For l =, there could be 3 differet p states, with m = -,, ad +. As the secod row i Fig. 9 shows, these states are essetially idetical i structure, ad are ust differetly orieted i space, thus eplaiig the 3-fold degeeracy of the kietic eergy see Eq. (63). This is ot quite true for 5 differet d states (l = ), show i the bottom row of Fig. 9, as well as states with higher l: despite their equal eergies, they differ ot oly by their special orietatio. The states with m = have gradiet oly i the directio, while the states with the ultimate values of m (m = l) chage oly gradually (as si l ) i the polar directio, while oscillatig i the azimuthal directio. The states with itermediate values of m provide a crossover betwee these two etremes, oscillatig i both directios, stroger ad stroger i the directio of as m is icreased. Still, the magetic quatum umber, surprisigly, does ot affect the eergy for ay l. Aother surprisig feature of the spherical harmoics follows from the compariso of Eq. (63) with the secod of classical relatios (5). These epressios coicide if we iterpret costat L l( l ), (3.78) as the value of the full agular mometum squared L = L (icludig its both ad compoets) i the eigestate with eigefuctio Y m l. O the other had, the structure of the azimuthal compoet F() of the wavefuctio is eactly the same as i D aially-symmetric problems, suggestig that Eq. (39) still gives correct values (i our ew otatio, L z = m) for the z-compoet of the agular mometum. If this is so, why for ay state with l >, (L z ) = m l is less tha L = l(l + )? I other words, what prevets the agular mometum vector to be fully aliged with ais z? Besides that issue, though the above aalysis of the spherical rotator is formally (mathematically) complete, it is as usatisfactory o the physics level as the harmoic oscillator aalysis i Sec..6. I particular, it does ot eplai the meaig of the etremely simple relatios for eigevalues of eergy ad agular mometum o the backdrop of rather complicated wavefuctios. We will obtai atural aswers to all these questios ad cocers i Sec. 5.6, but ow let us complete our survey of wave mechaics by etedig it to 3D motio i a arbitrary sphericallysymmetric potetial (55). I this case we have to use the full form of the Laplace operator i spherical coordiates. The variable separatio procedure is a evidet geeralizatio of what we have doe before, with the particular solutio R ( )Θ( ) F( ), (3.79) whose substitutio ito the statioary Schrödiger equatio yields mr R d dr r dr dr Θ d dθ d F si U ( r) E. (3.8) si d d si F d 64 The letter ames for states with differet values of l stem from the history of optical spectroscopy - for eample, letter s, used for l =, origially deoted the sharp optical lie series, etc. The sequece of the letters is as follows: s, p, d, f, g, h, ad further i the alphabetical order. Chapter 3 Page 4 of 56

148 Attractive Coulomb potetial It is evidet that the agular part (the two last terms i square brackets) separates from the radial part, ad for the former part we get Eq. (56) agai, with the oly chage, R r. This chage does ot affect the fact that the eigefuctios of that equatio are the spherical harmoics (7), ad the agular eigeeergy is give by Eq. (63), agai with the replacemet R r. This meas that for the radial fuctio, Eq. (8) gives the followig equatio, d dr r l l U r E mr ( ) dr dr ( ). (3.8) R Note that o iformatio about the magetic quatum umber m has ot crept ito the radial equatio (besides establishig the limitatio (6) for possible values of l), so that this equatio depeds oly o the latter quatum umber. The radial equatio becomes rather simple for U(r) =, ad may be used, for eample, to solve the eigeproblem for the free 3D motio of a particle iside the sphere of radius R. Leavig that problem for the reader s eercise, I will proceed to the most importat Bohr atom problem, i.e. of motio i the so-called attractive Coulomb potetial 65 C U ( r), r with C. (3.8) The atural scales of r ad E are, respectively, 66 C r ad E m mc mr. (3.83) I the ormalized uits E/E ad r/r, Eq. (8) looks simpler, d R dr l( l ) R R d d, (3.84) but ufortuately its eigefuctios may be called elemetary oly i the most geerous meaig of the word. With the adequate ormalizatio, R R r dr, (3.85), l ', l ' these (mutually orthogoal) fuctios may be preseted as 65 Historically, the solutio of this problem i 98, that reproduced the mai result (.8)-(.9) of the old quatum theory developed by N. Bohr i 9, without its restrictive assumptios, was the decisive step for the geeral acceptace of Schrödiger s wave mechaics. 66 These two scales are obtaied from relatios E /mr C/r, i.e. from the equality of the atural scales of the potetial ad kietic eergies, droppig all umerical coefficiets. For the most importat case of the hydroge atom, C = e /4, these scales are reduced, respectively, to the Bohr radius r B (.3) ad the Hartree eergy E H (.9). Note also that for a hydroge-like atom (or rather io), with C = Z(e /4 ), these two key parameters are rescaled as r = r B /Z, E = Z E H. Chapter 3 Page 4 of 56

149 / 3 l ( l )! r r, ( ) ep. 3 ( )! l r R l r Ll (3.86) r l r r r Bohr atom s radial fuctios q Here L ( ) are the so-called associated Laguerre polyomials, which may be calculated as p q q q d L p ( ) ( ) L p q ( ) q. (3.87) d Associated Laguerre polyomials from simple Laguerre polyomials L p () L p (). 67 I tur, the easiest way to obtai L p () is to use the followig Rodrigues formula: 68 p d p L p ( ) e e p. (3.88) d Notice that i cotrast with the associated Legedre fuctios P m l participatig i spherical harmoics, q L p are ust polyomials, ad those with small idices p ad q are ideed simple. Returig to Eq. (86), we see that the atural quatizatio of the radial equatio (84) has brought us a ew quatum umber (iteger). I order to uderstad its rage, we should otice that accordig to Eq. (88), the highest power of terms i polyomial L p+q is (p + q), ad hece, accordig to Eq. (87), that of L q p is p, so that of the highest power i the polyomial participatig i Eq. (86) is ( l ). Sice the power caot be egative (to avoid the uphysical divergece of wavefuctios at r ), the radial quatum umber has to obey the restrictio l +. Sice l, as we already kow, may take values l =,,,, we may coclude that may oly take values,,... (3.89) What makes this relatio importat is the followig, most surprisig result of the theory: the eigeeergies correspodig to wavefuctios (79), which are ideed with 3 quatum umbers: m R ( r) Y (, ), (3.9), l. m, l l Rodrigues formula for Laguerre polyomials deped oly o ad agree with Bohr s formula (.8): E C E m. (3.9) Because of this reaso, is usually called the pricipal quatum umber, ad the above relatio betwee it ad more subordiate l is rewritte as l. (3.9) Together with iequality (6), this gives us the most importat hierarchy of the 3 quatum umbers ivolved i the problem: l l m l, (3.93) Bohr atom s quatum umbers 67 I Eqs. (87)-(88), p ad q are o-egative itegers, with o relatio whatsoever to particle s mometum or electric charge. Sorry for this otatio, but it is absolutely commo, ad ca hardly result i ay cofusio. 68 Named after the same B. O. Rodrigues, ad belogig to the same class as his aother key result, Eq. (65). Chapter 3 Page 4 of 56

150 Groud state s radial fuctio Takig ito accout the (l +)-degeeracy related to the magetic umber m, ad usig the well-kow formula for the arithmetic progressio, 69 we see that each eergy level (9) has the followig orbital degeeracy: g (l ) l. (3.94) l l l Due to its importace for applicatios, let us spell out the quatum umber hierarchy of a few lowesteergy states, usig the traditioal otatio i which the value of is followed by the letter that deotes the value of l: : l (oe s state) m. (3.95) : 3 : l l l l l (oe s state) (three p states) (oe 3s state) (three 3p states) (five 3d states) m, m,. m, m,, m,,. (3.96) (3.97) Figure shows plots of the radial fuctios (86) of the listed states. The most importat of them is of course the groud (s) state with = ad hece E = - E /, whose radial fuctio (86) is ust r / r R, ( r) e, (3.98) 3 / r ad the agular distributio is uiform - see Eq. (74). The gap betwee the groud eergy ad the eergy E = - E /8 of the lowest ecited states (with = ) i a hydroge atom (i which E = E H 7. ev) is as large as ~ ev, so that their thermal ecitatio requires temperatures as high as ~ 5 K, ad the overwhelmig part of all hydroge atoms i the visible Uiverse are i their groud state. Sice atomic hydroge makes up about 75% of the ormal matter, we are very fortuate that such simple formulas as Eqs. (74) ad (98) describe the atomic states most frequetly met i Mother Nature! 7 The radial fuctios of the et states, s ad p, are also ot too comple: r r / r r r / r R, ( r) e, 3 / r r R,( r) e. (3.99) 3 / / r 3 r (Note agai that the former of these states (s) ca oly have a uiform agular distributio, while three p states have differet values of m =,, ad hece have differet agular distributios see Eq. (75) ad the secod row of Fig. 9.) The most importat tred here is a larger radius of decay of the epoet (r for = istead of r for = ), ad hece the radial etesio of the states. This tred is cofirmed by the followig geeral formula: 7 69 See, e.g., MA Eq. (.5a). 7 Forgettig for a miute about such ew dark clouds o the horizo of the moder physics as the hypothetical dark matter ad dark eergy. 7 Note that eve at the largest value of l, equal to ( -), term l(l + ) i Eq. () caot compesate term 3. Chapter 3 Page 43 of 56

151 r r 3 l( l ). (3.), l The secod importat tred is that at fied, the orbital quatum umber l determies how fast does the wavefuctio chage with r ear the origi, ad how much it oscillates i the radial directio at larger r. For eample, the s eigefuctio R, (r) is ovaishig at r =, ad makes oe wiggle (has oe root) i the radial directio, while eigefuctios p equal zero at r =, ad do ot oscillate at all i the radial directio. Istead, those wavefuctios always oscillate as fuctios of some agle see the secod row of Fig. 9. The same tred i clearly visible for = 3 (see Fig. ), ad cotiues for the higher values of. R 3 /, l r.5 s ( l ) R.5 p ( l ) 3 /, l r s ( l ) R / 3, l r.5 3 p ( l ) r/r 3d ( l ) r/r 3s ( l ) r/r Fig. 3.. The lowest radial fuctios of the Bohr atom problem. The iterpretatio of these results is that the states with l = l ma = may be viewed as aalogs of the circular motio of a particle i a plae whose orietatio defies the quatum umber m, with a almost fied radius r r ( ). O the other had, the best classical image of a s-state (l = ) is the purely radial motio of the particle to ad from the attractig ceter. (The latter image is especially imperfect, because the motio would eed to happe simultaeously i all radial directios.) The classical laguage becomes reasoable oly for the so-called Rydberg states, with >>, whose liear superpositios may be used to compose wave packets closely followig the classical, circular or elliptic traectories of the particle ust as was discussed i Sec.. for the free D motio. Chapter 3 Page 44 of 56

152 Besides Eq. (), mathematics gives us several other simple relatios for the radial fuctios R,l (ad, sice the spherical harmoics are ormalized to, for the eigefuctios as the whole), icludig those that we will use later i the course: 7 r, / l r r ll / ( l r 3, l r r, l, l ),. (3.) I particular, the first of them meas that for ay eigefuctio,l,m, with all its complicated radial ad agular depedecies, there is a simple relatio betwee the potetial ad full eergies: C E U C E,, l (3.) r r, l so that the average kietic eergy of the particle, T,l = E - U,l, is equal to E >. These simple results are i a sharp cotrast with the rather complicated epressios for the eigefuctios, ad motivate a search for more geeral methods of quatum mechaics, which would replace or at least complemet our brute-force (wave-mechaics) approach, to reveal their real ature. Such a approach will be the mai topic of the et chapter Atoms Before proceedig to that chapter, let me show that, rather strikigly, the classificatio of quatum umbers i the simple potetial well (8), carried out i the last sectio, together with very modest borrowigs from the further theory, allows a semi-quatitative eplaatio of the whole system of chemical elemets. The oly two additios we eed are the followig facts: (i) due to iteractio with relatively low-temperature eviromets, atoms ted to rela ito their lowest-eergy state, ad (ii) due to the Pauli priciple (valid for electros as Fermi particles), each orbital eigestate discussed above ca be occupied with electros with opposite spis. Of course, atomic electros do iteract, so that their quatitative descriptio requires quatum mechaics of multiparticle systems, which is rather comple. (Its mai cocepts will be discussed i Chapter 8.) However, the lio s share of this iteractio reduces to simple electrostatic screeig, i.e. the partial compesatio of the electric charge of the atomic ucleus, as felt by a particular electro, by other electros of the atom. This screeig chages the qualitative results (such as the eergy scale) dramatically; however, the quatum umber hierarchy, ad hece their classificatio, is ot affected. The system of atoms is most ofte preseted as the famous periodic table of chemical elemets, 73 whose simple versio is show i Fig., while Fig. presets a sequetial list of the elemets with their electro cofiguratios. The umbers i table s cells (ad the first colum i the list) are the 7 The first of these relatios may be also readily proved usig the Heller-Feyma theorem (see Chapter ); this proof is left for reader s eercise. Note also that the last of the epressios diverges at l =, i particular i the groud state of the system (with =, l = ). 73 Also called the Medeleev table, after D. Medeleev who put forward the cocept of the periodicity of chemical elemet properties as fuctios of Z pheomeologically i 869. (The eplaatio of the periodicity had to wait for 6 more years util the quatum mechaics formulatio i the late 9s.) Chapter 3 Page 45 of 56

153 atomic umbers Z, which physically are the umbers of protos i the atomic ucleus, ad hece the umbers of electros i the electrically eutral atom. The electro cofiguratio i Fig. follows the covetio already used i Eqs. (95)-(97), with the additioal upper ide showig the umber of electros with the idicated values of quatum umbers ad l. The lightest atom, with Z =, is hydroge (chemical symbol H) the oly atom for each the theory discussed i Sec. 6 is quatitatively correct. 74 Accordig to Eq. (9), the s groud state of its oly electro correspods to quatum umbers =, l =, ad m = see Eq. (96). I most versios of the periodic table, the cell of H is placed i the top left corer. I the et atom, helium (He, Z = ), the same orbital quatum state (s) is filled with two electros with differet spis. 75 Note that due to the twice higher electric charge of the ucleus, i.e. the twice higher value of costat C i Eq. (8), resultig i a 4-fold icrease of costat E (83), the bidig eergy of each electro is crudely 4 times higher tha that of the hydroge atom - though the electro iteractio decreases it by about 5% - see Sec. 7.. This is why takig oe electro away (i.e. positive ioizatio) of the helium atom requires a very high eergy, 3.4 ev, which is ot available i usual chemical reactios. O the other had, a eural helium atom caot bid oe more electro (i.e. form a egative io) either. As a result, helium, ad all other elemets with fully completed electro shells (sets of states with eigeeergies well separated from higher eergy levels) is a chemically iert oble gas, thus startig the whole right-most colum of the periodic table, committed to such elemets. H 3 Li Na 9 K 37 Rb 55 Cs 87 Fr 4 Be Mg Ca 38 Sr 56 Ba 88 Ra Property leged: alkali metals trasitio metals metalloids alkali-earth metals ometals haloges rare-earth metals other metals oble gases Sc Ti V Cr M Fe Co Ni Cu Z Y Zr Nb Mo Tc Ru Rh Pd Ag Cd Hf Ta W Re Os Ir Pt Au Hg Rf Db Sg Bh Hs Mt Ds Rg C 5 B 3 Al 3 Ga 49 I 8 Tl 3 Uut 6 C 4 Si 3 Ge 5 S 8 Pb 4 Fl 7 N 5 P 33 As 5 Sb 83 Bi 5 Uup 8 O 6 S 34 Se 5 Te 84 Po 6 Lv 9 F 7 Cl 35 Br 53 I 85 At 7 Uus He Ne 8 Ar 36 Kr 54 Xe 86 R 8 Uuo Lathaides: 57 La Actiides: 89 Ac 58 Ce 89 Ac 59 Pr 9 Th 6 Nd 9 Pa 6 Pm 9 U 6 Sm 93 Np 63 Eu 94 Pu 64 Gd 95 Am 65 Tb 96 Cm 66 Dy 97 Bk Fig.. The periodic table of elemets, showig their atomic umbers, as well as their basic physical/chemical properties at the so-called ambiet (meaig usual laboratory) coditios. 67 Ho 98 Cf 68 Er 99 Es 69 Tm Fm 7 Yb Md 7 Lu Lr 74 Besides very small fie-structure correctios to be discussed i Chapters 6 ad As will be discussed i detail i Chapter 8, electros of the same atom are actually idistiguishable, ad their quatum states are ot idepedet, ad frequetly etagled. These factors are importat for several properties of helium atoms (ad heavier elemets as well), especially for their respose to eteral fields. However, for the atom classificatio purposes, they are ot crucial. Chapter 3 Page 46 of 56

154 Atomic umber Atomic symbol Electro states Atomic umber Atomic symbol Electro states Atomic umber Atomic symbol Electro states Period Period 5 [Kr] shell, 77 Ir 4f 4 5d 7 6s plus: 78 Pt 4f 4 5d 9 6s H s 37 Rb 5s 79 Au 4f 4 5d 6s He s 38 Sr 5s 8 Hg 4f 4 5d 6s Period [He] shell, 39 Y 4d 5s 8 Tl 4f 4 5d 6s 6p plus: 4 Zr 4d 5s 8 Pb 4f 4 5d 6s 6p 3 Li s 4 Nb 4d 4 5s 83 Bi 4f 4 5d 6s 6p 3 4 Be s 4 Mo 4d 5 5s 84 Po 4f 4 5d 6s 6p 4 5 B s p 43 Tc 4d 6 5s 85 At 4f 4 5d 6s 6p 5 6 C s p 44 Ru 4d 7 5s 86 R 4f 4 5d 6s 6p 6 7 N s p 3 45 Rh 4d 8 5s [R] shell, 8 O s p 4 46 Pd 4d Period 7 plus: 9 F s p 5 47 Ag 4d 5s 87 Fr 7s Ne s p 6 48 Cd 4d 5s 88 Ra 7s Period 3 [Ne] shell, 49 I 4d 5s 5p 89 Ac 6d 7s plus: 5 S 4d 5s 5p 9 Th 6d 7s Na 3s 5 Sb 4d 5s 5p 3 9 Pa 5f 6d 7s Mg 3s 5 Te 4d 5s 5p 4 9 U 5f 3 6d 7s 3 Al 3s 3p 53 I 4d 5s 5p 5 93 Np 5f 4 6d 7s 4 Si 3s 3p 54 Xe 4d 5s 5p 6 94 Pu 5f 6 7s 5 P 3s 3p 3 [Xe] shell, 95 Am 5f 7 7s 6 S 3s 3p 4 Period 6 plus: 96 Cm 5f 7 6d 7s 7 Cl 3s 3p 5 55 Cs 6s 97 Bk 5f 9 7s 8 Ar 3s 3p 6 56 Ba 6s 98 Cf 5f 7s Period 4 [Ar] shell, 57 La 5d 6s 99 Es 5f 7s plus: 58 Ce 4f 5d 6s Fm 5f 7s 9 K 4s 59 Pr 4f 3 6s Md 5f 3 7s Ca 4s 6 Nd 4f 4 6s No 5f 4 7s Sc 3d 4s 6 Pm 4f 5 6s 3 Lr 5f 4 6d 7s Ti 3d 4s 6 Sm 4f 6 6s 4 Rf 5f 4 6d 7s 3 V 3d 3 4s 63 Eu 4f 7 6s 5 Db 5f 4 6d 3 7s 4 Cr 3d 4 4s 64 Gd 4f 7 5d 6s 6 Sg 5f 4 6d 4 7s 5 M 3d 5 4s 65 Tb 4f 9 6s 7 Bh 5f 4 6d 5 7s 6 Fe 3d 6 4s 66 Dy 4f 6s 8 Hs 5f 4 6d 6 7s 7 Co 3d 7 4s 67 Ho 4f 6s 9 Mt 5f 4 6d 7 7s 8 Ni 3d 8 4s 68 Er 4f 6s Ds 5f 4 6d 8 7s 9 Cu 3d 9 4s 69 Tm 4f 3 6s Rg 5f 4 6d 9 7s 3 Z 3d 4s 7 Yb 4f 4 6s C 5f 4 6d 7s 3 Ga 3d 4s 4p 7 Lu 4f 4 5d 6s 3 Uut 5f 4 6d 7s 7p 3 Ge 3d 4s 4p 7 Hf 4f 4 5d 6s 4 Fl 5f 4 6d 7s 7p 33 As 3d 4s 4p 3 73 Ta 4f 4 5d 3 6s 5 Uup 5f 4 6d 7s 7p 3 34 Se 3d 4s 4p 4 74 W 4f 4 5d 4 6s 6 Lv 5f 4 6d 7s 7p 4 35 Br 3d 4s 4p 5 75 Re 4f 4 5d 5 6s 7 Uus 5f 4 6d 7s 7p 5 36 Kr 3d 4s 4p 6 76 Os 4f 4 5d 6 6s 8 Uuo 5f 4 6d 7s 7p 6 Fig. 3.. Atomic electro cofiguratios. The upper ide shows the umber of electros i states with the idicated quatum umbers (the first digit) ad l (letter-coded as listed above). Chapter 3 Page 47 of 56

155 The situatio chages dramatically as we move to the et elemet, lithium (Li), with Z = 3 electros. Two of them are still accommodated by the ier shell = (listed i Fig. as the helium shell [He]), but the third oe has to reside i the et shell with = ad l =, i.e. i the s state. Accordig to Eq. (9), the bidig eergy of this electro is much lower, especially if we take ito accout that accordig to Eq. (), the s electros of the [He] shell are much closer to the ucleus ad almost completely compesate two thirds of its electric charge +3e. As a result, the s electro is reasoably well described by Eq. (99), with bidig eergy of ust 5.39 ev, so that a lithium atom ca give out that electro rather easily to either atoms of other elemets to form chemical compouds, or ito the commo coductio bad of solid state lithium - ad as a result it is a typical alkali metal. The similarity of chemical properties of lithium ad hydroge, with the chemical valece of oe, 76 places Li as the startig elemet of the secod period (row), with the first period limited to oly H ad He. I the et elemet, beryllium (Z = 4), the s state ( =, l = ) picks up oe more electro, with the opposite spi. Due to the higher electric charge of the ucleus, Q = 4e, with oly half of it compesated by s electros of the [He] shell, the bidig eergy of the s electros is higher tha i lithium, so that the ioizatio eergy icreases to 9.3 ev. As a result, beryllium is also chemically active but ot as active as lithium, with the valece of two, ad is also is metallic i its solid state phase, but does ot coduct electric curret as well as lithium. Movig i this way alog the secod row of the periodic table (from Z = 3 to Z = ), we see the gradual fillig of all 4 differet orbital states of the = shell, by electros each, with gradually growig ioizatio potetial (up to.6 ev i Ne with Z = ), i.e. the growig reluctace to have metallic coductace or form positive ios. However, the fial elemets of the row, such as oyge (O, with Z = 8) ad especially fluorie (F, with Z = 9) ca readily pick up etra electros to fill their p states, i.e. form egative ios. As a result, these elemets are chemically active, with the double valece for oyge ad sigle valece for fluorie. However, the fial elemet of this row, eo, has its = shell full, ad caot form a stable egative io. This is why it is a oble gas, like helium. Traditioally, i the periodic table it is placed right uder helium (Fig. ), to emphasize the similarity of their chemical ad physical properties. But this ecessitates makig a at least 6-cell gap i the st row. (Actually, the gap is ofte made larger, to accommodate et rows keep readig.) Period 3, i.e. the 3 rd row of the table starts eactly like period, with sodium (Na, with Z = ), also a chemically active alkali metal whose atom features electros fillig shells with = ad = (i Fig. collectively called the eo shell, [Ne]), plus oe electro i a 3s state ( = 3, l =, m = ), which may be reasoably well described by the hydroge atom theory see, e.g., the red trace o the last pael of Fig.. Naively we could epect that, accordig to Eq. (94), ad with the accout of double spi degeeracy, this period of the table should have = 3 = 8 elemets, with gradual fillig of two 3s states, si 3p states, ad te 3d states. However, here we ru ito a big surprise: after argo (Ar, with Z = 8), a relatively iert elemet with ioizatio eergy of 5.7 ev due to the fully filled 3s ad 3p states, the et elemet, potassium (K, with Z = 9) is a alkali metal agai! The reaso for that is the differece of the actual electro eergies from those of the hydroge atom, which is due mostly to iter-electro iteractios ad gradually accumulates with the growth of Z. It may be semi-quatitatively uderstood from the results of Sec. 6. I hydroge-like atoms, electro state eergies do ot deped o the quatum umber l (as well as m) see Eq. (9). However, the 76 Chemical valece is a relatively vague term describig the umber of atom s electros ivolved i chemical reactios. For the same atom, the umber may deped o the chemical compoud formed. Chapter 3 Page 48 of 56

156 orbital quatum umber does affect the wavefuctio of a electro. As Fig. shows, the larger l the less the probability for a electro to be close to the ucleus, where its positive charge is less compesated by other electros. As a result of this effect (ad also the relativistic correctios to be discussed i Sec. 6.3), electro s eergy grows with l. Actually, this effect was visible eve i period : it maifests itself i the fillig order (p states after s states). However, for potassium (K, with Z = 9) ad calcium (Ca, with Z = ), eergies of 3d states become so high that eergies of two 4s states (with opposite spis) are lower, ad they are filled first. As described by factor 3 i the square brackets of Eq. (), ad also by Eq. (), the effect of the pricipal umber o the distace from the ucleus is stroger tha that of l <, so that 4s wavefuctios of K ad Ca are relatively far from the ucleus, ad determie the chemical valece (equal to ad, correspodigly) of these elemets. The et atoms, from Sc (Z = ) to Z (Z = 3), with the gradually filled iteral 3d states, are the so-called trasitio metals whose (comparable) ioizatio eergies ad chemical properties are determied by 4s electros. This fact is the origi of the differece betwee various forms of the periodic table. I its most popular optio, show i Fig., K is used to start the et, period 4, ad the a ew period is started each time ad oly whe the first electro with the et pricipal quatum umber () appears. 77 This topology provides a very clear mappig o the chemical properties of the first elemet of each period (a alkali metal), as well as its last elemet (a oble gas). This also automatically meas makig gaps i all previous rows. Usually, this gap is made betwee the atoms with completely filled s states ad with the first electro i a p state, because here the properties of the elemets make a somewhat larger step. (For eample, the step from Be to B makes the material a isulator, but it is ot large eough to make a similar differece betwee Mg to Al.) As a result, elemets of the same colum have approimately similar chemical valece ad physical properties. However, to accommodate loger lowest rows, such presetatio is icoveiet, because the whole table would be too broad. This is why the so-called rare earths, icludig lathaides (with Z from 57 to 7, of the 6 th row, with gradual fillig of 4f ad 5d states) ad actiides (Z from 89 to 3, of the 7 th row, with gradual fillig of 5f ad 6d states), are preseted as outlet lies (Fig. ). This is quite acceptable for the purposes of stadard chemistry, because chemical properties of elemets withi each group are rather close. To summarize, the periodic table of elemets is ot periodic i the strict sese of the word. Nevertheless, it has had a eormous historic sigificace for chemistry, as well as atomic ad solid state physics, ad is still very coveiet for may purposes. For our course, the most importat aspect of its discussio is the surprisig possibility to describe, at least for classificatio purposes, such a comple multi-electro system as a atom as a set of quasi-idepedet electros i certai quatum states ideed with the same quatum umbers, l, ad m as those of the hydroge atom. This fact eables the use of various perturbatio theories, which give more quatitative descriptio of atomic properties. Some of these techiques will be reviewed i Chapters 6 ad 8 of this course Aother optio is to retur to the first colum as soo a atom has oe electro i s state (like it is i Cu, Ag, ad Au, i additio to the alkali metals). 78 For a bit more detailed (but still very succict) discussio of valece ad other chemical aspects of atomic structure, I ca recommed Chapter 5 of the classical tet by L. Paulig, Geeral Chemistry, Dover, 988. Chapter 3 Page 49 of 56

157 3.8. Eercise problems 3.. A particle of eergy E is icidet (i Fig. o the right, withi the plae of drawig) o a sharp potetial step: y, for, U ( r ) U, for. k Fid the particle reflectio probability R as a fuctio of the icidece agle ; sketch ad discuss the fuctio, for differet magitudes ad sigs of U. 3.. Use the fiite differece method with step h a / to calculate as may eigeeergies as possible, for a free particle cofied to the iterior of: (i) a square with side a; (ii) a cube with side a. For the square, repeat the calculatios, usig a fier step: h = a/3. Compare the results for differet h, with the eact formula. Hit: It is advisable to first solve (or review the solutio of :-) the similar D problem i Chapter, or start from readig about the fiite differece method. 79 Also, try to eploit problem s symmetry Use the variatioal method to estimate the groud state eergy of a particle of mass m, movig i a spherically-symmetric potetial 4 U r ar I the classical versio of the Ladau level problem discussed i Sec., the ceter of particle s orbit is a itegral of motio, determied by iitial coditios. Calculate the commutatio relatios betwee the quatum-mechaical operators correspodig to the Cartesia coordiates of the ceter, ad to the sum of their squares * Aalyze how are the Ladau levels (3.5) modified by a additioal costat electric field E, directed alog the particle plae. Cotemplate the physical meaig of your result, ad its B implicatios for the quatum Hall effect i a gate V g V g gate-defied Hall bar. (The area LW area of W such a bar [see Fig. 3.6 of the lecture otes] is D electro defied by metallic gate electrodes parallel to gas plae semicoductor the D electro gas plae - see Fig. o the right. gate The egative voltage V g, applied to the gates, chases the electros gas out of the cofiemet plae at the remaiig sample area.) 3.6. Aalyze how are the Ladau levels (5) modified if a D particle is cofied i a additioal D potetial well U() = m / 79 See, e.g., CM Sec. 8.5 or EM Sec..8. Chapter 3 Page 5 of 56

158 3.7. Fid the eigefuctios of a spiless, charged 3D particle movig i crossed (perpedicular), uiform electric ad magetic fields. For each eigefuctio, calculate the epectatio value of particle s velocity i the directio perpedicular to both fields, ad compare the result with the solutio of the correspodig classical problem. Hit: Geeralize Ladau s solutio for D particles, discussed i Sec Use the Bor approimatio to calculate the agular depedece ad the full cross-sectio of scatterig of a icidet plae wave, propagatig alog ais, by the followig pair of poit ihomogeeities: a a U ( r) W r z r z. Aalyze the results i detail. Derive the coditio of the Bor approimatio s validity for such deltafuctioal scatterers Use the Bor approimatio to calculate the differetial ad full cross-sectios of a spherical scatterer: U, for r R, U ( r ), otherwise. Aalyze both results, especially the agular depedece of d/d, i detail, for kr << ad kr >>. 3.. Use the Bor approimatio to calculate differetial ad full cross-sectios of electro scatterig by a screeed Coulomb field of a poit charge Ze, with electrostatic potetial Ze r r e, 4 r eglectig the spi iteractio effects, ad aalyzed their depedece o the screeig parameter. Compare the results with those give by the classical ( Rutherford ) formula 8 for the uscreeed Coulomb potetial ( ), ad formulate the coditio of Bor approimatio s validity i this limit. 3.. A quatum particle of mass m with electric charge Q is scattered by a localized distributed charge with a spherically-symmetric desity (r) ad zero total charge. Use the Bor approimatio to calculate the differetial cross-sectio of forward scatterig (with scatterig agle = ), ad evaluate it for scatterig of electros by a hydroge atom i its groud state. 3.. Reformulate the Bor approimatio for the D case. Use the result to fid the scatterig ad trasfer matrices of a rectagular scatterer U, for d /, U ( ), otherwise. 8 See, e.g., CM Sec. 3.7, i particular Eq. (3.7). Chapter 3 Page 5 of 56

159 Compare the results with the those of the eact calculatios carried out earlier i the course Use Eq. (88) to show that the Bragg rule for the diffractio wave maima, k = k + Q, where Q is ay vector of the reciprocal lattice defied by Eq. (), is valid ot oly for electromagetic waves, but also for o-relativistic quatum particle scatterig by a periodic (Bravais) lattice I the tight-bidig approimatio, calculate the eigestates ad eigevalues of three similar, weakly coupled quatum wells located i the vertices of a equilateral triagle Figure o the right shows a fragmet of a periodic D lattice, with ope ad solid poits showig the locatio of differet local potetials say, differet atoms. y a (i) Fid the reciprocal lattice ad the st Brilloui zoe; a (ii) Fid wave umber k of the moochromatic radiatio icidet alog ais, at which the lattice creates the first-order diffractio peak withi the [, y] plae, ad the directio towards this peak. (iii) Semi-qualitatively, describe the evolutio of the itesity of the peak if the local potetials represeted by the ope ad solid poits ted to each other For the D heagoal lattice (Fig. b): (i) fid the reciprocal lattice Q ad the st Brilloui zoe; (ii) use the tight-bidig approimatio to calculate the dispersio relatio E(q) for a D particle movig i a potetial with such periodicity, close to the eigeeergy of a aially-symmetric state quasi-localized at the potetial miima; (iii) aalyze ad sketch (or plot) the resultig dispersio relatio E(q) iside the st Brilloui zoe * Complete the tight-bidig approimatio calculatio of bad structure of the hoeycomb lattice, started i the ed of Sec. 4. Aalyze the results. Prove that the Dirac poits q D are located i the corers of the st Brilloui zoe, ad epress the velocity v, participatig i Eq. (), i terms of the couplig eergy. Show that the fial results do ot chage if the quasi-localized wavefuctios are ot aially-symmetric, but are proportioal to ep{i} - as they are, with =, for the p z electros of carbo atoms i graphee, which are resposible for its trasport properties Eamie basic properties of the so-called Waier fuctios defied as e i qr 3 ( r) cost ( r) d q, R BZ where q (r) is the Bloch wavefuctio (3.8), R is ay vector of the Bravais lattice, ad the itegratio over quasi-mometum q is eteded over ay (e.g., the first) Brilloui zoe Evaluate the log-rage electrostatic iteractio (the so-called Lodo dispersio force) betwee two similar, electrically-eutral but polarizable molecules, modelig them as isotropic 3D harmoic oscillators. q Chapter 3 Page 5 of 56

160 Hit: Usig the classical epressio for the iteractio betwee two electric dipoles, 8 try to preset the total Hamiltoia of the system as a sum of Hamiltoias of several idepedet harmoic oscillators, ad calculate their groud-state eergy as a fuctio of distace betwee the molecules. 3.. Use the variable separatio method to fid epressios for the eigefuctios ad the correspodig eigeeergies of a free D particle cofied iside a thi roud disk of radius R:, for ρ R, U, for R, where {, y, }. What is the level degeeracy? Calculate 5 lowest eergy levels with accuracy better tha %. 3.. Calculate the groud-state eergy of a D particle localized i a shallow flat-bottom potetial well U, for R U, with U., for R mr 3.. Spell out the eplicit form of spherical harmoics Y (, ) ad Y 4 (, ) Calculate ad i the groud state of the plaar ad spherical rotators of radius R. What ca you say about averages p ad p? 3.4. Accordig to the discussio i the begiig of Sec. 5, eigefuctios of a 3D harmoic oscillator may be calculated as products of three D Cartesia oscillators - see, i particular Eq. (4), with d = 3. However, accordig to the discussio i Sec. 3.6, wavefuctios of the type (9), proportioal to spherical harmoics Y m l, are also eigestates of this spherically-symmetric system. Represet: (i) the groud state of the oscillator, ad (ii) each of its lowest ecited states, take i the form (9), as liear combiatios of products of D oscillator wavefuctios. Also, calculate the degeeracy of th eergy level of the oscillator A spherical rotator (with r ( + y + z ) / = R = cost) of mass m is i the state with wavefuctio cost si. 3 Calculate the system s eergy * Calculate the eigefuctios ad the eergy spectrum of a 3D particle free to move iside a sphere of radius R: 8 See, e.g., EM Sec. 3.. Chapter 3 Page 53 of 56

161 , for r R, U, for R r. Calculate 5 lowest eergy levels with a % accuracy, ad idicate the degeeracy of each level. Hit: The solutio of this problem requires the so-called spherical Bessel fuctios l (), whose descriptio is available i most math hadbooks Fid the smallest value of depth U for that the spherical quatum well U, for r R, U, for R r, has a boud (localized) eigestate. Does such a state eist for a very arrow ad deep well U = -W(r), with a positive ad fiite W? 3.8. Calculate the smallest value of depth U for that the followig spherically-symmetric quatum well, U has a boud (localized) eigestate. r U r / R e, with U, R Hit: Try to itroduce the followig ew variables: f rr ad Ce -r/r, with a appropriate choice of costat C Calculate the lifetime of the lowest metastable state i the spherical-shell potetial U ( r) W ( r R), with W, i the limit of large W. Specify the limit of validity of your result Calculate the coditio at which a particle of mass m, movig i the field of a very thi spherically-symmetric shell, with U WrR, with W r, has at least oe localized ( boud ) statioary state. Compare the result with that for potetial U r W r, with W. Hit: Note that the first delta-fuctio is oe-dimesioal, while the secod oe is threedimesioal, so that parameters W ad W have differet dimesioalities A particle, movig i a cetral potetial U(r), with U(r) at r, has a statioary state with the followig wavefuctio: r Cr e cos, where C,, ad are costats. Calculate: 8 See, e.g., ay of the hadbooks recommeded i MA Sec. 6(ii)., Chapter 3 Page 54 of 56

162 (i) probabilities of all possible values of quatum umbers m ad l, (ii) the cofiig potetial, ad (iii) state s eergy Calculate the eergy spectrum of a particle movig i a mootoic, but otherwise arbitrary attractive cetral potetial U(r), i the approimatio of large orbital quatum umbers l. Formulate the quatitative coditio(s) of validity of your theory. Check that for the Coulomb potetial U(r) = -C/r, your result agrees with Eq. (9) A electro had bee i the groud state of a hydroge-like atom/io with uclear charge Ze, whe the charge suddely chaged to (Z + )e. 83 Calculate the probabilities for the electro of the chaged system to be: (i) i the groud state, ad (ii) i the lowest ecited state. Evaluate these probabilities for the particular case of the beta decay of tritium, with the formatio of a sigle-positive io of 3 He Calculate ad p i the groud state of a hydroge-like atom. Compare the results with Heiseberg s ucertaity relatio. What do these results tell about electro s velocity i the atom? Apply to Eq. (8) the Hellma-Feyma theorem (see Problem.4) to prove: (i) the first of Eqs. (3.), ad (ii) the fact that for a spiless particle i a arbitrary spherically-symmetric attractive potetial U(r), the groud state is always a s-state (with the orbital quatum umber l = ) For the groud state of a hydroge atom, calculate the epectatio values of E ad E, where E is the electric field created by the atom at distace r >> r from its ucleus. Iterpret the resultig relatio betwee E ad E (at the same observatio poit). 83 Such a fast chage happes, for eample, at the beta-decay, whe oe of ucleus euros suddely becomes a proto, emittig a high-eergy electro ad a eutrio which leave the system very fast (istatly o the atomic time scale), ad do ot participate i the atom trasitio s dyamics. Chapter 3 Page 55 of 56

163 This page is itetioally left blak Chapter 3 Page 56 of 56

164 Chapter 4. Bra-ket Formalism The obective of this chapter is a discussio of Dirac s bra-ket formalism of quatum mechaics, which ot oly overcomes some icoveieces of wave mechaics, but also allows a atural descriptio of such iteral properties of particles as their spi. I the course of discussio of the formalism I will give several simple eamples of its use, leavig more ivolved applicatios for the followig chapters. 4.. Motivatio We have see that wave mechaics gives may results of primary importace. Moreover, it is fully (or mostly) sufficiet for may applicatios, for eample, for solid state electroics ad device physics. However, i the course of our survey we have filed several grievaces about this approach. Let me briefly summarize these complaits: (i) Wave mechaics is focused o the spatial depedece of wavefuctios. O the other had, our attempts to aalyze the temporal evolutio of quatum systems withi this approach (beyod the trivial time behavior of the eigefuctios, described by Eq. (.6)), ru ito techical difficulties. For eample, we could derive Eq. (.59) describig time dyamics of the metastable state, or Eq. (.85) describig quatum oscillatios i coupled wells, oly for the simplest potetial profiles, though it is ituitively clear that these simple results should be commo for all problem of this kid. Derivig the equatios of such processes for arbitrary potetial profiles is possible usig perturbatio theories (to be reviewed i Chapter 6), but that i the wave mechaics laguage they would require very bulky formulas. (ii) The same is true cocerig other issues that are coceptually addressable withi wave mechaics, e.g., the Feyma path itegral approach, descriptio of couplig to eviromet, etc. Addressig them i wave mechaics would lead to formulas so bulky that I had (wisely :-) postpoed them util we have got a more compact formalism o had. (iii) I the discussio of several key problems (for eample the harmoic oscillator ad spherically-symmetric potetials) we have ru ito rather complicated eigefuctios coeistig with simple eergy spectra - that ifer some simple backgroud physics. It is very importat to get this physics revealed. (iv) I the wave mechaics postulates, formulated i Sec.., quatum mechaical operators of the coordiate ad mometum are treated very uequally see Eqs. (.6b). However, some key epressios, e.g., for the fudametal eigefuctio of a free-particle, p r ep i, (4.) or the harmoic oscillator s Hamiltoia, m H m ivite a similar treatmet of mometum ad coordiate. p r, (4.) K. Likharev

165 However, the strogest motivatio for a more geeral formalism comes from wave mechaics coceptual icapability to describe elemetary particles spis ad other iteral quatum degrees of freedom, such as quark flavors or lepto umbers. I this cotet, let us review the basic facts o spi (which is a very represetative ad eperimetally the most accessible of all iteral quatum umbers), to uderstad what a more geeral formalism should eplai - as a miimum. Figure shows the coceptual scheme of the simplest spi-revealig eperimet, first carried out by O. Ster ad W. Gerlach i 9. A collimated beam of electros is passed through a gap betwee poles of a strog maget, where the magetic field B, whose orietatio is take for ais z i Fig., is o-uiform, so that both B z ad db z /dz are ot equal to zero. As a result, the beam splits ito two parts of equal itesity. collimator z y N maget W = 5% electro source B z B z, z S W = 5% particle detectors Fig. 4.. The simplest Ster- Gerlach eperimet. This simplest eperimet ca be semi-quatitatively eplaied o classical, though somewhat pheomeological grouds by assumig that each electro has a itrisic, permaet magetic dipole momet m. Ideed, classical electrodyamics tells us that the potetial eergy U of a magetic dipole i a eteral magetic field is equal to (-m B), so that the force actig o the particle, F U m B, (4.3) has a ovaishig vertical compoet B z Fz mz B z mz. (4.4) z z Hece if we further postulate the eistece of two possible, discrete values of m z =, this eplais the Ster-Gerlach effect qualitatively, as a result of the icidet electros havig a radom sig, but similar magitude of m z. A quatitative eplaatio of the beam splittig agle requires the magitude of to be equal (or close) to the so-called Bohr mageto 3 e 3 J B.974. (4.5) m e T As we will see below, this value caot be eplaied by ay iteral motio of the electro, say its rotatio about ais z. Bohr mageto To my kowledge, the cocept of spi as a iteral rotatio of a particle was first suggested by R. Kroig, the a -year-old studet, i Jauary 95, a few moths before two other studets, G. Uhlebeck ad S. Goudsmit - to whom the idea is usually attributed. The cocept was the accepted ad developed quatitatively by W. Pauli. See, e.g., EM Sec. 5.4, i particular Eq. (5.). 3 A coveiet memoic rule is that it is close to K/T. I the Gaussia uits, B e/m e c Chapter 4 Page of 4

166 Much more importatly, this semi-classical laguage caot eplai the results of the followig set of multi-stage Ster-Gerlach eperimets, show i Fig. - eve qualitatively. I the first of the eperimets, the electro beam is first passed through a magetic field orieted (together with its gradiet) alog ais z, ust as i Fig.. The oe of the two resultig beams is absorbed (or otherwise removed from the setup), while the other oe is passed through a similar but -orieted field. The eperimet shows that this beam is split agai ito two compoets of equal itesity. A classical eplaatio of this eperimet would require a very uatural suggestio that the iitial electros had radom but discrete compoets of the magetic momet simultaeously i two directios, z ad. However, eve this assumptio caot eplai the results of the three-stage Ster-Gerlach eperimet show o the middle pael of Fig.. Here, the previous two-state setup is complemeted with oe more absorber ad oe more maget, ow with the z-orietatio agai. Completely couterituitively, it agai gives two beams of equal itesity, as if we have ot yet filtered out the electros with m z correspodig to the lower beam, i the first, z-stage. SG (z) % SG () 5% 5% absorber SG (z) SG (z) % SG () SG (z) % % % SG (z) 5% 5% Fig. 4.. Three multi-stage Ster-Gerlach eperimets. Boes SG ( ) deote magets similar to oe show i Fig., with the ais orieted i the idicated directio. The oly way to save the classical eplaatio here is to say that maybe, electros somehow iteract with the magetic field, so that the -polarized (o-absorbed) beam becomes spotaeously depolarized agai somewhere betwee magetic stages. But ay hope for such eplaatio is ruied by the cotrol eperimet show o the bottom pael of Fig., whose results idicate that o such depolarizatio happes. We will see below that all these (ad may more) results fid a atural eplaatio i the matri mechaics pioeered by W. Heiseberg, M. Bor ad P. Jorda i 95. However, the matri formalism is icoveiet for the solutio of most problems discussed i Chapters -3, ad for a time it was eclipsed by Schrödiger s wave mechaics, which had bee put forward ust a few moths later. However, very soo P. A. M. Dirac itroduced a more geeral bra-ket formalism, which provides a geeralizatio of both approaches ad proves their equivalece. Let me describe it. Chapter 4 Page 3 of 4

167 4.. States, state vectors, ad liear operators The basic otio of the geeral formulatio of quatum mechaics is the quatum state of a system. 4 To get some gut feelig of this otio, if a quatum state of a particle may be adequately described by wave mechaics, this descriptio is give by the correspodig wavefuctio (r, t). Note, however, the state as such is ot a mathematical obect (such as a fuctio), 5 ad ca participate i mathematical formulas oly as a poiter e.g., the ide of fuctio. O the other had, the wavefuctio is ot a state, but a mathematical obect (a comple fuctio of space ad time) givig a quatitative descriptio of the state - ust as the radius-vector as a fuctio of time is a mathematical obect describig the motio of a classical particle see Fig. 3. Similarly, i the Dirac formalism a certai quatum state is described by either of two mathematical obects, called the state vectors: the ket-vector ad bra-vector. 6 Oe should be cautios with the term vector here. Usual geometric vectors are defied i the usual geometric (say, Euclidea) space. I cotrast, bra- ad ket-vectors are defied i abstract Hilbert spaces of a give system, 7 ad, despite certai similarities with the geometric vectors, are ew mathematical obects, so that we eed ew rules for hadlig them. The primary rules are essetially postulates ad are ustified oly the correct descriptio/predictio of all eperimetal observatios their corollaries. While these is a geeral cosesus amog physicists what the corollaries are, there are may possible ways to carve from them the basic postulate sets. Just as i Sec.., I will ot try too hard to beat the umber of the postulates to the smallest possible miimum, tryig istead to keep their physical meaig trasparet. particle i state mathematical descriptio classical mechaics : r( t) wave mechaics :either bra - ket formalism :either ( r, t) or Ψ or * α ( r, t) Fig Particle s state ad its descriptios. (i) Ket-vectors. Let us start with ket-vectors - sometimes called ust kets for short. Perhaps the most importat property of the vectors cocers their liear superpositio. Namely, if several ketvectors describe possible states of a quatum system, the ay liear combiatio (superpositio) c, (4.6) Liear superpositio of ket-vectors 4 A attetive reader could otice my smugglig term system istead of particle which was used i the previous chapters. Ideed, the bra-ket formalism allows the descriptio of quatum systems much more comple tha a sigle spiless particle that is a typical (though ot the oly possible) subect of wave mechaics. 5 As was epressed icely by A. Peres, oe of pioeers of the quatum iformatio theory, quatum pheomea do ot occur i the Hilbert space, they occur i a laboratory. 6 Terms bra ad ket were suggested to reflect the fact that pair ad may be cosidered as the set of parts of combiatio (see Eq. () below), which remids a epressio i the usual agle brackets. 7 The Hilbert space of a give system is defied as the set of all its possible state vectors. As should be clear from this defiitio, it is ot advisable to speak about a Hilbert space of quatum states. Chapter 4 Page 4 of 4

168 where c are ay (possibly comple) c-umbers, also describes a possible state of the same system. (Oe may say that vector belogs to the same Hilbert space as all.) Actually, sice ket-vectors are ew mathematical obects, the eact meaig of the right-had part of Eq. (6) becomes clear oly after we have postulated the followig rules of summatio of these vectors,, (4.7) ' ' ad their multiplicatio by c-umbers: c c. (4.8) Note that i the set of wave mechaics postulates, statemets parallel to (7) ad (8) were uecessary, because wavefuctios are the usual (albeit comple) fuctios of space ad time, ad we kow from the usual algebra that such relatios are valid. As evidet from Eq. (6), the comple coefficiet c may be iterpreted as the weight of state i the liear superpositio. Oe importat particular case is c =, showig that state does ot participate i the superpositio. By the way, the correspodig term of sum (6), i.e. product Null-state vector, (4.9) Liear superpositio of bra-vectors Ier bra-ket product has a special ame: the ull-state vector. (It is importat to avoid cofusio betwee the ull-state correspodig to vector (9), ad the groud state of the system, which is frequetly deoted by ketvector. I some sese, the ull-state does ot eist at all, while the groud state does ad frequetly is the most importat quatum state of the system.) (ii) Bra-vectors ad ier ( scalar ) products. Bra-vectors, which obey the rules similar to Eqs. (7) ad (8), are ot ew, idepedet obects: if a ket-vector is kow, the correspodig bravector describes the same state. I other words, there is a uique dual correspodece betwee ad, 8 very similar (though ot idetical) to that betwee a wavefuctio ad its comple cougate *. The correspodece betwee these vectors is described by the followig rule: if a ket-vector of a liear superpositio is described by Eq. (6), the the correspodig bra-vector is * c c. (4.) The mathematical coveiece of usig two types of vectors, rather tha ust oe, becomes clear from the otio of their ier product (also called the short bracket): *. (4.) This is a (geerally, comple) 9 scalar, whose mai property is the liearity with respect to ay of its compoet vectors. For eample, if a liear superpositio is described by the ket-vector (6), the 8 Mathematicias like to say that the ket- ad bra-vectors of the same quatum system are defied i two isomorphic Hilbert spaces. 9 This is oe of the differeces of bra- ad ket-vectors from the usual (geometrical) vectors whose scalar product is always a real scalar. Chapter 4 Page 5 of 4

169 while if Eq. () is true, the c, (4.) * c. (4.3) I plai Eglish, c-umbers may be moved either ito, or out of the ier products. The secod key property of the ier product is *. (4.4) It is compatible with Eq. (); ideed, the comple cougatio of both parts of Eq. () gives: * * c c * *. (4.5) Fially, oe more rule: the ier product of the bra- ad ket-vectors describig the same state (called the orm squared) is real ad o-egative,. (4.6) I order to give the reader some feelig about the meaig of this rule: we will show below that if state may be described by wavefuctio (r, t), the Ier product s comple cougate State s orm squared * 3 d r. (4.7) Hece the role of the bra-ket is very similar to the comple cougatio of the wavefuctio, ad Eq. () emphasizes this similarity. (Note that, by covetio, there is o cougatio sig i the bra-part of the ier product; its role is played by the agular bracket iversio.) (iii) Operators. Oe more key otio of the Dirac formalism are quatum-mechaical liear operators. Just as for the operators discussed i wave mechaics, the fuctio of a operator is the geeratio of oe state from aother: if is a possible ket of the system, ad  is a legitimate operator, the the followig combiatio, Â, (4.8) is also a ket-vector describig a possible state of the system, i.e. a ket-vector i the same Hilbert space as the iitial vector. As follows from the adective liear, the mai rules goverig the operators is their liearity with respect to both ay superpositio of vectors: ad ay superpositio of operators: A c c A, (4.9) c A. (4.) c A Chapter 4 Page 6 of 4

170 Hermitia cougate operator Hermitia operator s defiitio Log bracket Log bracket s comple cougate These rules are evidetly similar to Eqs. (.53)-(.54) of wave mechaics. The above rules imply that a operator acts o the ket-vector o its right; however, a combiatio of the type  is also legitimate ad presets a ew bra-vector. It is importat that, geerally, this vector does ot represet the same state as ket-vector (8); istead, the bra-vector isomorphic to ket-vector (8) is Â. (4.) This statemet serves as the defiitio of the Hermitia cougate (or Hermitia adoit )  of the iitial operator Â. For a importat class of operators, called the Hermitia operators, the cougatio is icosequetial, i.e. for them A A. (4.) (This equality, as well as ay other operator equatio below, meas that these operators act similarly o ay bra- or ket-vector.) To proceed further, we eed a additioal postulate, called the associative aiom of multiplicatio: ito ay legitimate bra-ket epressio, ot icludig a eplicit summatio, we may isert or remove paretheses (ust i the ordiary product of scalars), meaig as usual that the operatio iside the paretheses is performed first. The first two eamples of this postulate are give by Eqs. (9) ad (), but the associative aiom is more geeral ad says, for eample: A A A, (4.3) This equality serves as the defiitio of the last form, called the log bracket (evidetly, also a scalar), with a operator sadwiched betwee a bra-vector ad a ket-vector. This defiitio, whe combied with the defiitio of the Hermitia cougate ad Eq. (4), yields a importat corollary: * * A A A A, (4.4) which is most frequetly rewritte as * A A. (4.5) The associative aiom also eables to readily eplore the followig defiitio of oe more, outer product of bra- ad ket-vectors: If we cosider c-umbers as a particular type of operators, the accordig to Eqs. () ad (), for them the Hermitia cougatio is equivalet to the simple comple cougatio, so that oly a real c-umber may be cosidered as a particular case of the Hermitia operator (). Here legitimate meas havig a clear sese i the bra-ket formalism. Some eamples of illegitimate epressios: A, A,,. Note, however, that the last two epressios may be legitimate if ad are states of differet systems, i.e. if their state vectors belog to differet Hilbert spaces. We will ru ito such tesor products of bra- ad ket vectors (sometimes deoted, respectively, as ad ) i Chapters 6-8. Chapter 4 Page 7 of 4

171 . (4.6) I cotrast to the ier product (), which is a scalar, this mathematical costruct is a operator. Ideed, the associative aiom allows us to remove paretheses i the followig epressio:. (4.7) But the last bra-ket is ust a scalar; hece the mathematical obect (6) actig o a ket-vector (i this case, ) gives a ew ket-vector, which is the essece of operator s actio. Very similarly, (4.8) - agai a typical operator s actio o a bra-vector. Now let us perform the followig calculatio. We may use the paretheses isertio ito the braket equality followig from Eq. (4), to trasform it to the followig form: *, (4.9) *. (4.3) Sice this equatio should be valid for ay vectors ad, its compariso with Eq. (5) gives the followig operator equality. (4.3) This is the cougate rule for outer products; it remids rule (4) for ier products, but ivolves the Hermitia (rather tha the usual comple) cougatio. The associative aiom is also valid for the operator multiplicatio : A B A B, AB A B, (4.3) showig that the actio of a operator product o a state vector is othig more tha the sequetial actio of the operads. However, we have to be rather careful with the operator products; geerally they do ot commute: A B B A. This is why the commutator, the operator defied as A, B AB B A is a very useful optio. Aother similar otio is the aticommutator:, (4.33) A, B AB B A. (4.34) Fially, the bra-ket formalism broadly uses two special operators: the ull operator defied by the followig relatios: Outer bra-ket product Outer product s Hermitia cougate Commutator Aticommutator A, B Aother popular otatio for the aticommutator is ; it will ot be used i these otes. Chapter 4 Page 8 of 4

172 Null operator,, (4.35) Idetity operator for a arbitrary state ; we may say that the ull operator kills ay state, turig it ito the ull-state. Aother elemetary operator is the idetity operator, which is also defied by its actio (or rather iactio :-) o a arbitrary state vector: I, I. (4.36) Epasio over basis vectors Basis vectors' orthoormality Epasio coefficiets as ier products 4.3. State basis ad matri represetatio While some operatios i quatum mechaics may be carried out i the geeral bra-ket formalism outlied above, most calculatios are doe for specific quatum systems that feature at least oe full ad orthoormal set {u} of states u, frequetly called a basis. These terms mea that ay state vector of the system may be represeted as a uique sum of the type (6) or () over its basis vectors: *, u, (4.37) u (so that, i particular, if is oe of the basis states, say u, the = ), ad that u u ' '. (4.38) For the systems that may be described by wave mechaics, eamples of the full orthoormal bases are represeted by ay orthoormal set of eigefuctios calculated i the previous 3 chapters as the simplest eample, see Eq. (.76). Due to the uiqueess of epasio (37), the full set of coefficiets gives a complete descriptio of state (i a fied basis {u}), ust as the usual Cartesia compoets A, A y, ad A z give a complete descriptio of a usual geometric 3D vector A (i a fied referece frame). Still, let me emphasize some differeces betwee the quatum-mechaical bra- ad ket-vectors ad the usual geometric vectors: (i) a basis set may have a large or eve ifiite umber of states u, ad (ii) the epasio coefficiets may be comple. With these reservatios i mid, the aalogy with geometric vectors may be pushed eve further. Let us ier-multiply both parts of the first of Eqs. (37) by a bra-vector u ad the trasform the relatio usig the liearity rules discussed i the previous sectio, ad Eq. (38): u u u u u, (4.39) ' ' Together with Eq. (4), this meas that ay of the epasio coefficiets i Eq. (37) may be preseted as a ier product: u ' ' *, u ; (4.4) these relatios are aalogs of equalities A = A of the usual vector algebra. Usig these importat relatios (which we will use o umerous occasios), epasios (37) may be rewritte as Chapter 4 Page 9 of 4

173 u u, u u, (4.4) A compariso of these relatios with Eq. (6) shows that the outer product defied as u u, (4.4) Proectio operator is a legitimate liear operator. Such a operator, actig o ay state vector of the type (37), sigles out ust oe of its compoets, for eample, u u u, (4.43) i.e. kills all compoets of the liear superpositio but oe. I the geometric aalogy, such operator proects the state vector o its ( th ) directio, hece its ame the proectio operator. Probably, the most importat property of the proectio operators, called the closure (or completeess) relatio, immediately follows from Eq. (4): their sum over the full basis is equivalet to the idetity operator: u u I. (4.44) This meas i particular that we may isert the left-had part of Eq. (44) ito ay bra-ket relatio, at ay place the trick that we will use agai ad agai. Let us see how epasios (37) trasform all the otios itroduced i the last sectio, startig from the short bra-ket () (the ier product of two state vectors): * * u u. (4.45), ' ' ' Besides the comple cougatio, this epressio is similar to the scalar product of the usual vectors. Now, let us eplore the log bra-ket (3):, ', ' ' ' ' ' A * u A * u A. (4.46) Here, the last step uses a very importat otio of matri elemets of the operator, defied as A u A u, ' ' * ' ' '. (4.47) As evidet from Eq. (46), the full set of the matri elemets completely characterizes the operator, ust as the full set of epasio coefficiets (4) fully characterizes a quatum state. The term matri meas, first of all, that it is coveiet to preset the full set of A as a square table (matri), with the liear dimesio equal to the umber of basis states u of the system uder the cosideratio, i.e. the size of its Hilbert space. As two simplest eamples, all matri elemets of the ull-operator, defied by Eqs. (35), are evidetly equal to zero (i ay basis), ad hece it may be preseted as a matri of zeros (the ull matri):......, (4.48) Closure relatio Operator s matri elemets Null matri Chapter 4 Page of 4

174 Idetity matri Matri elemet of a operator product Log bracket as a matri product Short bracket as a matri product while for the idetity operator Î, defied by Eqs. (36), we readily get i.e. its matri (called the idetity matri) is diagoal also i ay basis: I ' u I u u u, (4.49) ' ' '... I.... (4.5) The coveiece of the matri laguage eteds well beyod the presetatio of particular operators. For eample, let us use defiitio (47) to calculate matri elemets for a product of two operators: AB u AB u. (4.5) ( ) " " Here we ca use Eq. (44) for the first (but ot the last!) time, isertig the idetity operator betwee the two operators, ad the epressig it via a sum of proectio operators: ( AB) " u AB u " u AIB u " u A u ' u ' B u " ' ' A ' B '". (4.5) This result correspods to the stadard row by colum rule of calculatio of a arbitrary elemet of the matri product A A... A B... B AB A B B (4.53)... Hece the product of operators may be preseted (i a fied basis!) by that of their matrices (i the same basis). This is so coveiet that the same laguage is ofte used to preset ot oly the log bracket, but eve the simpler short bracket: A A... * * * A A' ',,... A A..., (4.54) ' * * *,,..., (4.55)... although these equalities require the use of o-square matrices: rows of (comple-cougate!) epasio coefficiets for the presetatio of bra-vectors, ad colums of these coefficiets for the presetatio of ket-vectors. With that, the mappig of states ad operators o matrices becomes completely geeral. Now let us have a look at the outer product operator (6). Its matri elemets are ust Chapter 4 Page of 4

175 * u u '. (4.56) These are elemets of a very special square matri, whose fillig requires the kowledge of ust N scalars (where N is the basis set size), rather tha N scalars as for a arbitrary operator. However, a simple geeralizatio of such outer product may preset a arbitrary operator. Ideed, let us isert two idetity operators (44), with differet summatio idices, o both sides of ay operator: A IAI u u A u ' u ', ' (4.57) ad use the associative aiom to rewrite this epressio as A u u A u ' u '. (4.58), ' But the epressio i the middle log bracket is ust the matri elemet (47), so that we may write ' ' A u A ' u '. (4.59), ' The reader has to agree that this formula, which is a atural geeralizatio of Eq. (44), is etremely elegat. Also ote the followig parallel: if we cosider the matri elemet defiitio (47) as some sort of aalog of Eq. (4), the Eq. (59) is a similar aalog of the epasio epressed by Eq. (37). The matri presetatio is so coveiet that it makes sese to move it by oe level lower from state vector products to bare state vectors resultig from operator s actio upo a give state. For eample, let us use Eq. (59) to preset the ket-vector (8) as ' A u A ' u ' u, ', ' A ' u '. (4.6) Accordig to Eq. (4), the last short bracket is ust, so that ' u A A ' ' ' ' u (4.6), ' ' But epressio i middle paretheses is ust the coefficiet of epasio (37) of the resultig ketvector (6) i the same basis, so that ' A ' '. (4.6) ' This result correspods to the usual rule of multiplicatio of a matri by a colum, so that we may represet ay ket-vector by its colum matri, with the operator actio lookig like ' A ' A A A (4.63) Absolutely similarly, the operator actio o the bra-vector (), represeted by its row-matri, is Operator s epressio via its matri elemets Chapter 4 Page of 4

176 Hermitia cougate s matri elemets Operator s eigestates ad eigevalues Hermitia operator s eigevalues A A... A A.... (4.64) * * * * ', ',...,,... By the way, Eq. (64) aturally raises the followig questio: what are the elemets of the matri i its right-had part, or more eactly, what is the relatio betwee the matri elemets of a operator ad its Hermitia cougate? The simplest way to get a aswer is to use Eq. (5) with two arbitrary states (say, u ad u ) of the same basis i the role of ad. Together with the orthoormality relatio (38), this immediately gives 3 * A ' A '. (4.65) Thus, the matri of the Hermitia cougate operator is the comple cougated ad trasposed matri of the iitial operator. This result eposes very clearly the essece of the Hermitia cougatio. It also shows that for the Hermitia operators, defied by Eq. (), * A, (4.66) ' A ' i.e. ay pair of their matri elemets, symmetric about the mai diagoal, should be comple cougate of each other. As a corollary, the mai-diagoal elemets have to be real: * A A, i.e. Im A. (4.67) (Matri (5) evidetly satisfies Eq. (66), so that the idetity operator is Hermitia.) I order to fully appreciate the special role played by Hermitia operators i the quatum theory, let us itroduce the key otios of eigestates a (described by their eigevectors a ad a ) ad eigevalues (c-umbers) A of a operator Â, defied by the equatio they have to satisfy: 4 Let us prove that eigevalues of ay Hermitia operator are real, 5 A A a A a. (4.68) * A, for,,..., N, (4.69) 3 For the sake of formula compactess, below I will use the shorthad otatio i which the operads of this equality are ust A ad A*. I believe that it leaves little chace for cofusio, because the Hermitia cougatio sig may pertai oly to a operator (or its matri), while the comple cougatio sig * to a scalar say a matri elemet. 4 This equatio should look familiar to the reader see the statioary Schrödiger equatio (.6), which was the focus of our studies i the first three chapters. We will see soo that that equatio is ust a particular (coordiate) represetatio of Eq. (66) for the Hamiltoia as the operator of eergy. 5 The reciprocal statemet is also true: if all eigevalues of a operator are real, it is Hermitia (i ay basis). This statemet may be readily proved by applyig Eq. (93) below to the case whe A kk = A k kk, with A k * = A k. Chapter 4 Page 3 of 4

177 while the eigestates correspodig to differet eigevalues are orthogoal: a a ', if A A. (4.7) ' The proof of both statemets is surprisigly simple. Let us ier-multiply both sides of Eq. (68) by bra-vector a. I the right-had part of the result, the eigevalue A, as a c-umber, may be take out of the bra-ket, givig a ' A a A a a. (4.7) This equality should hold for ay pair of eigestates, so that we may swap the idices i Eq. (7), ad comple-cougate the result: ' * ' ' ' * * a A a A a a. (4.7) Now usig Eqs. (4) ad (5), together with the Hermitia operator defiitio (), we may trasform Eq. (7) to the followig form: Hermitia operator s eigevectors a ' A * a A a a. (4.73) ' ' Subtractig this equatio from Eq. (7), we get * A A ' a ' a. (4.74) There are two possibilities to satisfy this equatio. If idices ad are equal (deote the same eigestate), the the bra-ket is the state s orm squared, ad caot be equal to zero. The the left paretheses (with = ) have to be zero, i.e. Eq. (69) is valid. O the other had, if ad correspod to differet eigestates, the paretheses caot equal zero (we have ust proved that all A are real!), ad hece the state vectors ideed by ad should be orthogoal, e.g., Eq. (7) is valid. As will be discussed below, these properties make Hermitia operators suitable for the descriptio of physical observables Chage of basis ad matri diagoalizatio From the discussio of last sectio, it may look that the matri laguage is fully similar to, ad i may istaces more coveiet tha the geeral bra-ket formalism. I particular, Eqs. (5), (54), (55) show that ay part of ay bra-ket epressio may be directly mapped o the similar matri epressio, with the oly slight icoveiece of usig ot oly colums, but also rows (with their elemets comple-cougated), for state vector presetatio. I this cotet, why do we eed the bra-ket laguage at all? The aswer is that the elemets of the matrices deped o the particular choice of the basis set, very much like the Cartesia compoets of a usual vector deped o the particular choice of referece frame orietatio (Fig. 4), ad very frequetly it is coveiet to use two or more differet basis sets for the same system. With this motivatio, let us study what happes if we chage from oe basis, {u}, to aother oe, {v} - both full ad orthoormal. First of all, let us prove that for each such pair of bases, there eists such a operator U that, first, Chapter 4 Page 4 of 4

178 Basis trasform Uitary operator s defiitio v U, (4.75) u ad, secod, U U U U I. (4.76) (Due to the last property, 6 U is called a uitary operator, ad Eq. (75), a uitary trasformatio.) y' y y α y ' ' ' Fig Trasformatio of compoets of a D vector at a referece frame rotatio. Uitary operator of basis trasform Cougate uitary trasform operator A very simple proof of both statemets may be achieved by costructio. Ideed, let us take - a evidet geeralizatio of Eq. (44). The ' U v ' u, (4.77) ' ' ' ' ' U u v u u v v, (4.78) so that Eq. (75) has bee proved. Now, applyig Eq. (3) to each term of sum (77), we get so that, ' ' ' ' ' U u ' v, (4.79) U U v u u v v v v v. (4.8) ' ' But accordig to the closure relatio (44), the last epressio is ust the idetity operator, q.e.d. 7 (The proof of the secod equality i Eq. (76) is absolutely similar.) As a by-product of our proof, we have also got aother importat epressio (79). It implies, i, ' ' ' Reciprocal basis trasform particular, that while, accordig to Eq. (77), operator U performs the trasform from the old basis u to the ew basis v, its Hermitia adoit U performs the reciprocal uitary trasform: U v u u. (4.8) ' ' ' 6 A alterative way to epress Eq. (76) is to write U U, but I will try to avoid this laguage. 7 Quod erat demostradum (Lat.) what eeded to be proved. Chapter 4 Page 5 of 4

179 Now, let us see how do the matri elemets of the uitary trasform operators look like. Geerally, as was stated above, operator s elemets deped o the basis we calculate them i, so we should be careful - iitially. For eample, let us calculate the elemets i basis {u}: U ' i u u U u ' u vk uk u ' u v '. (4.8) k Now performig a similar calculatio i basis {v}, we get U ' i v v U v ' v vk uk v ' u v '. (4.83) k Surprisigly, the result is the same! This is of course true for the Hermitia cougate of the uitary trasform operator as well: U ' i u U v u. (4.84) ' i v These epressios may be used, first of all, to rewrite Eq. (75) i a more direct form. Applyig the first of Eqs. (4) to state v of the ew basis, we get Similarly, the reciprocal trasform is v u u u v ' ' ' U ' u. (4.85) v v u ' ' U ' v. (4.86) These equatios are very coveiet for applicatios; we will use them already later i this sectio. Net, we may use Eqs. (83), (84) to epress the effect of the uitary trasform o epasio coefficiets (37) of vectors of a arbitrary state. I the old basis {u}, they are give by Eq. (4). Similarly, i the ew basis {v}, i v v. (4.87) Agai isertig the idetity operator i the form of closure (44), with iteral ide, ad the usig Eq. (84), we get i v v u u ' ' v u ' u ' U ' u ' U ' ' ' ' ' ' The reciprocal trasform is (of course) performed by matri elemets of operator U : i u U ' ' i v ' i u. (4.88). (4.89) Both structurally ad philosophically, these epressios are similar to the trasformatio of compoets of a usual vector at coordiate frame rotatio. For eample, i two dimesios (Fig. 4): Basis trasforms: matri form Chapter 4 Page 6 of 4

180 Matri elemets trasforms Operator/ matri trace ' cos si. si cos (4.9) y' y (I this aalogy, the equality to of the determiat of the rotatio matri i Eq. (9) correspods to the uitary property (76) of the uitary trasform operators.) Please pay attetio here: while the trasform (75) from the old basis {u} to the ew basis {v} is performed by the uitary operator, the chage (88) of a state vectors compoets at this trasformatio requires its Hermitia cougate. Actually, this is also atural from the poit of view of the geometric aalog of the uitary trasform (Fig. 4): if the ew referece frame {, y } is obtaied by a couterclockwise rotatio of the old frame {, y} by some agle, for the observer rotatig with the frame, vector (which is itself uchaged) rotates clockwise. Due to the aalogy betwee epressios (88) ad (89) o oe had, ad our old fried Eq. (6) o the other had, it is temptig to skip idices i our ew results by writig i v U, U. (4.9) i u Sice matri elemets of U ad U do ot deped o basis, such laguage is ot too bad; still, the symbolic Eq. (9) should ot be cofused with geuie (basis-idepedet) bra-ket equalities. Now let us use the same trick of idetity operator isertio, repeated twice, to fid the trasformatio rule for matri elemets of a arbitrary operator: A' v v A v ' v uk uk A i uk' uk' v ' U k Akk' i uu ; (4.9) k'' k k' k, k' absolutely similarly, we ca get A' i u U k Akk' i vu k''. (4.93) k, k' I the spirit of Eq. (9), we may preset these results symbolically as well, i a compact bra-ket form: i u, A U A U A U A U. (4.94) i v As a saity check, let us apply this result to the idetity operator: I i v i u i u i u i v i v U IU U U I i u (4.95) i u - as it should be. Oe more ivariat of the basis chage is the trace of ay operator, defied as the sum of the diagoal terms of its matri i a certai basis: Tr A Tr A A. (4.96) The (easy) proof of this fact, usig the relatios we have already discussed, is left for reader s eercise. So far, I have implied that both state bases {u} ad {v} are kow, ad the atural questio is where does this iformatio comes from i quatum mechaics of actual physical systems. To get a partial aswer to this questio, let us retur to Eq. (68) that defies eigestates ad eigevalues of a Chapter 4 Page 7 of 4

181 operator. Let us assume that the eigestates a of a certai operator  form a full ad orthoormal set, ad fid the matri elemets of the operator i the basis of these states. For that, it is sufficiet to iermultiply both sides of Eq. (68), writte for ide, by the bra-vector of a arbitrary state a of the same set: a A a a A a. (4.97) ' ' ' The left-had part is ust the matri elemet A we are lookig for, while the right had part is ust A. As a result, we see that the matri is diagoal, with the diagoal cosistig of eigevalues: A. (4.98) ' A I particular, i the eigestate basis (but ot ecessarily i a arbitrary basis!), A meas the same as A. Thus the most importat problem of fidig the eigevalues ad eigestates of a operator is equivalet to the diagoalizatio of its matri, 8 i.e. fidig the basis i which the correspodig operator acquires the diagoal form (98); the the diagoal elemets are the eigevalues, ad the basis itself is the desirable set of eigestates. Let us modify the above calculatio by ier-multiplyig Eq. (68) by a bra-vector of a differet basis say, the oe, deoted {u}, i which we kow the matri elemets A. The multiplicatio gives u k k ' A a u A a. (4.99) I the left-had part we ca (as usual :-) isert the idetity operator, betwee the operator ad the ketvector, ad the use the closure relatio (44), while i the right-had part, we ca move the eigevalue A out of the bra-ket, ad the isert a summatio over a ew ide, compesatig it with the proper Kroecker delta symbol: u k k' k' k' k ' A u u a A u a. (4.) Movig out the sig of summatio over k, ad usig defiitio (47) of the matri elemets, we get k ' kk ' A kk ' uk ' a k ' kk' A. (4.) But the set of such equalities, for all N possible values of ide k, is ust a system of liear, homogeeous equatios for ukow c-umbers u k a. But accordig to Eqs. (8)-(84), these umbers are othig else tha the matri elemets U k of a uitary matri providig the required trasformatio from the iitial basis {u} to the basis {a} that diagoalizes matri A. The system may be preseted i the matri form: A A A... A A A U... U, (4.) Matri elemets i eigestate basis Operator diagoalizatio 8 Note that epressio matri diagoalizatio is a commo ad coveiet, but dagerous argo. (A matri is ust a matri, a ordered set of c-umbers, ad caot be diagoalized.) It is OK to use this argo if you remember clearly what it actually meas see the defiitio above. Chapter 4 Page 8 of 4

182 ad the usual coditio of its cosistecy, Characteristic equatio for fidig eigevalues A A A... A A A , (4.3) Pauli matrices plays the role of the characteristic equatio of the system. This equatio has N roots A,; pluggig each of them back ito system (), we ca use it to fid N matri elemets U k (k =,, N) correspodig to this particular eigevalue. However, sice equatios (3) are homogeeous, they allow fidig U k oly to a costat multiplier. I order to esure their ormalizatio, i.e. the uitary character of matri U, we may use the coditio that all eigevectors are ormalized (ust as the basis vectors are): a a a u u a U, (4.4) k k for each. This ormalizatio completes the diagoalizatio. 9 Now (at last!) I ca give the reader some eamples. As a simple but very importat case, let us diagoalize the operators described (i a certai -fuctio basis {u}) by the so-called Pauli matrices i σ, σ y, σ z. (4.5) i Though itroduced by a physicist, with a specific purpose to describe electro s spi, these matrices have a geeral mathematical sigificace, because together with the idetity matri I, they provide a full, liearly-idepedet basis - meaig that a arbitrary matri may be preseted as A A a I aσ a yσ y azσ z, A A (4.6) with a uique set of 4 coefficiets a. Let us start with diagoalizig matri. For it, the characteristic equatio (3) is evidetly A k A k, k (4.7) ad has two roots, A, = ±. (Agai, the umberig is arbitrary!) The reader may readily check that the eigevalues of matrices y ad z are similar. However, the eigevectors of the operators correspodig to all these matrices are differet. To fid them for, let us plug its first eigevalue, A = +, back ito equatios (), writte for this particular case: u u a a u u a a,. (4.8) 9 A possible slight complicatio here are degeerate cases whe characteristic equatio gives certai equal eigevalues correspodig to differet eigevectors. I this case the requiremet of the mutual orthogoality of these states should be additioally eforced. Chapter 4 Page 9 of 4

183 The equatios are compatible (of course, because the used eigevalue A = + satisfies the characteristic equatio), ad ay of them gives u u a i. e. U. (4.9) a, U With that, the ormalizatio coditio (4) yields U U. (4.) Although the ormalizatio is isesitive to the simultaeous multiplicatio of U ad U by the same phase factor ep{i} with ay real, it is coveiet to keep the coefficiets real, for eample takig =, i.e. to get U U. (4.) Performig a absolutely similar calculatio for the secod characteristic value, A = -, we get U = -U, ad we may choose the commo phase to get U U, (4.) so that the whole uitary matri for diagoalizatio of the operator correspodig to is U U, (4.3) For what follows, it will be coveiet to have this result epressed i the ket-relatio form see Eqs. (85)-(86): a U u U u u u, a U u U u u u, (4.4) u U a U a a a, u U a U a a a, (4.5) These results are already sufficiet to uderstad the Ster-Gerlach eperimets described i Sec. - with two additioal postulates. The first of them is that particle s iteractio with eteral magetic field may be described by the followig vector operator of the dipole magetic momet: m S, (4.6) where the coefficiet, specific for every particle type, is called the gyromagetic ratio, ad Ŝ is the vector operator of spi. For the so-called spi-½ particles (icludig the electro), this operator may be represeted, i the so-called z-basis, by the followig 3D vector of the Pauli matrices (5): Uitary matri diagoalizig Magetic momet operator Note that though this particular uitary matri is Hermitia, this is ot true for a arbitrary choice of phases. This is the key poit i the electro s spi descriptio, developed by W. Pauli i For a electro, with its egative charge q = -e, the gyromagetic ratio is egative: e = -g e e/m e, where g e is the dimesioless g-factor. Due to quatum electrodyamics effects, the factor is slightly higher tha : g e = ( + / + ).3934, where e /4 c /37 is the fie structure ( Sommerfeld ) costat. Chapter 4 Page of 4

184 Spi-½ matri Quatum measuremet potulate i S y y z z i y z z y σ σ σ, (4.7) ad,y,z are the usual Cartesia uit vectors i 3D space. (I the quatum-mechaics sese, they are ust c-umbers, or rather c-vectors.) The z-basis, i which Eq. (77) is valid, is defied as a orthoormal basis of two states, frequetly deoted a, i which the z-compoet of the vector operator of spi is diagoal, with eigevalues +/ ad -/. Note that we do ot uderstad what eactly these states are, 3 but loosely associate them with a certai iteral rotatio of the electro about z-ais, with either positive or egative agular mometum compoet S z. However, ay attempt to use such classical iterpretatio for quatitative predictios rus ito fudametal difficulties see Sec. 5.7 below. The secod ew postulate describes the geeral relatio betwee the bra-ket formalism ad eperimet. 4 Namely, i quatum mechaics, each real observable A is represeted by a Hermitia operator A A, ad a result of its measuremet i a quatum state, described by a liear superpositio of the eigestates a of the operator, a, with a, (4.8) may be oly oe of correspodig eigevalues A. 5 If state (8) ad all eigestates are ormalized to uity, the the probability of outcome A is 6 W, a a, (4.9) * a a, (4.) This relatio is evidetly a geeralizatio of Eq. (.) i wave mechaics. As a saity check, let us assume that the set of eigestates a is full, ad calculate the sum of all the probabilities: W a a I. (4.) Now returig to the Ster-Gerlach eperimet, coceptually the descriptio of the first (zorieted) eperimet show i Fig. is the hardest for us, because the statistical esemble describig the upolarized electro beam at its iput is mied ( icoheret ), ad caot be described by a pure 3 If you thik about it, word uderstad typically meas that we ca eplai a ew, more comple otio i terms of those discussed earlier ad cosidered kow. I our eample, we caot epress the spi states by some wavefuctio (r), or ay other mathematical otio discussed earlier. The bra-ket formalism has bee iveted eactly to eable mathematical aalysis of such ew quatum states. 4 Here agai, ust like i Sec.., the statemet implies the abstract (mathematical) otio of ideal eperimets, postpoig the discussio of real (physical) measuremets util Sec As a remider, i the ed of Sec. 3 we have already proved that such eigestates correspodig to differet A are orthogoal. If ay of these values is degeerate, i.e. correspods to several differet eigestates, they should be also selected orthogoal, i order for Eq. (8) to be valid. 6 This key relatio, i particular, eplais the most commo term for the (geerally, comple) coefficiets, the probability amplitudes. Chapter 4 Page of 4

185 ( coheret ) superpositio of the type (6) that have bee the subect of our studies so far. (We will discuss the mied esembles i Chapter 7.) However, it is ituitively clear that its results, ad i particular Eq. (6), are compatible with the descriptio of its two output beams as sets of electros i pure states ad, respectively. The absorber followig that first stage (Fig. ) ust takes all spi-dow electros out of the picture, producig a output beam of polarized electros i a pure state. For such beam, probabilities () are W = ad W =. This is certaily compatible with the result of the cotrol eperimet show o the bottom pael of Fig. : the repeated SG (z) stage does ot split such a beam, keepig the probabilities the same. Now let us discuss the double Ster-Gerlach eperimet show o the top pael of Fig.. For that, let us preset the z-polarized beam i aother basis of two states (I will deote them as ad ) i which, by defiitio, the matri of operator Ŝ is diagoal. But this is eactly the set we called a, i the matri diagoalizatio problem solved above. O the other had, states ad are eactly what we called u, i that problem, because i this basis, matrices z ad hece S z are diagoal. Hece, i applicatio to the electro spi problem, we may rewrite Eqs. (4)-(5) as,, (4.),, (4.3) Curretly, for us the first of Eqs. (3) is most importat, because it shows that the quatum state of electros eterig the SG () stage may be preseted as a coheret superpositio of electros with S = +/ ad S = -/. Notice that the beams have equal probability amplitude moduli, so that accordig to Eq. (), the split beams ad have equal itesities, i accordace with eperimet. (The mius sig before the secod ket-vector is of o cosequece here, though it may have a impact o outcome of other eperimets for eample if the ad beams are brought together agai.) Now, let us discuss the most mysterious (from the classical poit of view) multi-stage SG eperimet show o the middle pael of Fig.. After the secod absorber has take out all electros i, say, the state, the remaiig electros i state are passed to the fial, SG (z), stage. But accordig to the first of Eqs. (), this state may be preseted as a (coheret) liear superpositio of the ad states, with equal amplitudes. The stage separates these two states ito separate beams, with equal probabilities W = W = ½ to fid a electro i each of them, thus eplaiig the eperimetal results. To coclude our discussio of the multistage Ster-Gerlach eperimet, let me ote that though it caot be eplaied i terms of wave mechaics (which operates with scalar de Broglie waves), it has a aalogy i classical theories of vector fields, such as the classical electrodyamics. Let a plae electromagetic wave propagate perpedicular to the plae of drawig, ad pass through liear polarizer. Similarly to the iitial SG (z) stages (icludig the followig absorbers) show i Fig., the polarizer produces a wave liearly polarized i oe directio the vertical directio i Fig. 3. Its electric field vector has o horizotal compoet, as may be revealed by wave s full absorptio i a perpedicular polarizer 3. However, let us pass the wave through polarizer first. I this case, the output wave does acquire a horizotal compoet, as ca be, agai, revealed by passig it through polarizer 3. If agles betwee polarizatio directio ad, ad betwee ad 3, are both equal /4, each polarizer reduces the wave amplitude by a factor of, ad hece itesity by a factor of, eactly like i the multistage Relatio betwee eigevectors of operators S ad S z Chapter 4 Page of 4

186 SG eperimet, with polarizer playig the role of the SG () stage. The oly differece is that the ecessary agle is /4, rather tha by / for the Ster-Gerlach eperimet. I quatum electrodyamics (see Chapter 9 below), which cofirms the classical predictios for this eperimet, this differece is eplaied by that betwee the iteger spi of the electromagetic field quata, photos, ad the halfiteger spi of electros. 3 Fig Light polarizatio sequece similar to the 3-stage Ster-Gerlach eperimet show o the middle pael of Fig.. Epectatio value as a log bracket 4.5. Observables: Epectatio values ad ucertaities After this particular (ad hopefully epirig) eample, let us discuss the geeral relatio betwee the Dirac formalism ad eperimet i more detail. The epectatio value of a observable over ay statistical esemble (ot ecessarily coheret) may be always calculated usig the geeral rule (.37). For the particular case of a coheret superpositio (8), we ca combie that defiitio with Eq. () ad the secod of Eqs. (8), ad the use Eqs. (59) ad (98) to write A A W * A a A a a a A a a. (4.4) Now usig the completeess relatio (44) twice, with idices ad, we arrive at a very simple ad importat formula 7 A, ' A. (4.5) This is a clear aalog of the wave-mechaics formula (.3) ad as we will see i the et chapter, may be used to derive it. A huge advatage of Eq. (5) is that it does ot eplicitly ivolve the eigevector set of the correspodig operator, ad allows the calculatio to be performed i ay coveiet basis. 8 For eample, let us cosider a arbitrary state of spi-½, ad calculate the epectatio values of its compoets. The calculatios are easiest i the z-basis, because we kow the operators of the compoets i that basis see Eq. (7). Represetig the ket- ad bra-vectors of our state as liear superpositios of vectors of the basis states ad, * *. (4.6), ' ' 7 This equality reveals the full beauty of Dirac s otatio. Ideed, iitially the quatum-mechaical brackets ust remided the agular brackets used for statistical averagig. Now we see that i this particular (but most importat) case, the agular brackets of these two types may be ideed equal to each other! 8 Note that Eq. () may be rewritte i the form similar to Eq. (5): W, where a a is the operator (4) of proectio upo the th eigestate a. Chapter 4 Page 3 of 4

187 Chapter 4 Page 4 of 4 ad pluggig these epressios to Eq. (5) writte for observable S z, we get. * * * * * * z z z z z z S S S S S S (4.7) Now there are two equivalet ways (both very simple :-) to calculate the bra-kets i this epressio. The first oe is to represet each of them i the matri form i the z-basis, i which bra- ad ket-vectors of states ad are, respectively, matri-rows (, ) ad (, ), or the similar matricolums. Aother (perhaps more elegat) way is to use the geeral Eq. (59), for the z-basis, to write,, z y S i S S. (4.8) For our particular calculatio, we may plug the last of these epressios ito Eq. (7), ad to use the orthoormality coditios (9):,. (4.9) Both calculatios give (of course) the same result: * * z S. (4.3) This particular result might be also obtaied usig Eq. () for probabilities W = * ad W = *: * * W W S z. (4.3) The formal way (7), based o usig Eq. (5), has, however, a advatage of beig applicable, without ay chage, to fidig the observables whose operators are ot diagoal i the z-basis, as well. I particular, absolutely similar calculatios give, * * * * * * S S S S S (4.3), * * * * * * i S S S S S y y y y y (4.33) Similarly, we ca epress, via the same coefficiets ad, the r.m.s. fluctuatios of all spi compoets. For eample, let us have a good look at the spi state. Accordig to Eq. (6), i this state = ad =, so that Eqs. (3)-(33) yield:, y z S S S. (4.34) Now let us use the same Eq. (5) to calculate the spi compoet ucertaities. Accordig to Eqs. (5) ad (7), operators of spi compoet squared are equal to (/) Î, so that the geeral Eq. (.33) yields Spi-½ compoet operators

188 z z z z (4.35a) S S S S I, S S S S I, (4.35b) S y S y S y S y I. (4.35c) While Eqs. (34) ad (35a) are compatible with the classical otio of the spi beig defiitely i the state, this correspodece should ot be overstretched to the iterpretatio of this state as a certai (z) orietatio of electro s magetic momet m, because such classical picture caot eplai Eqs. (35b) ad (35c). The best (but still imprecise!) classical image I ca offer is the magetic momet m orieted, o the average, i the z-directio, but still havig - ad y-compoets strogly wobblig about their zero average values. It is straightforward to verify that i the -polarized ad y-polarized states the situatio is similar, with the correspodig chage of idices. Thus, i either state may all 3 compoets of the spi have eact values. Let me show that this is ot ust a occasioal fact, but reflects the most profoud property of quatum mechaics, the ucertaity relatios. Cosider observables, A ad B, that may be measured i the same quatum state. There are two possibilities here. If operators correspodig to the observables commute,, B A, (4.36) the all the matri elemets of the commutator i ay orthogoal basis (i particular, i the basis of eigestates a of operator A) are also zero. From here, we get A, B a a AB a a BA a a. (4.37) ' I the first bra-ket of the middle epressio, let us act by operator  o the bra-vector, while i the secod oe, o the ket-vector. Accordig to Eq. (68), such actio turs operators ito the correspodig eigevalues, so that we get ' A a B a ' A' a B a ' A A' a B a '. (4.38) This meas that if eigestates of operator  are o-degeerate (i.e. A A if ), the matri of operator B has to be diagoal i basis a, i.e., the eigestate sets of operators  ad B coicide. Such pairs of observables, that share their eigestates, are called compatible. For eample, i wave mechaics of a particle, mometum (.6) ad the kietic eergy (.7) are compatible, sharig eigefuctios (.9). Now we see that this is ot occasioal, because each Cartesia compoet of the kietic eergy is proportioal to the square of the correspodig compoet of the mometum, ad ay operator commutes with a arbitrary power of itself: A, A A, AA... A AAA... A AA... A A '. (4.39) Chapter 4 Page 5 of 4

189 Now, what if operators  ad B do ot commute? The the followig geeral ucertaity relatio is valid: 9 AB A, B. (4.4) The proof of Eq. (4) may be divided ito two steps, the first of which proves the so-called Schwartz iequality: 3. (4.4) The proof may be started by usig postulate (6) - that the orm of ay legitimate state of the system caot be egative. Let us apply this postulate to the state with the followig ket-vector:, (4.4) Geeral ucertaity relatio Schwartz iequality where ad are possible, o-ull states of the system, so that the deomiator i Eq. (4) is ot equal to zero. For this case, Eq. (6) gives. (4.43) Opeig the paretheses, we get. (4.44) After the cacellatio of oe ier product i the omiator ad deomiator of the last term, it cacels with the rd (or 3 rd ) term, provig the Schwartz iequality (4). Now let us apply this iequality to states  ~ ad B~, (4.45) where, i both relatios, is the same (but otherwise arbitrary) possible state of the system, ad the deviatios operators are defied similarly to observable deviatios (see Sec..), for eample, ~ A A A. (4.46) With this substitutio, ad takig ito accout that the observable operators  ad B are Hermitia, Eq. (4) yields ~ ~ ~ ~ A B AB. (4.47) 9 Note that both sides of Eq. (4) are state-specific; the ucertaity relatio statemet is that this iequality should be valid for ay possible quatum state of the system. 3 This iequality is the quatum-mechaical aalog of the usual vector algebra result. Chapter 4 Page 6 of 4

190 Commutatio relatio for spi-/ compoet operators Sice state is arbitrary, we may use Eq. (5) to rewrite this relatio as a operator iequality: ~ ~ AB AB. (4.48) Actually, this is already a ucertaity relatio, eve better (stroger) tha its stadard form (4); moreover, it is more coveiet i some cases. I order to proceed to Eq. (4), we eed a couple more steps. First, let us otice that the operator product i Eq. (48) may be recast as ~ ~ ~ ~ i ~ ~ AB A, B C, where C i A, B. (4.49) Ay aticommutator of Hermitia operators, icludig that i Eq. (49), is a Hermitia operator, ad its eigevalues are purely real, so that its epectatio value (i ay state) is also purely real. O the other had, the commutator part of Eq. (49) is ust C ~ i A ~, B i A A B B ib B A A i AB BA i A, B. (4.5) Secod, accordig to Eqs. (5) ad (65), the Hermitia cougate of ay product of Hermitia operators  ad B is ust the product of swapped operators. Usig the fact, we may write C i A B i AB, ( ) i ( BA ) iba iab ia, B C, (4.5) so that operator Ĉ is also Hermitia, i.e. its eigevalues are also real, ad thus its average is purely real as well. As a result, the square of the average of the operator product (49) may be preseted as ~ ~ ~ ~ AB A, B C. (4.5) Sice the first term i the right-had part of this equality caot be egative,, A B ~ ~ i AB C, (4.53) ad we ca cotiue Eq. (48) as ~ ~ AB AB A, B, (4.54) thus provig Eq. (4). For the particular case of operators ad p (or a similar pair of operators for aother Cartesia coordiate), we ca readily combie Eq. (4) with Eq. (.4b) ad to prove the origial Heiseberg s ucertaity relatio (.3). For the spi-/ operators defied by Eq. (7), it is straightforward (ad highly recommeded to the reader) to show that S, S is, y z (4.55) with similar relatios for other pairs of idices take i the correct order (from to y to z to, etc.). As a result, the ucertaity relatios (4) for spi-/ particles, otably icludig electros, are Chapter 4 Page 7 of 4

191 S S y S z, etc. (4.56) I particular, i the state, the right-had part of this relatio equals (/), ad either of the ucertaities S, S y ca equal zero. As a remider, our direct calculatio earlier i this sectio has show that each of these ucertaities is equal to /, i.e. their product equals to the lowest value allowed by the ucertaity relatio (56). I this aspect, the spi-polarized states are similar to the Gaussia wave packets studied i Sec... Ucertaity relatios for spi-/ compoets 4.6. Quatum dyamics: Three pictures So far i this chapter, I shied away from the discussio of system dyamics, implyig that the bra- ad ket-vectors of the system are their sapshots at a certai istat t. Now we are sufficietly prepared to eamie their time depedece. Oe of the most beautiful features of quatum mechaics is that the time evolutio may be described usig either of three alterative pictures, givig eactly the same fial results for epectatio values of all observables. From the stadpoit of our wave mechaics eperiece, the Schrödiger picture is the most atural. I this picture, the operators correspodig to time-idepedet observables (e.g., to the Hamiltoia fuctio H of a isolated system) are also costat, while the bra- ad ket-vectors of the quatum state of the system evolve i time as t) ( t ) u ( t, t ), ( t) u( t, t ) ( t ), (4.57) ( where u ( t, t ) is the time-evolutio operator, which obeys the followig differetial equatio: i u H u, (4.58) where Ĥ is the Hamiltoia operator of the system (that is always Hermitia, H H ), ad the dot meas the differetiatio is over argumet t, but ot t. While this equatio is a very atural replacemet of the wave-mechaical equatio (.5), ad is also frequetly called the Schrödiger equatio, 3 it still should be cosidered as a ew, more geeral postulate, which fids its fial ustificatio (as it is usual i physics) i the agreemet betwee its corollaries with eperimet - more eactly, i havig ot a sigle credible cotradictio with eperimet. Startig the discussio of Eqs. (57)-(58), let us first cosider the case of a system described by a time-idepedet Hamiltoia, whose eigestates a ad eigevalues E obey Eq. (68), 3 H a E a, (4.59) ad hece are also time-idepedet. (Similarly to the wavefuctios defied by Eq. (.6), a are called the statioary states of the system.) Let us use Eqs. (57)-(59) to calculate the law of time evolutio of the epasio coefficiets, defied by Eq. (8), i the statioary state basis: Schrödiger equatio of operator evolutio 3 Moreover, we will be able to derive Eq. (.5) from Eq. (54) see Sec Here I itetioally use ide rather tha to emphasize the special role played by the special role of the statioary eigestates a i quatum dyamics. Chapter 4 Page 8 of 4

192 Time evolutio of statioary states Pauli Hamiltoia: operator Pauli Hamiltoia: z-basis matri ( t) d dt a a ( t) Hu ( t, t i d dt a ) ( t ) u( t, t E i ) ( t a ) u( t, t a ) ( t u( t, t ) ) ( t E i ) a ( t) i E. (4.6) This is the same simple equatio as Eq. (.59), ad its itegratio yields a similar result cf. Eq. (.6), ust with the iitial time t rather tha : i ( t) ( t ) ep E t t. (4.6) I order to illustrate how does this result work, let us cosider spi-½ dyamics i a timeidepedet, uiform eteral magetic field B, takig its directio for ais z. To costruct the system s Hamiltoia, we may apply the correspodece priciple to the classical epressio for the eergy of a magetic momet m i the eteral magetic field B, 33 U m B. (4.6) I quatum mechaics, the operator correspodig to the momet m is give by Eq. (6) (suggested by W. Pauli), so that the spi-field iteractio is described by the so-called Pauli Hamiltoia: H m B S B BS z, (4.63) where Ŝ z is the operator of the z-compoet of electro s spi. Accordig to Eq. (7), i the z-basis of states ad, the matri of operator (63) is B H σ z Ω σ z, with Ω B. (4.64) The costat so defied coicides with the classical frequecy of the precessio of a symmetric top, with a agular mometum S ad magetic momet m = S, about ais z, iduced by eteral torque = mb: 34 mb Ω B. (4.65a) S S For a electro, with its egative gyromagetic ratio e = -g e e/m e, eglectig the mior differece betwee factors g e ad, we get e B, (4.65b) i.e. the frequecy s magitude coicides with that of the cyclotro frequecy c see Eq. (3.48). I order to apply the geeral Eq. (6), at this stage we would eed to fid the eigestates a ad eigeeergies E of our Hamiltoia. However, with our (smart :-) choice of the directio of ais z, the Hamiltoia matri is already diagoal: m e 33 See, e.g., EM Eq. (5.). As a remider, we have already used this epressio for the derivatio of Eq. (3). 34 See, e.g., CM Sec. 6.5, i particular Eq. (6.7), ad EM Sec. 5.5, i particular Eq. (5.4) ad its discussio. Chapter 4 Page 9 of 4

193 H σ z, (4.66) meaig that ad are the eigestates of the system, with eigeeergies, respectively, E ad E. (4.67) (Note that their differece, ΔE E E Ω B, (4.68) correspods to the classical eergy mb of flippig the magetic dipole with momet m = /, orieted alog the directio of field B. 35 ) With that, Eq. (6) immediately yields followig epressios for the time evolutio of the epasio coefficiets: i i ( t) ( t )ep t t, ( t) ( t )ep t t, (4.69) allowig a ready calculatio of time evolutio of the epectatio values of ay observable. I particular, we ca calculate the epectatio value of S z as a fuctio of time by applyig Eq. (3) to a arbitrary time momet t: S * * * * ( t) ( t) ( t) ( t) ( t) () () () () z S z (). (4.7) Thus the epectatio value of the spi compoet parallel to the applied magetic field remais costat, regardless of the iitial state of the system. However, this is ot true for the compoets perpedicular to the field. For eample, Eq. (3), applied to momet t, gives S ( t) * * * t t t t e * i( tt ) i( tt ) e. (4.7) Clearly, this epressio describes siusoidal oscillatios with frequecy (65). The amplitude ad phase of these oscillatios deped o iitial coditios. Ideed, solvig Eqs. (3)-(33) for the epasio coefficiet products, we get relatios * * t t S t i S t, t t S t i S t (4.7) y y Spi-½ i magetic field: eigeeergies valid for ay time t. Pluggig their values for t = ito Eq. (7), we get S ( t) S i tt ) S i S e S i S y cost S si t. y ( i( tt) y e (4.73) A absolutely similar calculatio usig Eq. (33) gives 35 Note also that if the product B is positive, so is, so that E is egative, while E is positive. This is i the correspodece with the classical picture of a magetic dipole m havig egative potetial eergy whe it is aliged with the eteral magetic field B see Eq. (6). Chapter 4 Page 3 of 4

194 S y cost S si. ( t) S t (4.74) y These formulas show, for eample, if at momet t = the spi s state was, i.e. S () = S y () =, the the amplitude of oscillatio of the both lateral compoet of spi vaishes. O the other had, if the spi was iitially i state, i.e. had the defiite, maimum possible value of S, equal to / (i classics, we would say the spi / was orieted i directio ), the both epectatio values S ad S y oscillate i time 36 with this amplitude, with the phase shift / betwee them. These formulas may be iterpreted as the torque-iduced precessio of the Cartesia compoets of the spi vector of legth S = /, cofied i plae [, y], with classical frequecy = B about ais z (couterclockwise if B > ). Thus, the gyromagetic ratio is ust the agular frequecy of the torque-iduced precessio of spi (about field s directio) per uit magetic field; for electros, e.76 s - T - ; for protos, the ratio is much smaller because of their larger mass: p s - T -, ad for larger spi-½ uclei, may be much smaller still e.g., s - T - for the 57 Fe ucleus. 37 Note, however, that this classical laguage does ot describe large quatum-mechaical ucertaities of these observables, which are abset i the classical picture of the precessio at least whe it starts from a defiite orietatio of the agular mometum vector. Now let us retur to the discussio of the geeral Schrödiger equatio (58) ad prove the followig fasciatig fact: it is possible to write the geeral solutio of this operator equatio. I the easiest case whe the Hamiltoia is time-idepedet, this solutio is a eact aalog of Eq. (6), i (, ) (, )ep i u t t u t t H t t Iep H t t. (4.75) To start its proof we should, first of all, uderstad what does a fuctio (i this case, the epoet) of a operator mea. I the operator (ad matri) algebra, such fuctios are defied by their Taylor epasios; i particular, Eq. (75) meas that u ( t, t ) I I k! i Ht k! i H k i! i 3! ( ) 3 3 t t H t t H ( t t )..., 3 (4.76) where, 3 H HH H HHH, etc. Workig with such series of operator products is ot as hard as oe could imagie, due to their regular structure. For eample, let us differetiate Eq. (76) over t: 36 This is oe more (hopefully, redudat :-) illustratio of the differece betwee averagig over the statistical esemble ad over time: i Eqs. (7), (73)-(74), ad quite a few relatios below, oly the former averagig has bee performed, so the results are still fuctios of time. 37 Such composite particles as uclei (ad, from the poit of view of high-eergy physics, eve such hadros as protos) may be characterized by a certai et spi (ad hece by certai ) oly if durig the cosidered process their iteral degrees of freedom remai i a certai (usually, groud) quatum state. Chapter 4 Page 3 of 4

195 u ( t, t )! i H i H I!! i i H H ( t t ) i! 3! H 3( t t... t t H ( t t )... Hu ( t, t ), i 3 ) i (4.77) so that the differetial equatio (58) is ideed satisfied. O the other had, Eq. (75) also satisfies the iitial coditio u ( t, ) t u ( t, t ) I, (4.78) which immediately follows from the defiitio (57) of the evolutio operator, so it is ideed the (uique) solutio for the time evolutio operator i the Schrödiger picture. Now let us allow operator Ĥ to be a fuctio of time, but with the coditio that its values (i fact, operators) at differet istats commute with each other: H ( t' ), H ( t" ), for ay t', t". (4.79) (A importat eample of such a Hamiltoia is that of a particle uder the effect of a classical, timedepedet force F(t): H F F( t) r. (4.8) Ideed, the radius-vector operator r does ot deped eplicitly o time ad hece commutes with itself, as well as with c-umbers F(t ) ad F(t ).) I this case it is sufficiet to replace, i all above formulas, product H ( t ) with the correspodig itegral over time; i particular, Eq. (75) is geeralized as t t i (, ) ep u t t I H ( t' ) dt'. t This replacemet meas that the first form of Eq. (76) should be replaced with (4.8) Evolutio operator: eplicit epressio k k t k t t t i (, ) ( ) i u t t... ( ) ( )... I H t' dt' I dt dt dtk H t H t H ( tk ). k k! (4.8) t k t t t The proof that the first form of Eq. (8) satisfies Eq. (58) is absolutely similar to the oe carried out above. We may ow use Eq. (8) to show that the time-evolutio operator is uitary at ay momet, eve for the time-depedet Hamiltoia. Ideed, from that formula, t t i i u ( t, t ) u t t I H t' dt' I H (, ) ep ( ) ep ( t" ) dt". (4.83) t t Sice each of the epoets may be preseted with the Taylor series (8), ad, thaks to Eq. (79), differet compoets of these sums may be swapped at will, epressio (83) may be maipulated eactly as the product of c-umber epoets, i particular rewritte it as Chapter 4 Page 3 of 4

196 u ( t, t ) u ( t, t i ) Iep t t H ( t' ) dt' t t H ( t" ) dt" Iep{} I. This property esures, i particular, that the system state s ormalizatio does ot deped o time: (4.84) ( t) ( t) ( t ) ( ) u t,t u ( t,t ) ( t ) ( t ) ( t ). (4.85) The most difficult cases for the eplicit solutio of Eq. (58) are those whe Eq. (79) is violated. 38 It may be prove that i these cases the itegral limits i the last form of Eq. (8) should be trucated, givig the so-called Dyso series k t k i u ( t, t )... ( ) ( )... I dtdt dtk H t H t H ( tk ). k t t t t t (4.86) Sice we would ot have time to use this relatio i our course, I will skip its proof. 39 Let me ow retur to the geeral discussio of quatum dyamics to outlie its alterative, Heiseberg picture. For that, let us recall that accordig to Eq. (5), i quatum mechaics the epectatio value of ay observable A is a log bra-ket. Below we will see that other quatities (say, the rates of quatum trasitios betwee pairs of differet states, say ad ) may also be measured i eperimet; the most geeral form for all such measurable quatities is the followig log bracket: Â. (4.87) As has bee discussed above, i the Schrödiger picture the bra- ad ket-vectors of the states are timedepedet, while the variable operators stay costat (if the correspodig variables do ot eplicitly deped o time), so that Eq. (87), applied to momet t, may be preseted as ( t) A S ( t), (4.88) where ide S emphasizes the Schrödiger picture. Let us apply to the bra- ad ket-vectors i this epressio the evolutio law (57): A ( t ) u ( t, t ) A u ( t, t ) ( t ). (4.89) This equality meas that if we form a log bracket with bra- ad ket-vectors of the iitial-time states, together with the followig time-depedet Heiseberg operator 4 S Heiseberg operator Log bracket i the Heiseberg picture A H ( t) u ( t, t ) A u ( t, t ) u ( t, t ) A ( t ) u ( t, t ), (4.9) all eperimetally measurable results will remai the same as i the Schrödiger picture: S H A ( t ) A ( t, t ) ( ). (4.9) H t 38 We will ru ito such situatios i Chapter 7, but will ot eed to apply Eq. (86). 39 It may be foud, for eample, i Chapter 5 of J. Sakurai s tetbook see Refereces. 4 Note this relatio is similar i structure to the symbolic Eqs. (94). Chapter 4 Page 33 of 4

197 Chapter 4 Page 34 of 4 Let us see how does the Heiseberg picture work for the same simple (but very importat!) problem of the spi-½ precessio i a z-orieted magetic field, described (i the z-basis) by the Hamiltoia matri (64). I that basis, Eq. (58) for the time-evolutio operator reads u u u u u u u u u u u u i. (4.9) We see that i this simple case the equatios for differet matri elemets of the evolutio operator matri are decoupled, ad readily solvable, usig the uiversal iitial coditio (78): 4. si σ Icos,) u( / / t i t e e t z t i t i (4.93) Now we ca use Eq. (9) to fid the Heiseberg-picture operators of spi compoets. Droppig ide H for brevity (the Heiseberg-picture operators are clearly marked by their depedece o time ayway), we get. () si S () cos S si σ cos σ,) u(,)σ ( u,) ()u(,)s ( u ) ( S y / / / / t t t t e e e e e e t t t t t y t i t i t i t i t i t i (4.94) Absolutely similar calculatios of the other spi compoets yield t t t t ie ie t y y y t i t i () si S () cos S si σ cos σ ) ( S, (4.95). () σ ) ( S z z z S t (4.96) A practical advatage of these formulas is that they describe system s evolutio for arbitrary iitial coditios, thus makig the aalysis of the iitial state effects very simple. Ideed, sice i the Heiseberg picture the epectatio values of observables are calculated usig Eq. (9) (with = ), with time-idepedet bra- ad ket vectors, such averagig of Eqs. (94)-(96) immediately returs us to Eqs. (7), (73), ad (74), obtaied i the Schrödiger picture. Moreover, these equatios for the Heiseberg operators formally coicide with the classical equatios of the torque-iduced precessio for c-umber variables. (I the et chapter, we will see that the same eact mappig is valid for the Heiseberg picture of the orbital motio.) 4 We could of course use this equatio result, together with Eq. (57), to obtai all the above results for this system withi the Schrödiger picture. I our simple case, the use of Eqs. (6) for this purpose was more straightforward, but i some cases (e.g., for time-depedet Hamiltoias) a eplicit calculatio of the timeevolutio matri may be the oly practicable way to proceed.

198 Heiseberg equatio of motio I order to see that the last fact is by o meas a coicidece, let us combie Eqs. (58) ad (9) to form a eplicit differetial equatio of the Heiseberg operator evolutio. For that, let us differetiate Eq. (9) over time: d dt A H u AS u A Su u u u AS. (4.97) t t t Pluggig i the derivatives of the time evolutio operator from Eq. (58) ad its Hermitia cougate, ad multiplyig both parts of the equatio by i, we get d AS i A H u HASu u u u AS Hu. (4.98a) dt t If for the Schrödiger-picture Hamiltoia the coditio similar to Eq. (79) is satisfied, the, accordig to Eqs. (77) or (8), the Hamiltoia commutes with the time evolutio operator ad its Hermitia cougate, ad may be swapped with ay of them. 4 Hece, we may rewrite Eq. (98a) as i d dt A H A A Hu A S u i u u u A u H S i u S S t t u u A u, H Now usig the defiitio (9) agai, for both terms i the right-had part, we may write S. (4.98b) d A i A i A H H H H, dt t. (4.99) This is the so-called Heiseberg equatio of motio. 43 Let us see how does this equatio look for the same problem of spi-½ precessio i a z- orieted, time-idepedet magetic field, described i the z-basis by the Hamiltoia matri (64), which does ot deped o time. I this basis, Eq. (99) for the vector operator of spi reads 44 S S Ω S S S i, Ω. S S S - S S (4.) Oce agai, the equatios for differet matri elemets are decoupled, ad their solutio is elemetary: S S t S() cost, S t S () it it t S () e, S t S () e. cost, (4.) 4 Due to the same reaso, H H u H Su u uh S H S ; this is why the ide of the Hamiltoia operator may be dropped i Eqs. (98)-(99). 43 Reportedly, this equatio was derived by P. A. M. Dirac, who was so geerous that he himself gave the ame of his colleague to this key result, because Heiseberg was sayig somethig like this. 44 Usig commutatio relatios (55), this equatio may be readily geeralized to the case of arbitrary magetic field B(t) ad arbitrary state basis the eercise highly recommeded to the reader. Chapter 4 Page 35 of 4

199 Accordig to Eq. (9), the iitial values of the Heiseberg-picture matri elemets are ust the Schrödiger-picture oes, so that usig Eq. (7) we may rewrite this solutio i either of two forms: S t) it e it e it ie it ie ( y z z it e it e, z with i y. (4.) The simplicity of the last epressio is spectacular. (Remember, it covers ay iitial coditios, ad all 3 spatial compoets of spi!) O the other had, for some purposes the former epressio may be more coveiet; i particular, its Cartesia compoets immediately give our earlier results (94)- (96). Oe of advatages is that the Heiseberg picture is that it provides a more clear lik betwee the classical ad quatum mechaics. Ideed, aalytical classical mechaics may be used to derive the followig equatio of time evolutio of a arbitrary fuctio A(q, p, t) of geeralized coordiates ad mometa of the system, ad time: 45 da A A, H, (4.3) dt t where H is the classical Hamiltoia fuctio of the system, ad {..,..} is the so-called Poisso bracket defied, for two arbitrary fuctios A(q, p, t) ad B(q, p, t), as A B A B A,. (4.4) B p q q p Comparig Eq. (3) with Eq. (99), we see that the correspodece betwee the classical ad quatum mechaics (i the Heiseberg picture) is provided by the followig symbolic relatio 46 i A, B A, B. (4.5) Poisso bracket Classical vs. quatum mechaics 45 See, e.g., CM Eq. (.7). Also, please ecuse my use, for the Poisso bracket, the same (traditioal) symbol {, } as for the aticommutator. We will ot ru ito the Poisso brackets agai i the course, leavig very little chace for cofusio. 46 Sice we have ru ito the commutator of Heiseberg-picture operators, let me ote emphasize agai that the values of such a operator at differet momets of time ofte do ot commute. Perhaps the simplest eample is the operator of coordiate of a free D particle, with Hamiltoia H p / m. Ideed, i this case Eq. (99) yields equatios i, H ip / m ad i p p, H, with simple solutios (similar to those for classical motio of the correspodig observables): p t cost p, t p t / m, so that, t, p t / m, p t / m i t / m, if t. S S Chapter 4 Page 36 of 4

200 Iteractio evolutio operator This relatio may be used, i particular, for fidig appropriate operators for system s observables, if their form is ot immediately evidet from the correspodece priciple. We will develop this argumetatio further i the et chapter where we revisit the wave mechaics, ad also i Chapter 9. Fially, let us discuss oe more alterative picture of quatum dyamics. It is also attributed to P. A. M. Dirac, ad is called either the Dirac picture, or (more frequetly) the iteractio picture. The last ame stems from the fact that this picture is very useful for the perturbative (approimate) approaches to systems whose Hamiltoias may be partitioed ito two parts, H H H it, (4.6) where Ĥ is the sum of relatively simple Hamiltoias of o-iteractig compoet sub-systems, while their secod term i Eq. (6) represets their weak iteractio. (Note, however, that the relatios i the balace of this sectio are eact ad ot based o these assumptios.) I this case, it is atural to cosider, together with the geuie uitary operator u t, t of the time evolutio of the system, which obeys Eq. (58), a similarly defied uitary operator of evolutio of the uperturbed system described by Hamiltoia Ĥ aloe: i u H, (4.7) u ad also the followig iteractio evolutio operator, u I u u. (4.8) The sese of this defiitio becomes more clear if we isert the reciprocal relatio, ad its Hermitia cougate, u u u u u u u I ito the basic Eq. (9) which is valid i ay picture: A ( t ) u ( t, t ) A u ( t, t S ) ( t ), (4.9) u u u u, (4.) I I ( t ) u I t, t u t, t A u t, t u t, t ( t ) S I. (4.) This relatio shows that all calculatios of the observable epectatio values ad trasitio rates (i.e. all the results of quatum mechaics that may be eperimetally verified) are epressed by the followig formula, with the stadard bra-ket structure (87), A ( ) I t A ( t) ( t), (4.) if we assume that both the state vectors ad operators evolve i time, with the vectors evolvig due to the iteractio operator û I, I I Iteractio picture: state vectors Iteractio picture: operators t) ( t ) u ( t, t ), ( t) u ( t, t ) ( t ), (4.3) I ( I I I while the operators evolutio beig govered by the uperturbed operator û : t, t A u t t A ( t) u I. (4.4) S, Chapter 4 Page 37 of 4

201 These relatios describe the iteractio picture of quatum dyamics. Let me defer a eample of its coveiece util the perturbative aalysis of ope quatum systems i Sec. 7.6, ad here ed the discussio with a proof that the iteractio evolutio operator satisfies the Schrödiger equatio, i u H u, (4.5) i which Ĥ is the iteractio Hamiltoia trasformed i accordace with rule (4): I I t u t, t H u t t I it, I H I. (4.6) The proof is very straightforward: first usig defiitio (8), ad the Eqs. (58) ad the Hermitia cougate of Eq. (7), we may write iu I i d dt H u u i u u u u H u u u Hu H u u u H H it u u u H u u H itu H u u H u u H itu. u (4.7) Sice û may be preseted as a itegral of Ĥ (similar to Eq. (8) relatig û ad Ĥ ), these operators commute, so that the paretheses i the last form of Eq. (7) vaish. Now pluggig û from Eq. (9), we get the equatio, I u H u u I u H u u it it I i u, (4.8) that is equivalet to the combiatio of Eqs. (5) ad (6). Equatio (5) shows that if the eergy scale of iteractio H it is much weaker tha the backgroud eergy H, operators ûi ad u I, ad hece the state vectors (3) evolve relatively slowly. Such a eclusio of fast backgroud oscillatios is especially coveiet for the perturbative approaches to comple iteractig systems, i particular to the ope quatum systems that weakly iteract with their eviromet see Sec Eercise problems 4.. Let ad be two possible quatum states of the same system, ad  be a liear operator. Which of the followig epressios are legitimate (i. e. have a well-defied meaig) withi the bra-ket formalism? *  5.  6.  7.  Â. 4.. Prove that if  ad B are liear operators, the: (i) A A ; (ii) ia ia (iv) operators A A ad A A are Hermitia. AB B A ; ; (iii) Chapter 4 Page 38 of 4

202 4.3. Prove that for ay liear operators A, B, C, D, A B, CD A B, C D AC B, D A, C DB C A, D B Calculate all possible biary products (for, =, y, z) of the Pauli matrices (5), i σ, σ y, σ z, i ad their commutators ad aticommutators (defied similarly to those of the correspodig operators). Preset the results usig the Kroecker delta ad Levi-Civita permutatio symbols Calculate the followig epressios, (i) (c), ad the (ii) (bi + c), for the scalar product c of the Pauli matri vector + y y + z z by a arbitrary c-umber vector c, where is a iteger, ad b is a arbitrary scalar c-umber. Hit: For task (ii), you may like to use the biomial theorem, 48 ad the trasform the result i a way eablig you to use the same theorem backwards * Use the results of the previous problem to derive Eqs. (.65)-(.66) for the trasparecy T of a system of N similar, equidistat, delta-fuctioal tuel barriers Use result of Problem 5 to spell out the followig the followig matri: ep{i}, where is the vector of Pauli matrices, is a c-umber vector of uit legth, ad is a c-umber scalar Use the result of Problem 5(ii) to calculate ep{a}, where A is a arbitrary matri Epress elemets of matri B = ep{a} eplicitly via those of the matri A. Spell out your result for the followig matrices: a a i i A, A', a a i i with real a ad. 4.. Prove that for arbitrary square matrices A ad B, Tr (AB) Tr (BA). Is each diagoal elemet (AB) ecessarily equal to (BA)? 47 See, e.g., MA Eqs. (3.) ad (3.). 48 See, e.g. MA Eq. (.9). Chapter 4 Page 39 of 4

203 4.. Prove that the matri trace of a arbitrary operator does ot chage at a arbitrary uitary trasformatio. 4.. Prove that for ay two full ad orthoormal bases u, v of the same Hilbert space, Tr u v v u. ' 4.3. Is the D scatterig matri S, defied by Eq. (33), uitary? What about the D trasfer matri T defied by Eq. (34)? 4.4. Calculate the trace of the followig matri: i a σepi b σ ep, where is the Pauli matri vector, while a ad b are usual (c-umber) geometric vectors Let A be eigevalues of some operator Â. Epress the followig two sums, A, A, via the matri elemets A of this operator i a arbitrary basis Calculate z of a two-level system i a quatum state with the followig ket-vector: cost, where (, ) ad (, ) are eigestates of the Pauli matrices z ad, respectively. Hit: Double-check whether the solutio you are givig is geeral A electro is fully polarized i the positive z-directio. Calculate the probabilities of the alterative outcomes of a perfect Ster-Gerlach eperimet with the magetic field B orieted i the directio of some ais, performed o this electro A perfect Ster-Gerlach istrumet makes a sigle-shot measuremet of the followig combiatio, (S + S z )/, of two spi compoets of a z-polarized electro; after that, compoet S z of the same particle is measured. What are the possible outcomes of these measuremets ad their probabilities? 4.9. I a certai basis, the Hamiltoia of a spi-½ (two-level) system is described by matri H E ad the operator of some observable A, by matri A., with E E E, ' Chapter 4 Page 4 of 4

204 For the system s state with the eergy equal eactly to E, fid the possible results of measuremets of observable A ad the probabilities of the correspodig measuremet outcomes. 4.. * States u,,3 form a orthoormal basis of a system with Hamiltoia H u u u u u u h.c., 3 3 where is a real costat, ad h.c. meas the Hermitia cougate of the previous epressio. Calculate its statioary states ad eergy levels. Ca you relate this system with ay other(s) discussed earlier i the course? 4.. Suggest a Hamiltoia describig particle s dyamics i a ifiite D set of similar quatum wells i the tight-bidig approimatio, i the bra-ket formalism, ad verify that is yields the correct dispersio relatio (.6). 4.. Calculate eigevectors ad eigevalues of the followig matrices: A, 4.3. Fid eigevalues of the followig matri: B A a σ a σ a σ a σ, where a,y,z are real c-umbers (scalars), ad σ,y,z are the Pauli matrices. Sketch the depedece of the eigevalues o parameter a z, with a ad a y fied. Compare the result with Fig Derive a differetial equatio for the time evolutio of the epectatio value of a observable, usig both the Schrödiger picture ad the Heiseberg picture of quatum mechaics At t =, a spi-½ particle, whose iteractio with a eteral field is described by Hamiltoia H a σ a σ a σ a σ, (where a,y,z are real ad costat c-umbers, ad, y, z are the operators that, i the z-basis, are represeted by the Pauli matrices σ,y,z ), was i state, oe of two eigestates of operator z. Use the Schrödiger picture equatios to calculate the time evolutio of: (i) the ket-vector of the system (i ay statioary basis you like), (ii) the probabilities to fid the system i states ad, ad (iii) the epectatio values of all 3 spatial compoets ( S, etc.) of the spi vector operator S ( / ) σ. Aalyze ad iterpret the results for the particular case a y = a z =. y y y y z z z z Chapter 4 Page 4 of 4

205 4.6. For the same system as i the previous problem, use the Heiseberg picture equatios to calculate the time evolutio of: (i) all three spatial compoets ( S, etc.) of the spi operator Ŝ H (t), (ii) the epectatio values of the spi compoets. Compare the latter results with those of the previous problem For the same system as i two last problems, calculate the matri elemets of operator z i the basis of eigestates a, a. Hit: I cotrast to the cited problems, the aswer evidetly does ot deped o the iitial coditios I the Schrödiger picture of quatum mechaics, three operators satisfy the followig commutatio relatio: A, B. C What is their relatio i the Heiseberg picture (at the same time istat)? 4.9. A spi-½ particle is placed ito a magetic field B(t), which is a arbitrary fuctio of time. Derive the differetial equatios describig the time evolutio of: (i) the vector operator Ŝ of particle s spi (i the Heiseberg picture), ad (ii) the epectatio value S of spi s vector. Cotemplate the relative merits of the latter equatio for the descriptio of a sigle spi ad of a large collectio of similar, o-iteractig spis * Prove the Bloch theorem give by either Eq. (3.7) or Eq. (3.8). Hit: Cosider the traslatio operator T R, defied by the followig result of its actio o a arbitrary fuctio f(r): T f ( r) f ( r R), R where R is a arbitrary vector of the Bravais lattice (3.6). I particular, aalyze the commutatio properties of the operator, ad apply them to a eigefuctio (r) of the statioary Schrödiger equatio for a particle i a 3D periodic potetial described by Eq. (3.5). Chapter 4 Page 4 of 4

206 Chapter 5. Some Eactly Solvable Problems This describes several simplest but importat applicatios of the bra-ket formalism, otably icludig a few wave-mechaics problems we have already started to discuss i Chapters ad Two-level systems I the course of discussio of the bra-ket formalism i the last chapter, we have already cosidered several eamples of how it works for electro s spi. We have see, i particular, that i magetic field the electro has eigeeergies (4.67), i.e. two eergy levels. As will be show later i the course, such two-eergy-level picture is valid ot oly for electros ad other spi-½ elemetary particles (such as muos ad eutrios), but also may give a good approimatio for other importat quatum systems. For eample, as was already metioed i Chapter, two eergy levels are sufficiet for a approimate descriptio of dyamics of two weakly coupled quatum wells (Sec..6), ad of level aticrossig i the weak-potetial approimatio of the bad theory (Sec..7). Such two-level systems (alteratively called spi-½-like systems) are owadays the focus of additioal attetio i the view of prospects of their possible use for iformatio processig ad ecryptio. (I the last cotet, to be discussed i Sec. 8.5, a two-level system is usually called a qubit.) This is why before proceedig to other problems, let us summarize i brief what we have already leared about properties ad dyamics of two-level systems, i a more coveiet laguage. Accordig to the geeral Eq. (4.6), a ket- (or bra-) vector of a arbitrary pure (coheret) state of such a system may be preseted, at ay istat, as a liear combiatio of two basis vectors, for eample, (5.) ad hece is completely described by two comple coefficiets (c-umbers) say, ad. These two umbers are ot completely arbitrary; they are restricted by the ormalizatio coditio. If the basis vectors are ormalized, the to have the system i some basis state with a % probability, we eed W * * * *. (5.) This requiremet is automatically satisfied if we take the moduli of ad equal to the sie ad cosie of the same (real) agle. Thus we ca write, for eample, i i( ) cos e, si e. (5.3) Moreover, accordig to the geeral Eq. (4.5), if we deal with ust oe system, the commo phase factor ep{i} is uimportat for calculatio of ay epectatio values, ad we ca take =, so that Eq. (3) is reduced to To recall why this coditio is crucial, please revisit the begiig of Sec..3. Note also that, i particular, the mutual phase shifts betwee differet qubits are very importat for quatum iformatio processig (see Chapter 7 below), so that most discussios of these applicatios have to start from Eq. (3) rather tha Eq. (4). K. Likharev

207 i cos, si e. (5.4) The reaso why the argumet of sie ad cosie fuctios is usually take i the form /, becomes clear from Fig. a: Eq. (4) coveietly maps each state o a certai represetatio poit of a uit-radius Bloch sphere, with polar agle ad azimuthal agle. I particular, state (with = ad = ) correspods to the North Pole of the sphere ( = ), while state (with = ad = ), to its South pole ( = ). 3 Similarly, states ad, described by Eqs. (4.), i.e. havig = / ad = /, correspod to poits with = / ad to, respectively, = ad =. Two more special poits (deoted i Fig. a as ad ) are also located o sphere s equator (at = / ad = /); it is easy to check that they correspod to the eigestates of matri y (i the same z-basis). I order to uderstad why such mutually perpedicular locatio of these three special poit pairs o the Bloch sphere is ot occasioal, let us plug Eqs. (4) ito Eqs. (4.3)-(4.33) for the epectatio values of spi compoets. The result is S si cos, S y si si, S z cos, (5.5) showig that the radius-vector of the represetatio poit o the sphere is (after multiplicatio by /) ust the epectatio value of the spi vector S. Bloch sphere represetatio of state z (a) z (b) z (c) y y B B Fig. 5.. Bloch sphere: (a) otatio, ad presetatio of spi precessio i magetic fields directed alog: (b) ais z, ad (c) ais. Now let us see how does the represetatio poit moves i various cases. First of all, accordig to Eqs. (4.57)-(4.58), i the absece of a eteral field (whe the Hamiltoia operator is equal to zero ad hece the time-evolutio operator is costat) the poit does ot move at all. Now, if we apply to a electro a magetic field directed alog ais z, the, accordig to Eqs. (4.), the Heiseberg operator of S z (ad hece the epectatio value S z ) remais costat, while agle i Eq. (5) evolves Named after the same F. Bloch who has pioeered the eergy bad theory that was discussed i Chapters I the quatum iformatio literature, ket-vectors ad of these two states of a qubit are usually deoted as ( quatum oe ) ad ( quatum zero ). Chapter 5 Page of 5

208 i time as t + cost. This meas that the torque-iduced precessio of the spi i a costat field B = z B is described by a circular rotatio of the represetatio poit about ais z (i Fig. b, i the horizotal plae) with the classical precessio frequecy. This is essetially the classical picture of rotatio of the agular mometum vector about the precessio ais z, with both its legth ad the z- compoet coserved. 4 It is straightforward to repeat all calculatios of Sec. 4.6 for a field of a differet orietatio ad prove the (virtually evidet) result that the represetatio poit performs a similar rotatio about the field directio. (Fig. c shows such rotatio for a -directed field.) Fially, ote that it is sufficiet to tur off the field to stop the precessio istatly. (Sice Eq. (4.58) is the first-order differetial equatio, the represetatio poit has o effective iertia. 5 ) Hece chagig the directio ad magitude of the eteral field, it is possible to move spi s represetatio poit to ay positio o the Bloch sphere. (I Chapter 6 we will eamie aother method of maipulatig the poit positio, that is based o eteral rf field ad is more coveiet for some two-level systems.) I the cotet of quatum iformatio, this meas that i the absece of ucotrollable iteractio with eviromet, it is possible to prepare a qubit i ay pure quatum state, ad the keep it uchaged. From here it is clear that a qubit is very much differet from ad a classical bistable system used to store sigle bits of iformatio such as the voltage state of a usual SRAM cell (a positivefeedback loop of two trasistor-based iverters). As Eq. (4) shows, qubit s state is determied by two idepedet, cotiuous parameters ad, so it may store much more iformatio tha oe bit. (The differece is eve more spectacular i qubit systems, to be discussed i Sec. 8.5.) However, classical bistable systems, due to their oliearity, are stable with respect to small perturbatios, so that their operatio is rather robust with respect to uitetioal iteractio with their eviromet. I cotrast, qubit s state may be readily disturbed (i.e. its represetatio poit o the Bloch sphere shifted) by eve mior perturbatios, ad does ot have a iteral state stabilizatio mechaism. 6 Due to this reaso, qubit-based systems are rather vulerable to eviromet-iduced drifts, icludig dephasig ad relaatio effects, which will be discussed i Chapter Revisitig wave mechaics I order to use the bra-ket formalism for the descriptio of the orbital motio of a particle as a whole, we have to either rewrite or eve modify some of its formulas for the case of observables with cotiuous spectrum of eigevalues. (Oe eample we already kow well are the mometum ad kietic eergy of a free particle.) I that case, all the above epressios for states, their bra- ad ket-vectors, ad eigevalues, should be stripped of discrete idices, like ide i the key equatio (68) that determies eigestates ad eigevalues of observable A. For that, Eq. (68) may be rewritte i the form 4 Still, it is crucial to appreciate the differece betwee the epectatio values (5), i.e. c-umbers, ad the actual observables S, S y, ad S z which are described i quatum mechaics by operators. For eample, accordig to Eq. (4.56), for ay positio o the Bloch sphere, it is impossible to have eact values of Cartesia compoets, as it is i the classical picture. 5 The same is true for the agular mometum L at the classical torque-iduced precessio see, e.g., CM Sec.6.5 ad i particular Eq. (6.7). 6 I this aspect as well, the iformatio processig systems based o qubits are closer to classical aalog computers rather the classical digital oes. Chapter 5 Page 3 of 5

209 A a A. (5.6) A a A More essetially, all sums over such cotiuous eigestate sets should be replaced by itegrals. For eample, for a full ad orthoormal set of eigestates (6), the closure relatio (4.44) should be replaced with da a A a A I, (5.7) where the itegral should be take over the whole iterval of possible values of observable A. Applyig this relatio to the ket-vector of a arbitrary state (geerally, ot a eigestate of operator  ), we get I da a a da a a. (5.8) A This itegral replaces sum (4.37) for the represetatio of a arbitrary ket-vector as a epasio over eigestates of a operator. For the particular case whe = a A, this relatio requires 7 a A a A' A A A ( A A' ); (5.9) this formula replaces the orthoormality coditio (4.38). Accordig to Eq. (8), i the cotiuous case the bra-ket a A still plays the role of the coefficiet whose modulus squared determies state a A s probability see the last form of Eq. (4.). However, i the cotiuous spectrum case the probability to fid the system eactly i a particular state is ifiitesimal. Istead we should speak about the probability desity w(a) a A to fid the observable withi a small iterval da about a certai value A. The coefficiet i that relatio may be foud by makig the similar chage from summatio to itegratio (without ay additioal coefficiets) i the ormalizatio coditio (4.): da a A a. (5.) Sice the total probability of the system to be i some state should also equal w ( A) da, this meas that A w( A) a A a A a A. (5.) Now let us see how we ca calculate epectatio values of cotiuous observables, i.e. their esemble averages. If we speak about the same observable A whose eigestates are used as the basis (or ay compatible observable), everythig is simple. Isertig Eq. () ito the geeral statistical relatio which is ust the evidet cotiuous versio of Eq. (.37), we get Presetig this epressio as a double itegral, A A w( A) AdA, (5.) a A a da. (5.3) A A A A da da' a A ( A A' ) a, (5.4) 7 Notice that i the cotrast to the discrete spectrum case, the dimesioality of the bra- ad ket-vectors so ormalized is differet from. A' Cotiuous spectrum: closure relatio Cotiuous spectrum: state orthoormality Cotiuous spectrum: probability desity Chapter 5 Page 4 of 5

210 ad usig the cotiuous-spectrum versio of Eq. (4.), a A A a A' A ( A A' ), (5.5) Epectatio value we may write A da da' a a A a a A, (5.6) A A A' A' Wavefuctio as ier product so that Eq. (4.5) remais valid i the cotiuous-spectrum case without ay chages. The situatio is a bit more complicated for the epectatio values of operators that do ot commute with the base-creatig operator, because the matri of such a operators i that may ot be diagoal. We will cosider (ad overcome :-) this techical difficulty very soo, but otherwise we are ready for the discussio of wave mechaics. (For the otatio simplicity I will discuss its D versio; the geeralizatio to the D ad 3D cases is straightforward.) Let us cosider what is called the coordiate represetatio, postulatig the (ituitively almost evidet) eistece of a quatum state basis, whose with ket-vectors will be called, correspodig to a certai, eactly defied value of particle s coordiate. Writig the followig evidet idetity:, (5.7) ad comparig this relatio with Eq. (6), we see that they do ot cotradict each other if we assume that i the left-had part of this equatio is cosidered as the coordiate operator whose actio o a ket- (or bra-) vector is ust its multiplicatio by c-umber. (This looks like a proof, but is actually a separate, idepedet postulate, o matter how plausible.) I this coordiate represetatio, the ier product a A (t) becomes (t), ad Eq. () takes the form * w(, t) ( t) ( t) ( t) ( t). (5.8) Comparig this formula with the basic postulate (.) of wave mechaics, we see that they coicide if the Schrödiger s wavefuctio of time-evolvig state (t) is idetified with that bra-ket: 8 (, t) ( t). (5.9) This key formula provides the coectio betwee the bra-ket formalism ad wave mechaics, ad should ot be too surprisig for the (thoughtful :-) reader. Ideed, Eqs. (4.45) shows that ay ier product of vectors describig two states is a measure of their coicidece - ust as the scalar product of two geometric vectors. (The orthoormality coditio (4.38) is a particular maifestatio of this fact.) I this laguage, value (9) of wavefuctio at poit ad momet t characterizes how much of a particular coordiate does the state cotai at that particular istace. (Of course this iformal laguage is too crude to describe the fact that (, t) is a comple fuctio, which has ot oly a modulus, but also a phase.) 8 I do ot quite like epressios like used i some papers ad eve tetbooks. Of course, oe is free to replace with ay other letter ( icludig) to deote a quatum state, but the it is better ot to use the same letter to deote the wavefuctio, i.e. a ier product of two state vectors, to avoid cofusio. Chapter 5 Page 5 of 5

211 Let us rewrite the most importat formulas of the bra-ket formalism (so far, i the Schrödiger picture) i the wave mechaics otatio. I particular, let us use Eq. (9) to calculate the (partial) time derivative of the wavefuctio, multiplied by the usual coefficiet i: i i (t). (5.) t t Sice the coordiate operator does ot deped o time eplicitly, its eigestates are statioary, ad we ca swap the time derivative ad the time-idepedet ket-vector ad hece. Makig use of the Schrödiger-picture equatios (4.57) ad (4.58), ad the isertig the idetity operator i the cotiuous form (7) of the closure relatio, writte for the coordiate eigestates, d' ' ' I, (5.) we may cotiue to develop the right-had part of Eq. () as i t ( t) i u ( t, t ) ( ) (, ) ( ) t Hu t t t H ( t) t d' H ' ' ( t) d' H ' Ψ ( ', t). (5.) For a geeral Hamiltoia operator, we have to stop here, because if it does ot commute with the coordiate operator, its matri i the -basis is ot diagoal, ad itegral () caot be worked out eplicitly. However, there eists a broad set of space-local operators 9 whose argumets iclude ust oe value of the spatial coordiate, for which we ca move ket-vector to the right A ' ( ', t) A ( ', t) ' A (, t) ( ' ). (5.3) where operator  i the last two forms should be uderstood as its coordiate represetatio that is defied by Eq. (3) - if it is valid for a particular operator. For eample, cosider the D versio of operator (.4), p H U (, t), (5.4) m which was the basis of all our discussios i Chapter. Its potetial-eergy part commutes with operator, so its matri i the -basis is diagoal, meaig that this part of Hamiltoia (4) is clearly local, with its coordiate represetatio give merely by the c-umber fuctio U(,t). The situatio with the kietic eergy, which is a fuctio of mometum operator p, ot commutig with, is less evidet. Let me show that this operator is also local, ad i the same shot derive (the D versio of) Eq. (.6), if we postulate the commutatio relatio (.4): p p ii. (5.5) Spacelocal operators 9 Of all the operators we will ecouter i this course, oly the statistical operator ŵ is substatially o-local see Sec. 7.. I the secod equality, I have use Eq. (9) for variable. Chapter 5 Page 6 of 5

212 For that, let us cosider the followig matri elemet, p p '. O oe had, we may use Eq. (5) to write p p ' i I ' i ' i ( ' ). (5.6) O the other had, sice ' ' ' ad, we ca write p ' p ' p ' p p ' '. (5.7) Comparig Eqs. (6) ad (7), we may write p ' ( ') i (5.8a) ' Thus p is a local operator. Sice Eq. (8a) may be rewritte as p ' i ( '), (5.8b) its compariso with Eq. (3) shows that the formula used so much i Chapter, p i, (5.9) is ideed valid, but oly for the coordiate represetatio of the mometum operator. (Later i this sectio we will see that i a alterative, mometum represetatio, this operator looks completely differetly.) It is straightforward to show (ad virtually evidet) that ay operator f ( p ) is local as well, with its coordiate represetatio beig f i. (5.3) I particular, this pertais to the kietic eergy operator i Eq. (4), so that Eq. () is reduced to the Schrödiger equatio i its familiar wave-mechaics form (.8), if by Ĥ we mea its coordiate represetatio: H i U (, t) U (, t). (5.3) m m Now let us retur, as was promised, to operators that do ot commute with operator, ad hece do ot have to share its cotiuous spectrum. Ier-multiplyig both parts of the geeral Eq. (4.68) by ket-vector, ad isertig ito the left-had part the idetity operator i form (), we get d' A ' ' a A a, (5.3) The equivalece of the two forms of Eq. (8) may be readily prove, for eample, by compariso of their effect o ay differetiable fuctio f(, ), usig its Taylor epasio over argumet at poit = a simple but good eercise for the reader. Chapter 5 Page 7 of 5

213 i.e., usig the wavefuctio defiitio (9), d ' A ' ( ', t) A (, t). (5.33) If the operator A is space-local, i.e. satisfies Eq. (3), the this result is immediately reduced to A (, t) A (, t), (5.34) (where the left-had part implies the coordiate represetatio of the operator), eve if the operator does ot commute with operator. The most importat case of this coordiate-represetatio form of the eigeproblem (4.68) is the familiar Eq. (.6) for eigevalues E of eergy. Hece, the eergy spectrum of a system (that, as we kow very well from the first chapters of the course, may be discrete) is othig more tha the set of eigevalues of its Hamiltoia operator a very importat coclusio ideed. The operator locality also simplifies the epressio for its epectatio value. Ideed, pluggig the completeess relatio i the form () ito the geeral Eq. (4.5) twice (writte i the first case for ad i the secod case for ), we get * A d d' ( t) A ' ' ( t) d d' (, t) A ' Now, Eq. (3) reduces this result to ust ' dd' (, t) A (, t) (, t) A ( ', t). (5.35) * A * (, t) d. (5.36) i.e. to Eq. (.3), which we had to postulate i Chapter. So, we have essetially derived all basic relatios of wave mechaics from the bra-ket formalism, which will also allow us to get some importat ew results i that area. Before doig that, let us have a look at a pair of very iterestig relatios, together called the Ehrefest theorem. I order to derive them, let us calculate the followig commutator: 3, p p p p p. (5.37) Rewritig Heiseberg s commutatio relatio (5) as p p i, (5.38) we ca use it twice i the first term of the right-had part of Eq. (37) to sequetially move the mometum operators to the left: p p p ip p p ip p p i ip p p ip. (5.39) Operator s eigestates ad eigevalues I some systems of quatum mechaics postulates, the Schrödiger equatio (.8) itself is cosidered as a sort of eigestate/eigevalue problem (34) for operator i/t. Notice that such costruct is very close to that of the mometum operator -i/, ad similar argumets may be give for both epressios, startig from the ivariace of the quatum state of a free particle with respect to traslatios i time ad space, respectively. 3 It is ot importat whether we speak about the Schrödiger or Heiseberg picture here. Ideed, if three operators i the former picture are related as [ A S, BS ] = Ĉ S, the accordig to Eq. (4.9), i the latter picture,,, A B U A U U B U U A UU B U U B UU A U U A B U U C U C. H H H H H H H H S S S H Chapter 5 Page 8 of 5

214 Chapter 5 Page 9 of 5 The first term of the result cacels with the secod term of Eq. (37), so that the commutator is rather simple:., p i p (5.4) Let us use this equality to calculate the Heiseberg-picture equatio of motio for operator, applyig the geeral Heiseberg equatio (4.99) to the orbital motio, whe the Hamiltoia has the form (3), with time-idepedet potetial U(): 4. ( ),, U m p i H i dt d (5.4) The potetial eergy operator commutes with the coordiate operator. Thus, the right-had part of Eq. (4) is proportioal to commutator (4):. m p dt d (5.4) I that operator equality, we readily recogize the classical relatio betwee particle s mometum ad is velocity. Now let us see what does a similar procedure give for the mometum s derivative:. ( ),, U m p p i H p i dt p d (5.43) The kietic eergy operator commutes with the mometum operator, ad hece may be dropped from the right-had part of this equatio. I order to calculate the remaiig commutator of the mometum ad potetial eergy, let us use the fact that ay smooth potetial profile may be represeted by its Taylor epasio: k k k k U k U! ( ), (5.44) where the derivatives of U should be uderstood as c-umbers (evaluated at = ), so that we may write k k k k k k k k k p p U k p U k U p.....!,! ( ), times times. (5.45) Applyig Eq. (38) k times to the last term i the paretheses, eactly as we did it i Eq. (39), we get. )! (! ( ), k k k k k k k k U k i ik U k U p (5.46) But the last sum is ust the Taylor epasio of the derivative U/. Ideed,, )! ( '!! ' ' ' ' ' ' k k k k k k k k k k' k' k U k U k U k' U (5.47) 4 Sice this Hamiltoia is time-idepedet, we may replace the partial derivative over time t with the full oe. Heiseberg equatio for coordiate

215 where at the last step I have replaced the otatio of the summatio ide from k to k -. As a result, Eq. (43) yields: dp U ( ). (5.48) dt This equatio agai coicides with the classical equatio of motio! Discussig spi dyamics i Sec. 4.6 ad 5., we have already see that this is very typical of the Heiseberg picture. Moreover, averagig Eqs. (4) ad (48) over the iitial state (as Eq. (4.9) prescribes 5 ), we get similar results for the epectatio values: 6 d p d p U,. (5.49) dt m dt However, it is importat to remember that the equivalece betwee these quatum-mechaical equatios ad similar equatios of classical mechaics is superficial, ad the degree of the similarity betwee the two mechaics very much depeds o the problem. As oe etreme, let us cosider the case whe a particle s state, at ay momet betwee t ad t, may be accurately represeted by oe, relatively arrow wave packet. The we may iterpret Eqs. (49) as equatios of essetially classical motio for the wave packet s ceter, i accordace with the correspodece priciple. However, eve i this case it is importat to remember about the purely quatum mechaical effects of ovaishig wave packet width ad its spreadig i time, which were discussed i Sec... I the opposite etreme, Eqs. (49), though valid, may tell almost othig about system s dyamics. Maybe the most apparet eample is the leaky quatum well that was discussed i Sec..5 - see Fig..8 ad its discussio. Sice both the potetial U() ad the iitial state are symmetric relative to poit =, the right-had parts of both Eqs. (49) idetically equal zero. Of course, the result (that average values of both mometum ad coordiate stay equal zero at all times) is correct, but it does ot tell us too much about the rich dyamics of the system (the fiite lifetime of the metastable state, the formatio of two wave packets, their waveform ad propagatio speed), ad about the importat isight the solutio gives for the quatum measuremet theory. Aother similar eample is the bad theory (Sec..7), with its purely quatum effect of the allowed eergy bads ad forbidde gaps, of which Eq. (49) gives o clue. To summarize, the Ehrefest theorem is importat as a illustratio of the correspodece priciple, but its predictive power should ot be eaggerated. Now we are ready to patch some holes left durig our studies of wave mechaics i Chapters - 3. First of all, I have promised you to develop a more balaced view at the moochromatic de Broglie waves (4.), which would be more respectful to the evidet r p symmetry of the coordiate ad mometum. Let us do this for the D case whe the wave may be preseted as 7 Heiseberg equatio for mometum Ehrefest theorem 5 Ideed, actig eactly as at derivatio of Eq. (36), for a space-local Heiseberg operator we get * A (, t ) A ( t, t )Ψ(, t ) d. t Ψ H 6 The set of equatios (49) costitute the Ehrefest theorem. 7 From this poit o, for the sake of brevity I will drop ide i the otatio of the mometum ust as it was doe i Chapter. Chapter 5 Page of 5

216 p p ( ) a p epi, for all. (5.5) Let us have a good look at this fuctio. Sice it satisfies equatio (34) for the D mometum operator p i /, p p p, ` (5.5) p p is a eigefuctio of the mometum operator. But this meas that we ca also write Eq. (6) for the correspodig ket-vector: ad accordig to Eq. (9) the wavefuctio may be preseted as p p p p, (5.5) ( ) p. (5.53) p. Epressio (53) is quite remarkable i its p symmetry - which may be pursued further o. Before doig that, however, we have to discuss ormalizatio of such fuctios. Ideed, i this case, the probability desity w() (8) is costat, so that its itegral * w( ) d p ( ) p ( ) d (5.54) diverges if a p. Earlier i the course, we discussed two ways to avoid this divergece. Oe is to use a very large but fiite itegratio volume see Eq. (.3). Aother way to avoid the divergece is to form a wave packet of the type (.), possibly of a very large legth ad very arrow spread of mometa p. The itegral (54) may be required to equal without ay coceptual problem. However, both these methods violate the p symmetry, ad hece are icoveiet for our curret purposes. Istead, let us cotiue to idetify the bra- ad ket-vectors a A ad a A of the geeral theory, developed i the begiig of this sectio, with eigevectors p ad p of mometum ust as we have already doe i Eq. (5). The the ormalizatio coditio (9) becomes p p' ( p p'). (5.55) Isertig the idetity operator i the form () (with the itegratio variable replaced by ) ito the left-had side of this equatio, we ca traslate this ormalizatio rule to the wavefuctio laguage: d p Now usig Eq. (5), this requiremet turs ito the followig coditio: p' * d p ( ) p' ( ) ( p p' ). (5.56) a * p a p' ( p p' ) epi d a p ( p p' ) ( p p' ), (5.57) so that, fially, a p = ep{i}/() /, where is a arbitrary (real) phase, ad Eq. (5) becomes 8 8 Repeatig the calculatio for each Cartesia compoet of a plae moochromatic wave of arbitrary dimesioality d, we get p = () -d/ ep{i(pr/ + )}. Chapter 5 Page of 5

217 p p ( ) epi (5.58) / As was metioed above, for fiite-legth wave packets such ormalizatio is ot really ecessary. However, frequetly it makes sese to keep the pre-epoetial coefficiet i Eq. (58) eve for wave packets, because of the followig reaso. Let us form a wave packet of the type (.), based o wavefuctios (58) - takig = for the otatio brevity, because it may be icorporated ito fuctio (p): p ( ) p i dp ( )ep /. (5.59) From the mathematical poit of view, this is ust the equatio of a D Fourier spatial trasform, ad its reciprocal is p ( p) ( )ep i d /. (5.6) These epressios are completely symmetrical, ad preset the same wave packet; this is why fuctios () ad (p) are frequetly called, respectively, the coordiate (-) ad mometum (p-) represetatios of the (same) state of the particle. Usig Eqs. (53) ad (58), they may be preseted i a eve more maifestly symmetric form, ( ) ( p) p dp, ( p) ( ) p d, (5.6) i which the scalar products satisfy the basic postulate (4.4) of the bra-ket formalism: p * p ep i p /. (5.6) We already kow that i the -represetatio, i.e. i the usual wave mechaics, the coordiate operator is reduced to the multiplicatio by, ad the mometum operator is proportioal to a derivative over :, i p i i. (5.63) It is atural to guess that i the p-represetatio, the epressios for operators would be reciprocal:, i p i p i p p, (5.64) p with the differece i oe sig oly, due to the opposite sigs of the Fourier epoets i Eqs. (59) ad (6). The proof of Eqs. (64) is straightforward; for eample, actig by the mometum operator to wavefuctio (59), we get p ( ) i ( ) ( p) i / p p ( p)ep i dp, / p epi dp (5.65) Wave packet i reciprocal represetatios Mometum ad coordiate operators i reciprocal represetatios Chapter 5 Page of 5

218 ad similarly for operator actig o fuctio (p). Hece, the actio of the operators (63) o wavefuctio (i.e. state s -represetatio) gives the same results as the actio of operators (64) o fuctio (i.e. its p-represetatio). It is iterestig to have oe more, differet look at this coordiate-to-mometum duality. For that, otice that accordig to Eqs. (4.8)-(4.84), we may cosider the bra-ket p as a elemet of the (ifiite-size) matri U p of the uitary trasform from the -basis to p-basis. Now let us derive the operator trasform rule that would be a cotiuous versio of Eq. (4.9). Say, we wat to calculate a matri elemet of some operator i the p-represetatio: p A p'. (5.66) Isertig two idetity operators () ito this bra-ket, ad the usig Eq. (53) ad its comple cougate, ad also Eq. (3) (agai, valid oly for space-local operators!), we get * p A p' d d' p A ' ' p' d d' ( ) A ' ( ' ) d p ep ( ) p'' d' i ' Aepi p For eample, for the mometum operator itself, this relatio yields: p' p ep p' d i Aepi. (5.67) p p' p' ( p' p) p p p' d ep i i epi epi d p' ( p' p). (5.68) Due to Eq. (5), this result is equivalet to the secod of Eqs. (64). A atural questio arises: why is the mometum represetatio used much less frequetly tha the coordiate represetatio - i.e., the wave mechaics? The aswer is purely practical: besides the special case of the harmoic oscillator (to be revisited i Sec. 4 below), the orbital motio Hamiltoia (3) is ot p symmetric, with the potetial eergy U() beig typically a more comple fuctio tha the kietic eergy, which is quadratic i mometum. Because of that, it is easier for problem solutio to keep the potetial eergy operator ust a wavefuctio multiplier, as it is i the coordiate represetatio. The most sigificat eceptio of this rule is the motio i a periodic potetial, especially i the presece of additioal eteral force F(t), which may result i the effects discussed i Secs..8 ad.9 (the Bloch oscillatios, Ladau-Zeer tuelig etc.). Ideed, i this case the dispersio relatio E (q), typically rather ivolved, plays the role of the effective kietic eergy, while the effective potetial eergy U ef = F(t) i the field of the additioal force is a simple fuctio of. This is why discussios of the listed ad more comple issues of the bad theory (such as quasiparticle scatterig, mobility, diffusio, etc.) i solid state physics theory are most typically based o the mometum represetatio Feyma s path itegrals As has bee already metioed, eve withi the realm of wave mechaics, the bra-ket laguage allows to streamlie some calculatios that would be very bulky usig the otatio used i Chapters -3. Probably the best eample i the famous alterative, path itegral formulatio of quatum mechaics, Chapter 5 Page 3 of 5

219 developed i 948 by R. Feyma. 9 I will review this importat cocept - admittedly cuttig oe math corer for brevity. (This shortcut will be clearly marked.) Let us ier-multiply both parts of Eq. (4.57), which is essetially the defiitio of the timeevolutio operator, by the bra-vector of state, t) u ( t, t ) ( t ), (5.69) ( isert the idetity operator before the ket-vector i the right-had part, ad the use the closure coditio i the form of Eq. (), with replaced with : Accordig to Eq. (9), this equality may be preseted as ( t) d u ( t, t ) ( t ). (5.7) (, t) d u ( t, t ) (, t ). (5.7) Comparig this epressio with Eq. (.44), we see that the bra-ket i this relatio is othig else tha the D propagator, which was discussed i Sec..: t u ( t, t ) G(, t;, ). (5.7) As a remider, we have already calculated the propagator for a free particle see Eq. (.49). Now let us break the time segmet [t, t] ito N (for the time beig, ot ecessarily equal) parts by isertig (N ) itermediate poits (Fig. ) t t... t... t t, (5.73) ad rewrite the time evolutio operator i the form u ( t, t ) u ( t, t ) u ( t, t )... u ( t, t ) u ( t, ), (5.74) N N N t k whose correctess is evidet from the very defiitio (4.57) of the operator. Pluggig Eq. (74) ito Eq. (7), let us isert the idetity operator, agai i the form () but writte for k rather tha, betwee each two partial evolutio operators icludig time argumet t k. The result is N N G(, t; )... (, ) (, )..., t d d d u t t u t t u ( t, t ). (5.75) N N N N N N N k N N t t... t... tn tn k t Fig. 5.. Time partitio ad coordiate otatio at the iitial stage of the Feyma s path itegral derivatio. 9 Accordig to Feyma s memories, his work was motivated by a mysterious remark by P. A. M. Dirac i his pioeerig 93 tetbook o quatum mechaics. For a more thorough discussio of the path-itegral approach, see the famous tet R. Feyma ad A. Hibbs, Quatum Mechaics ad Path Itegrals first published i 965. (For its latest editio by Dover i, the book was emeded by D. Styler.) For a more recet moograph that reviews more applicatios, see L. Schulma, Techiques ad Applicatios of Path Itegratio, Wiley, 98. Chapter 5 Page 4 of 5

220 The physical sese of each itegratio variable k is the wavefuctio s argumet at time t k - see Fig.. The key Feyma s breakthrough was the realizatio that if all itervals are similar ad sufficietly small, t k t k- = d, all the partial bra-kets participatig i Eq. (75) may be readily epressed via Eq. (.49), eve if the particle is ot free, but moves i a statioary potetial profile U(). To show that, let us use either Eq. (4.75) or Eq. (4.8), which, for a small time iterval d, give the same result: i i p u ( d, ) ep Hd ep d U m d. (5.76) Geerally, a epoet of a sum of two operators may be treated as that of c-umber argumets, ad i particular factored ito a product of two epoets, oly if the operators commute. (Ideed, i this case we ca use all the stadard algebra for epoets of c-umber argumets.) I our case, this is ot so, because operator p does ot commute with, ad hece with U( ). However, it may be show that for a ifiitesimal time iterval d, the ovaishig commutator p d, U ( ) d, (5.77) m proportioal to (d), is so small that i the first approimatio i d its effects may be igored. As a result, we may factor the right-had part i Eq. (76) by writig i p i u ( d, ) d d U ( ep ep ) d. (5.78) m (This approimatio is very much similar i spirit to the rectagle-formula approimatio for a usual D itegral, which i also asymptotically impeachable.) Sice the secod epoetial fuctio i the right-had part of Eq. (78) commutes with the coordiate operator, we ca move it out of each partial bra-ket participatig i Eq. (75), with U() turig ito a c-umber fuctio: i p i d u( d, ) d ep d ep U ( ) d. (5.79) m But the remaiig bra-ket is ust the propagator of a free particle, ad we ca use Eq. (.49) for it: i p m m( d) d ep d epi. (5.8) m id d As the result, the full propagator (75) takes the form N m m( d) U ( ) G(, t;, t ) lim d dn dn.. d ep i i d.(5.8) N id k d / N / A strict proof of this ituitively evidet statemet would take more space ad time tha I ca afford. Chapter 5 Page 5 of 5

221 At N ad hece d (t t )/N, the sum uder the epoet i this epressio teds to a itegral: N k i m d d U ( ) t k i d t t m d d U ( ) d, (5.8) ad the epressio i square brackets is ust the particle s Lagragia fuctio L. The itegral of the fuctio over time is the classical actio S calculated alog a particular path (). 3 As a result, defiig the (D) path itegral as m (...) D[ ( )] lim d.. (...), d d d N N N id we ca brig our result to a superficially simple form N / (5.83a) i G(, t;, t ) ep S ( ) D[ ( )]. (5.83b) The ame path itegral for the mathematical costruct (83a) may be readily eplaied if we keep the umber N of time itervals large but fiite, ad also approimate each of the eclosed itegrals by a sum over M >> discrete poits alog the coordiate ais (Fig. 3a). d (a) (b) D path itegral: defiitio D propagator via path itegral M t t N t t Fig Several D classical paths i (a) the discrete approimatio ad (b) the cotiuous limit. The the path itegral is a product of (N - ) sums correspodig to differet values of time, each of them with M terms, each of the terms represetig the fuctio uder the itegral at a particular spatial poit. Multiplyig those (N ) sums, we get a sum of (N - )M terms, each evaluatig the fuctio at a specific spatial-temporal poit [, ]. These terms may be ow grouped to represet all possible differet cotiuous classical paths [] from the iitial poit [,t ] to the fiite poit [,t]. It is evidet that the last iterpretatio remais true eve i the cotiuous limit N, M see Fig. 3b. Why does such represetatio of the sum has sese? This is because i the classical limit the particle follows ust a certai path, correspodig to the miimum of actio S. Hece, for all close traectories, the differece (S S cl ) is proportioal to the square of the deviatio from the classical traectory. Hece, for a quasiclassical motio, with S cl >>, there is a substatial buch of close traectories, with (S S cl ) <<, that give similar cotributios to the path itegral. O the other had, See, e.g., CM Sec... 3 See, e.g., CM Sec. 9.. Chapter 5 Page 6 of 5

222 strogly o-classical traectories, with (S S cl ) >>, give phases S/ rapidly oscillatig from oe traectory to the et oe, ad their cotributios to the path itegral are averaged out. 4 As a result, for the quasiclassical motio, the propagator s epoet may be evaluated o the classical path: t i i m d Gcl ep cl ep U ( ) d. S t d (5.84) The sum of the kietic ad potetial eergies is the full eergy E of the particle, that remais costat for motio i a statioary potetial U(), so that we may rewrite the epressio uder the itegral as 5 m d d d U ( ) d m Ed m d Ed. (5.85) d d d With that replacemet, Eq. (83b) yields i d i i i Gcl ep m dep E( t t ) ep p( ) dep E( t t ), d (5.86) where p is the classical mometum of the particle. But (at least, leavig the pre-epoetial factor aloe) this is eactly the WKB approimatio result that was derived ad studied i detail i Chapter! Oe may questio the value of a calculatio that yields the results that could be readily obtaied from Schrödiger s wave mechaics. The Feyma s approach, is ideed ot used too ofte, but it has its merits. First, it has a importat philosophical (ad hece heuristic) value. Ideed, Eq. (83) may be iterpreted by sayig that the essece of quatum mechaics is the eploratio, by the system, of all possible paths (), each of them classical-like i the sese that the particle s coordiate ad velocity d/d (ad hece its mometum) are eactly defied simultaeously at each poit. The resultig cotributios to the path itegral are added up coheretly to form the fial propagator G, ad via it, the fial probability W G of the particle propagatio from [,t ] to [,t]. Of course, as the scale of actio (i.e. of the eergy-by-time product) of the motio decreases ad becomes comparable to, more ad more paths produce substatial cotributio to this sum, ad hece to W, esurig a larger ad larger differece betwee the quatum ad classical properties of the system. Secod, the path itegral provides a ustificatio for some simple eplaatios of quatum pheomea. A typical eample is the quatum iterferece effects discussed i Sec. 3. see, e.g., Fig. 3. ad the correspodig tet. At that discussio, we used the Huyges priciple to argue that at the two-slit iterferece, the WKB approimatio might be restricted of effects by two paths that pass through differet slits, but otherwise cosistig of straight-lie segmets. To have aother look at that assumptio, let us geeralize the path itegral to multi-dimesioal geometries. Fortuately, the simple structure of Eq. (83b) makes such geeralizatio virtually evidet: 4 This fact may be proved by epadig the differece (S S cl ) i the Taylor series i path variatios (leavig oly the leadig quadratic terms) ad workig out the resultig Gaussia itegrals. It is iterestig that the itegratio, together with the pre-epoetial coefficiet i Eq. (83a), gives eactly the pre-epoetial factor that we have already foud i Sec..4 whe refiig the WKB approimatio. 5 The same trick is ofte used i aalytical classical mechaics say, for provig the Hamilto priciple, ad for the derivatio of the Hamilto Jacobi equatios (see, e.g. CM Secs..3-4). Chapter 5 Page 7 of 5

223 S G( r, t; r t t, t ) dr Lr, d d i ep S t t r( ) D[ r( )], m dr d U ( r) d. (5.87) where defiitio (83a) of the path itegral should be also modified correspodigly. (I will ot go ito these techical details.) For the Youg-type eperimet (Fig. 3.), where a classical particle could reach the detector oly after passig through oe of the slits, the classical paths are the straight-lie segmets show i Fig. 3., ad if they are much loger tha the de Broglie wavelegth, the propagator may be well approimated by the sum of two itegrals of Ld = ip(r)dr/ - as it was doe i Sec. 3.. Last but ot least, the path itegral allows simple solutios of some problems that would be hard to get by other methods. As the simplest eample, let us cosider the problem of tuelig i multidimesioal space, sketched i Fig. 4 for the D case - ust for graphics simplicity. Here, potetial U(, y) has the saddle shape. (Aother helpful image is a moutai path betwee two summits, i Fig. 4 located o the top ad at the bottom of the drawig.) A particle of eergy E may move classically i the left ad right regios with U(, y) < E, but ca pass from oe of these regios to aother oe oly via the quatum-mechaical tuelig uder the pass. Let us calculate the trasparecy of this tuel barrier i the WKB approimatio, igorig the possible pre-epoetial factor. 3D propagator via the path itegral y U E U E U E U E r U E r U E Fig Saddle-type D potetial profile ad the istato traectory of a particle of eergy E (dashed lie, schematically). Accordig to the evidet multi-dimesioal geeralizatio Eq. (86), for the classically forbidde regio, where E < U(, y), the cotributios to propagator (87) are proportioal to r i ep κ ( r) drep E( t t ), (5.88) r where the magitude of vector at each poit may be calculated ust i the D case - see, e.g., Eq. (.97), ( r) U ( r) E, (5.89) m while its directio should be tagetial to the path traectory i space. Now the path itegral is actually much simpler tha i the classically-allowed regio, because the spatial epoets are purely real ad there is o comple iterferece betwee them. Because of the mius sig i the epoet, the largest Chapter 5 Page 8 of 5

224 3D tuelig i WKB limit cotributio to G evidetly comes from the traectory (or rather a arrow budle of traectories) for which the fuctioal r r κ( r' ) dr' (5.9) has the smallest value, ad the barrier trasmissio coefficiet may be calculated as r T G ep κ( r' ) dr', (5.9) r where r ad r are certai poits o the opposite classical turig-poit surfaces: U(r) = U(r ) = E. 6 Thus the tuelig problem is reduced to fidig the traectory (icludig poits r ad r ) that coects the two surfaces ad miimizes fuctioal (9). This is of course a well-kow problem of the calculus of variatios, 7 but it is iterestig that the path itegral provides a simple alterative way of solvig it. Let us cosider a auiliary problem of particle s motio i a potetial profile U iv (r) that is iverted relative to particle s eergy E, i.e. is defied by the followig equality: U ( r) E E U ( r). (5.9) iv As was discussed above, at fied eergy E, the path itegral for the WKB motio i the classically allowed regio of potetial U iv (,y) (that coicides with the classically forbidde regio of the origial problem) is domiated by the classical traectory correspodig to the miimum of where k iv should be determied from the relatio S p ( r' ) dr' k ( r' ) dr, (5.93) iv r r iv kiv ( r) E U m iv r r ( r). iv (5.94) But comparig Eqs. (89), (9), ad (94), we see that k iv = κ at each poit of space! This meas that the tuelig path (i the WKB limit) correspods to the classical (so-called istato) 8 traectory of the same particle i the iverted potetial U iv (r). If the iitial poit r is fied, this traectory may be readily foud by the meas of classical mechaics. (Note that the iitial velocity of the istato lauched from poit r should be zero, because by the classical turig poit defiitio: U iv (r ) = U(r ) = E.) Thus the problem is reduced to a simpler task of maimizig the trasparecy (9) over the positio of r o the equipotetial surface U(r ) = E. Moreover, for may symmetric potetials, the positio of this poit may be readily guessed without calculatios. 6 Oe ca argue that the pre-epoetial coefficiet i Eq. (9) should be close to, ust like i Eq. (.7), especially if the potetial is smooth i the sese of Eq. (.7), where is the coordiate alog the traectory. 7 For a cocise itroductio to the field see, e.g., I. Gelfad ad S. Fomi, Calculus of Variatios, Dover,, or L. Elsgolc, Calculus of Variatios, Dover, 7. 8 I quatum field theory, the istato cocept may be formulated somewhat differetly, ad has more comple applicatios - see, e.g. R. Raarama, Solitos ad Istatos, North Hollad, 987. Chapter 5 Page 9 of 5

225 Note that besides the calculatio of barrier trasparecy, the istato traectory has oe more importat implicatio: the so-called traversal time t of the classical motio alog it, i the iverted potetial, defied by Eq. (94), plays the role of the most importat (though ot the oly oe) time scale of particle s tuelig uder the potetial barrier Revisitig harmoic oscillator Let us retur to the D harmoic oscillator, i.e. ay system described by Hamiltoia (.5) with potetial eergy (.): p m H. (5.95) m I Sec.. we have used the brute-force (wave-mechaics) approach to aalyze the eigefuctios () ad eigevalues E of this Hamiltoia, ad foud that, ufortuately, that approach required relatively comple math that obscures the physics of these statioary ( Fock ) states. Now let us use the bra-ket formalism to make the properties of these states much more trasparet, usig very simple calculatios. First, itroducig ormalized (dimesioless) operators of coordiates ad mometum: 3 p,, (5.96) m where (/m ) / is the atural coordiate scale ( the r.m.s. spread of groud-state wavefuctio) which was discussed i detail i Sec.., we ca preset Hamiltoia (95) i a very simple ad p symmetric form: Now, let us itroduce a ew operator H. (5.97) Harmoic oscillator: Hamiltoia a / m p i i. m (5.98a) Sice both operators ad correspod to real observables, i.e. have real eigevalues ad hece are Hermitia (self-adoit), the Hermitia cougate of operator â is simply its comple cougate: Creatioaihilatio operators: defiitio a / m p i i, m Solvig the system of two equatios (98) for ad, we may readily get reciprocal relatios (5.98b) 9 See, e.g., M. Buttiker ad R. Ladauer, Phys. Rev. Lett. 49, 739 (98), ad refereces therei. 3 This ormalizatio is ot really ecessary, it ust makes the followig calculatios less bulky - ad thus more aesthetically appealig. Chapter 5 Page of 5

226 Creatioaihilatio operators: commutatio relatio Hamiltoia ad umber operators a a, a a. (5.99) i Our Hamiltoia (97) icludes squares of these operators. Calculatig them, we have to be careful to avoid swappig the ew operators, because they do ot commute. Ideed, for the ormalized operators (96), Eq. (.4) gives,, p ii, (5.) m so that Eqs. (98) yield i a, a i, i,, I. (5.) With such due cautio, Eq. (99) gives a a aa a a, a a aa a a. (5.) Pluggig these epressios back ito Eq. (97), we get H aa a a. (5.3) This epressio is elegat eough, but may be recast ito a eve more coveiet form. For that, let us rewrite the commutatio relatio () as ad plug it ito Eq. (3). The result is H a a I N where, i the last form, oe more (evidetly, Hermitia) operator, a a a a I (5.4) I, (5.5) N a a, (5.6) has bee itroduced. Sice, accordig to Eq. (5), operators Ĥ ad N differ oly by the additio of a idetity operator ad the multiplicatio by a c-umber, these operators commute. Hece, accordig to the geeral argumets of Sec. 4.5, they share the set of statioary (Fock) eigestates, ad we ca write the eigeproblem for the ew operator as N N, (5.7) where N are some eigevalues that, accordig to Eq. (5), determie also the eergy spectrum of the oscillator: E N. (5.8) Chapter 5 Page of 5

227 So far, we kow oly that all eigevalues N are real, but ot much more. I order to calculate them, let us carry out the followig calculatio - spledid i its simplicity ad efficiecy. Cosider the result of actio of operator N o the ket-vector â. Usig the defiitio (6) ad the associative rule, we may write N a a a a a aa. (5.9) Now usig the commutatio relatio (4), ad the Eq. (7), we may cotiue as a aa a a a I a Let us summarize the result of this calculatio: N I a N N a. (5.) N a N a. (5.) Performig a absolutely similar calculatio with operator â, we ca also get aother formula: a N a N. (5.) It is time to stop calculatios ad traslate these results ito plai Eglish: if is a eigeket of operator N with eigevalue N, the â ad â are also eigekets of that operator, with eigevalues (N + ), ad (N - ), respectively. This statemet may be preseted with the ladder diagram show i Fig. 5. eigeket a â a a a... a a a a a... eigevalue of N N N N Fig Hierarchy (the ladder diagram ) of eigestates of a D harmoic oscillator. Arrows show the actios of the creatio ad aihilatio operators o the eigestates. Operator â moves the system a step up the ladder, while operator â brigs it oe step dow. I other words, the former operator creates a ew ecitatio of the system, 3 while the latter operator kills ( aihilates ) such ecitatio. This is why â is called the creatio operator, ad â, the aihilatio operator. I its tur, accordig to Eq. (7), operator N does ot chage the state of the system, but couts its positio o the ladder: N N. (5.3) N 3 For the electromagetic field oscillators, such ecitatios are called photos; for the mechaical wave field oscillators, phoos, etc. Chapter 5 Page of 5

228 This is why N is called the umber operator, i our curret cotet meaig the umber of the elemetary ecitatios of the oscillator. This calculatio still eeds a completio. Ideed, we still do ot kow whether the ladder show i Fig. 5 shows all eigestates of the oscillator, ad what eactly the umbers N are. Fasciatig eough, both questios may be aswered by eplorig a sigle parado. Let us start with some state (step of the ladder), ad keep goig dow it, applyig operator â agai ad agai. Each time, eigevalue N is decreased by oe, so that evetually it should become egative. However, this caot happe, because ay real eigestate, icludig the states preseted by kets d â ad, should have a positive orm see Eq. (4.6). Comparig the orms,, d a a N N, (5.4) we see that the both of them caot be positive simultaeously if N is egative. The way toward the resolutio of this parado is to otice that the actio of the creatio ad aihilatio operators o the statioary states may cosist i ot oly their promotio to the et step of the ladder diagram, but also by their multiplicatio by some c-umbers: a A, a A'. (5.5) (Liear relatios () ad () clearly allow that.) Let us calculate coefficiets A assumig, for coveiece, that all eigestates, icludig states ad ( -), are ormalized:, From here, we get A = (N ) /, i.e. a A a * a A A * A N A N * A. (5.6) / i N e, (5.7) where is a arbitrary real phase. Now let us cosider what happes if all umbers N are itegers. (Because of the defiitio of N, give by Eq. (7), it is coveiet to call these itegers, i.e. by the same letter as the correspodig eigestate.) The whe we have come dow to state with =, a attempt to make oe more step dow gives a. (5.8) But i accordace with Eq. (4.9), the state i the right-had part of this equatio is the ull-state, i.e. does ot eist. 3 This gives the (oly kow :-) resolutio of the state ladder parado: the ladder has the lowest step with N = =. As a by-product of our discussio, we have obtaied a very importat relatio N =, which meas, i particular, that the state ladder icludes all eigestates of the oscillator. Pluggig this relatio ito Eq. (8), we see that the full spectrum of eigeeergies of the harmoic oscillator is described by the simple formula 3 Please ote agai the radical differece betwee the ull-state i the right-had part of Eq. (8) ad the state described by ket-vector i the left-had side of that relatio. The latter state does eist ad, moreover, presets the most importat, groud state of the system, with = - see Eq. (.69). Chapter 5 Page 3 of 5

229 E,,,..., (5.9) which was already discussed i Sec... It is rather remarkable that the bra-ket formalism has allowed us to derive it without calculatio of the correspodig (rather cumbersome) wavefuctios () see Eqs. (.79). Moreover, the formalism may be also used to calculate virtually ay bra-ket pertaiig to the oscillator, without usig (). I order to illustrate that, let us first calculate A participatig i the latter of relatios (5). This ca be doe absolutely similarly to the above calculatio of A ; otherwise, sice we already kow that A = (N ) / = /, we may otice that accordig to Eqs. (6) ad (5), the eigeproblem (7), that i our ew otatio for N becomes N, (5.) may be rewritte as ' a a a A A A. (5.) Comparig the first ad the last form of this equality, we see that A - = /A = /, i.e. A = ( + ) / ep(i ). Takig all phases ad equal to zero for simplicity, we may reduce Eqs. (5) to their fial, stadard form 33 / /, a a. (5.) Now we ca use these formulas to calculate, for eample, the matri elemets of operator i the Fock state basis: ' ' ' a a ' a ' a. (5.3) To complete the calculatio, we may ow use Eqs. () ad the Fock state orthoormality: The result is ' (5.4) '. Up ad dow the Fock state ladder ' / ', ', ', ', / ( ) / / ( ) /. m Actig absolutely similarly, for the mometum bra-kets we get a similar epressio: (5.5) Coordiate s matri elemets ' / m p i ', ', / ( ) /. (5.6) Hece the matrices of both operators i the Fock-state basis have oly two diagoals, adacet to the mai diagoal; all other elemets (icludig the diagoal oes) are zeros. 33 A useful memoic rule is that the c-umber coefficiet i ay of these relatios is equal to the square root of the largest umber of the two states it relates. Chapter 5 Page 4 of 5

230 Matri elemets of higher powers of these operators, as well as their products, may be hadled similarly, though the higher is the power, the bulkier is the result. For eample, 34 ' ' ' " " " ', " ', " ", ", " ( ) / ( )( ) / ', ', ( ) ',. ( " ) / ( " ) / / ( ) / (5.7) For applicatios, the most importat of these matri elemets are those o its mai diagoal: ( ). (5.8) This epressio shows, i particular, that the epectatio value of oscillator s potetial eergy i -th Fock state is m U. (5.9) This is eactly ½ of the total eergy (9) of the oscillator. As a saity check, a absolutely similar calculatio of the kietic eergy shows that p m m p, i.e. both partial eergies equal E /, ust as i a classical oscillator. 35 (5.3) 5.5. The Glauber ad squeezed states There is evidetly a huge differece betwee a quatum statioary (Fock) state of the oscillator ad its classical state. Ideed, let us write the classical Hamilto equatios of motio of the oscillator (usig capital letters to distiguish the classical variables from argumets of quatum wavefuctios): P U X, P m X. m (5.3) O the phase plae with Cartesia coordiates ad p (Fig. 6), these equatios describe clockwise rotatio of the represetatio poit {X(t), P(t)} alog a elliptic traectory startig from the iitial poit {X(), P()}. (The ormalizatio of mometum by m, similar to the oe performed by the secod of Eqs. (96), makes the traectory pleasigly circular, with a costat radius equal to oscillatio s amplitude A, reflectig the costat full eergy 34 The first lie of Eq. (7), evidetly valid for ay time-idepedet system, is the simplest of the so-called sum rules, which will be repeatedly discussed below. 35 Still ote that operators of the partial (potetial ad kietic) eergies do ot commute with either each other or with the full-eergy (Hamiltoia) operator, so that the Fock states are ot their eigestates. Chapter 5 Page 5 of 5

231 P( t) P() m E c A, A X ( t) cost X () m m, (5.3) determied by the iitial coditios.) For the forthcomig compariso with quatum states, it is coveiet to describe this classical solutio by the followig dimesioless comple variable ( t ) P( t) ( ), X t i m (5.33) which is essetially the stadard comple-umber represetatio of system s positio o the D phase plae, with A/. With this defiitio, Eqs. (3) are coveietly merged ito oe equatio, i, (5.34) with a evidet, very simple solutio i t ( t) () e, (5.35) where the costat () may be comple, ad is ust the (ormalized) classical comple amplitude of oscillatios. 36 This equatio describes siusoidal oscillatios of both X(t) Re[(t)] ad P Im[(t)], with a phase shift of / betwee them. p / m P/ m A X / Fig Schematic represetatio of various states of a harmoic oscillator o the phase plae. The bold black poit represets a classical state, with the dashed lie showig its traectory. (Very imperfect) classical images of the Fock states with =,, ad are show i blue, while the blurred red spot is the (equally schematic) Glauber state s image. Fially, the mageta elliptical spot is a classical image of a squeezed groud state. Arrows show the directio of states evolutio i time. O the other had, accordig to the basic Eqs. (4.57)-(4.58), the time depedece of a Fock state, as of a statioary state of the oscillator, is limited to the phase factor ep{-ie t/} ot i observables, but rather i the wavefuctio, ad a result, gives time-idepedet epectatio values of, p, or of ay fuctio thereof. (Moreover, as Eqs. (5) ad (6) show, = p =.) Takig ito 36 See, e.g., CM Chapter 4, especially Eqs. (4.4) ad Fig. 4.9 ad its discussio. Chapter 5 Page 6 of 5

232 Glauber state i coordiate represetatio accout Eqs. (9) ad (3), the closest (though very imperfect) geometric image 37 for such a state o the phase plae is a blurred circle of radius A = ( + ) /, alog which the wavefuctio is uiformly spread as a wave see the blue rigs i Fig. 6. For the groud state ( = ), with wavefuctio (.69), a better image is a blurred roud spot, of radius ~, at the origi. However, the Fock states are ot the oly possible quatum states of the oscillator: accordig to the basic Eq. (4.6), a state described by ket-vector (5.36) with ay set of (comple) c-umbers, is also its legitimate state, subect oly to the ormalizatio coditio =, givig. (5.37) It is atural to ask: ca we select coefficiets i such a special way that the state properties would be closer to the classical oes; i particular the epectatio values ad p of coordiate ad mometum would evolve i time ust as the classical values X(t) ad P(t), while the ucertaities of these observables would be time-idepedet ad the same as i the groud state: m /, m p m /, (5.38) with the smallest possible value of the ucertaity product, p = /. 38 Let me show that such a Glauber state, 39 which is schematically represeted i Fig. 6 by a blurred red spot aroud the classical poit {X(t), P(t)}, is ideed possible. Coceptually the simplest way to fid the correspodig coefficiets would be to calculate, p, ad p for a arbitrary set of, ad the try to optimize these coefficiets to reach our goal. However, this problem may be solved much easier usig wave mechaics. Ideed, let us cosider the followig wavefuctio m P( t) (, t) C ep X ( t) i. (5.39) Its compariso with Eqs. (.6) ad (.69) shows that this is ust a Gaussia wave packet with the average mometum P ad the coordiate width give by Eq. (38), but shifted alog ais by X. 37 I have to cofess that such geometric mappig of a quatum state o the phase plae [, p] is ot eactly defied; you may thik about colored areas i Fig. 6 as regios of pairs {, p} most probably obtaied i measuremets. A quatitative defiitio of such a mappig will be give i Sec. 7.3 usig the Wiger fuctio, though, as we will see, eve such imagig defiitio has certai iteral cotradictios. Still such cartoos may have cosiderable cogitive/heuristic value, if their limitatios are kept i mid. 38 I the quatum theory of measuremets, Eqs. (38) are frequetly referred to as the stadard quatum limit. 39 Named after R. J. Glauber who studied these states i detail i 965, though they had bee discussed i brief by E. Schrödiger as early as i 96. Aother popular ame, coheret, for the Glauber states is very misleadig, because all the quatum states we have studied so far (icludig the Fock states) may be preseted as coheret (pure) superpositios of the basis states. Chapter 5 Page 7 of 5

233 Hece, this wavefuctio satisfies all the above requiremets, ad a straightforward (though a bit bulky) differetiatio over ad t shows it also satisfies oscillator s Schrödiger equatio, provided that that fuctios X(t) ad P(t) satisfy classical Eqs. (3). This fact is true eve for a more geeral situatio whe the oscillator, iitially i its groud state 4 comes uder effect of a classical force F(t). (Evidetly, for its descriptio its is sufficiet to add this fuctio to the right-had part of the secod of Eqs. (3).) Moreover, the electromagetic radiatio formed i good (sigle-mode) lasers is also i the Glauber state. (As will be discussed i Chapter 9, the eperimetal formatio of Fock states, with the oly eceptio of =, i.e. the groud state, is much harder.) This is why the Glauber states are so importat. Though Eq. (39) gives the full wave-mechaics descriptio of a Glauber state, there is a substatial place for the bra-ket formalism here too. For eample, i order to calculate the correspodig coefficiets i epasio (36), * d ( ) ( d, (5.4) ) we would eed to use ot oly the simple Eq. (39), but also the Fock state wavefuctios (), which are ot very appealig see Eq. (.79). Istead, this calculatio may be readily doe i the bra-ket formalism, givig us oe importat byproduct result. Let us start from epressig the double shift of the groud state (by X ad P), that has led us to Eq. (39), i the operator laguage. Forgettig about the P for a miute, let us fid a traslatio operator T X that produces the desirable shift of coordiate by X of a arbitrary wavefuctio () say represeted as the stadard wave packet (59). Evidetly, the result of its actio, i the coordiate represetatio, is p( X ) T X ( ) ( X ) p i dp ( )ep. (5.4) / Hece, the shift may be achieved by the multiplicatio of each Fourier compoet of the packet, with mometum p, by ep{-ipx/}. This gives us a hit that the geeral form of the traslatio operator, valid i ay represetatio, should be px T X ep i. (5.4) The proof of this formula is provided by the fact that ay operator is uiquely determied by the set of its matri elemets i ay full ad orthogoal basis, i particular the basis of mometum states p. Accordig to Eq. (4), the aalog of Eq. (3) for the p-represetatio, applied to the traslatio operator (which is evidetly local), is px p T X p' ( p' ) ( p p' )ep i ( p), (5.43) so that operator (4) does eactly the ob we eed it to. X-traslatio operator 4 As will be discussed i Chapter 7, the groud state may be readily formed, for eample, by providig a weak couplig of the oscillator to a low-temperature (k B T << ) eviromet. Chapter 5 Page 8 of 5

234 -traslatio operator The operator that provides the shift of mometum by P is absolutely similar - with the opposite sig uder the epoet, due to the opposite sig of the epoet i the reciprocal Fourier trasform, so that the simultaeous shift by both X ad P may be achieved by the followig traslatio operator: P px T epi. (5.44) As we already kow, for a harmoic oscillator the creatio-aihilatio operators are more atural, so that we may use Eqs. (96) ad (99) to recast Eq. (44) as Glauber state as groud state s traslatio ep * T a a, with * T ep a αa, (5.45) where the c-umber (geerally, a fuctio of time) is defied by Eq. (33). Now, accordig to Eq. (39), we may form the Glauber state s ket-vector ust as T. (5.46) This formula looks ice ad simple, but makig practical calculatios (say, calculatig epectatio values of variables) with the traslatio operator (44) is ot too easy because of its epoet-of-operators form. Fortuately, it turs out that a much simpler represetatio for the Glauber state is possible. To show tha, let us start with the followig geeral (ad very useful) property of epoetial fuctios of operators: if A, B I, (5.47) (where  ad B are arbitrary operators, ad is a c-umber), the 4 Let us apply Eqs. (47)-(48) to two cases, both with First, let us take * A a a, A B ep A B I. ep (5.48) A T (5.49) so that ep, ep A. T B I, the Eq. (47) is valid with =, ad Eq. (48) yields T Î, (5.5) T This equality meas that the traslatio operator is uitary ot a big surprise, because if we shift a classical poit o the comple phase plae by (+) ad the by (-), we certaily must come back to the iitial positio. Relatio (5) meas merely that this fact is also true for ay quatum state as well. Secod, let us take B a ; i order to verify Eq. (47) ad fid the correspodig, let us calculate the commutator. Usig, at the due stage of calculatio, Eq. (4), we get * A, B a - a, a a, a I, (5.5) 4 The proof of Eq. (48) may be readily achieved by epadig operator f ( ) ep A B ep A i the Taylor series i the c-umber parameter, ad the evaluatig the result at =. Chapter 5 Page 9 of 5

235 so that i this case =, ad Eq. (48) yields T a T a I. (5.5) We have approached the summit of this beautiful calculatio. Let us cosider operator T T a T. (5.53) Usig Eq. (5), we may reduce this epressio to â T, while the applicatio of Eq. (5) to the same epressio yields T a T. Hece, we get the followig operator equality a T T a T, (5.54) which may be applied to ay state. Now actig by these two operators o the groud state ad usig the facts that â is the ull-state, while T, we fially get a very simple ad elegat result: 4 â. (5.55) Thus ay Glauber state is ust oe of eigestates of the aihilatio operator, amely the oe with the eigevalue equal to parameter, i.e. to the comple represetatio (33) of the classical state which is the ceter of the Glauber state s distributio. 43 This fact makes the calculatios of the Glauber state properties much simpler. As the simplest eample, let us use Eq. (55) to fid i the Glauber state: a a a a. (5.56) I the first term i the paretheses, we ca apply Eq. (55) directly, while i the secod term, we ca * use the bra-couterpart of that relatio, a. Now assumig that the Glauber state is ormalized, =, ad usig Eq. (33), we get * * X, (5.57) Actig absolutely similarly, we may readily eted this saity check to verify that p = P, ad that ad p ideed obey Eq. (38). As a more thorough saity check, let us use Eq. (55) to re-calculate Glauber state s wavefuctio (39). Ier-multiplyig both sides of that relatio by bra-vector, ad usig defiitio (98a) of the aihilatio operator, we get Glauber state as operator a s eigestate 4 It is also rather couter-ituitive. Ideed, accordig to Eq. (), the aihilatio operator â, actig o a Fock state, beats it dow to the lower-eergy state ( ) see Eq. (9). However, accordig to Eq. (55), its actio o a Glauber state does ot lead to the state chage ad hece to a eergy decrease! The resolutio of this parado may be achieved via represetatio of the Glauber state as a series of Fock states see Eq. (65) below. Operator â ideed trasfers each Fock compoet to a lower-eergy state, but it also re-weighs each term of the epasio, so that the complete eergy of the Glauber state remais costat. 43 Note that the spectrum of eigevalues of eigeproblem (55) is cotiuous it may be ay comple umber! Chapter 5 Page 3 of 5

236 p i m. (5.58) Sice is the bra-vector of the eigestate of the Hermitia operator, they may be swapped, with the operator givig its eigevalue ; actig o that bra-vector by the (local!) operator of mometum, we have to use it i the coordiate represetatio (63). As a result, we get m. (5.59) But is othig else tha the Glauber state s wavefuctio, so that Eq. (53) gives for it a firstorder differetial equatio. (5.6) m Chasig ad to the opposite sides of the equatio, ad usig defiitio (33) of parameter, we may brig this equatio to a form m P X i m. (5.6) Itegratig both parts, we retur to Eq. (39) that had bee derived by wave-mechaics meas. Now that we ca use Eq. (55) for fidig coefficiets i the epasio (36) of the Glauber state i series over the Fock states. Pluggig Eq. (36) ito each side of Eq. (55), usig the first of Eq. () i the left-had part, ad requirig the coefficiets at each ket-vector i both parts to be equal, we get the followig recurrece relatio for the coefficiets:. / (5.6) ( ) Assumig some value of, ad applyig the relatio sequetially for =,, etc., we get. (5.63) / (!) Now we ca fid from the ormalizatio requiremet (37), gettig. (5.64)! I this sum, we may readily recogize the Taylor epasio of fuctio ep{ }, so that the fial result (besides a arbitrary commo phase multiplier) is Glauber state as Fock states superpositio ep. (5.65) / (!) Chapter 5 Page 3 of 5

237 It meas i particular that the probability W * of fidig the system eergy o -th eergy level (9) obeys the well-kow Poisso distributio (Fig. 7):.8 W Fig The Poisso distributio for several values of. Note that W are defied oly for iteger values of ; lies are oly guides for the eye. W! e, (5.66) Poisso distributio where i our particular case. (5.67) For applicatios, perhaps the most importat mathematical property of this distributio is / ; (5.68) r.m.s. fluctuatio ote also that at, ad hece, the Poisso distributio approaches the Gaussia ( ormal ) oe. Now let us discuss the evolutio of the Glauber state i time. I the Schrödiger laguage, it is completely described by dyamics (3) of the c-umber shifts X(t) ad P(t) participatig i wavefuctio (39). Note agai that, i cotrast to the spread of the wave packet of a free particle, discussed i Sec.., i the harmoic oscillator the Gaussia packet of special width (38) does ot spread at all! A alterative ad equivalet way of dyamics descriptio is to use the Heiseberg equatio of motio. As Eqs. (4) ad (48) tell us, such equatios for Heiseberg operators of coordiate ad mometum they have to be similar to the classical equatio (3): p H, H p H m H. (5.69) m Now usig Eqs. (98), for the Heiseberg-picture creatio ad aihilatio operators we get equatios a i a, a i a, (5.7) H H that are completely similar for the classical equatio (34) for the c-umber parameter ad its comple cougate, ad hece have the solutios idetical to Eq. (35): H H Chapter 5 Page 3 of 5

238 Squeezed groud state Squeezig operator it i t a ( t) a () e, a ( t) a ( e. (5.7) H H H H ) As was discussed i Sec. 4.6, such equatios are very coveiet because they eable simple calculatio of time evolutio of observables for ay iitial state of the oscillator (Fock, Glauber, or ay other) usig Eq. (4.9). Applied to a Glauber state (), such calculatio gives the same results as have already bee derived earlier i this sectio, i particular cofirms that the Gaussia wave packet of the special width (38) does ot spread i time. Now let us cosider what happes if the iitial wave packet is still Gaussia, but has a differet width, say < /. As we already kow from Sec.., the mometum spread p will be correspodigly larger, still with the smallest ucertaity product: p = /. Such squeezed groud state s, with zero epectatio values of ad p, may be geerated from the Fock/Glauber groud state: s S s, (5.7a) usig the so-called squeezig operator, S * s ep s a a sa a, (5.7b) which depeds o a comple c-umber parameter s = re i. Parameter s modulus r determies the squeezig degree; it is straightforward to use Eq. (7) for checkig that if s is real ( =, = r), the / / m m r e m r e, p r e r e, so that p. (5.73) O the phase plae (Fig. 6), this state, with r >, may be represeted by a oval spot squeezed alog ais (hece the state s ame) ad stretched alog ais p; the same formulas but with r < describe the opposite squeezig. O the other had, phase of the squeeze parameter s determies the agle / of oval s tur about the phase plae origi see the mageta ellipse i Fig. 6; if, Eqs. (73) are valid for variables {, p } obtaied from {, p} via clockwise rotatio by that agle. For ay of such origi-cetered squeezed states, time evolutio is reduced to a icrease of the agle with rate, i.e. to the clockwise rotatio of the ellipse, without its deformatio, with agular velocity see the mageta arrows i Fig. 6. As a result, ucertaities ad p oscillate i time with double frequecy, while their product is costat at its miimal possible value /. Such squeezed groud states have importat implicatios for quatum measuremets (see Sec. 7.7 below) ad may be formed, for eample, by parametric ecitatio of the oscillator, 44 with a parameter modulatio depth close to, but still below the threshold of parametric oscillatios ecitatio. Ufortuately, I do have time for a further discussio of this iterestig topic, 45 but still eed to metio 44 For a discussio ad classical theory of this effect, see, e.g., CM Sec See, e.g., Chapter 7 i C. Gerry ad P. Kight, Itroductory Quatum Optics, Cambridge U. Press, 5, ad the spectacular measuremets of the Glauber ad squeezed states of electromagetic (light) oscillators by G. Breitebach et al., Nature 387, 47 (997), very large (te-fold) squeezig i such oscillators by H. Vahlbruch et al., Phys. Rev. Lett., 336 (8); ad recet first measuremets of the (so far, slight) squeezig i mechaical resoators, with eigefrequecy / as low as 3.6 MHz, by E. Wollma et al., Sciece 349, 95 (5). Chapter 5 Page 33 of 5

239 a more geeral class of squeezed states, cetered to a arbitrary poit {X, P} rather tha the origi, that may be formed by a additioal actio of the displacemet operator (44) o the squeezed groud state (7). Calculatios similar to those that led us from Eq. (45) to Eq. (55), but ow for the product operator T S s rater tha bare T, show that such a geeral squeezed state is a eigestate of the followig mied operator with eigevalue b a i cosh r a e sih r, (5.74a) * i cosh r e sih r. (5.74b) For the particular case =, Eq. (74b) yields =, i.e. the actio of operator (74a) o the squeezed groud state s with the same r ad yields the ull-state, thus geeralizig Eq. (8), which is valid for the usual (o-squeezed) groud state Revisitig spherically-symmetric problems Oe more blak spot to fill has bee left i our study of wave mechaics of spherically-3d symmetric systems i Sec Ideed, while the eigefuctios describig aially-symmetric D systems, ad the azimuthal compoets of those i spherically-symmetric 3D systems, are very simple, e im m, m,,,... (5.75) / the polar compoets of the eigefuctios i the latter case (i.e., of spherical harmoics) iclude the associate Legedre fuctios P m l (cos) that may be epressed via elemetary fuctios oly idirectly see Eqs. (3.65) ad (3.68). This makes all the calculatios less tha trasparet ad, i particular, does ot allow a clear isight ito the origi of the very simple eigevalue spectrum see, e.g., Eq. (3.63). The bra-ket formalism, applied to the agular mometum operator, allows oe to get such isight, ad also produces a very coveiet tool for may calculatios ivolvig spherically-symmetric potetials. Let us start from usig the correspodece priciple to spell out the quatum-mechaical operator of the orbital agular mometum L rp of a poit particle: y z L r p y z, i.e. L yp z zp y, etc., (5.76) p p p y z From this defiitio, we ca readily calculate the commutatio relatios for all Cartesia compoets of operators L, r, ad p ; for eample, L, y yp zp, y z p, y iz, (5.77) z y etc. Usig the sequetial umberig of coordiate aes ( = r, etc.), the summary of these calculatios may be preseted i similar, compact (ad beautiful!) forms: y Agular mometum operator Chapter 5 Page 34 of 5

240 Key commutatio relatios L, r ' ir " '", L, p ' ip " '", L, L i L ' " '", (5.78) where each of idices ad ad may take ay of values,, ad 3, is the complemetary ide of the same set (ot equal to either or ), ad is the Levi-Civita symbol (or permutatio symbol ). 46 Also itroducig i the atural way a (scalar!) operator of the observable L = L, Operator of L y z L L L L, (5.79) it is straightforward to check that this operator commutes with each of the Cartesia compoets:,. L (5.8) L This result, at the first sight, may seem to cotradict the last of Eqs. (78). Ideed, have t we leared i Sec. 4.5 that commutig operators (e.g., L ad ay of L ) share their eigestate sets? If yes, should t that mea that this set has to be commo for all 4 operators? 47 The resolutio i this parado may be foud i the coditio that was metioed ust after Eq. (4.38), but (sorry!) ot sufficietly emphasized there. Accordig to that relatio, if a operator has degeerate eigestates (i.e. if A = A eve for ), they should ot be ecessarily shared by aother compatible operator. This is eactly the situatio with the orbital agular mometum operators, that may be schematically show at a Ve diagram (Fig. 8): 48 the set of eigestates of operator L is highly degeerate, 49 ad is broader tha those of the compoet operators L (that, as will be show below, are o-degeerate util we cosider particle s spi). L Lz Ly Fig Ve diagram showig (schematically) the partitioig of the set of eigestates of operator L. Each ier sector correspods to the states shared with oe of Cartesia compoet operators L, while the outer (shaded) rig presets the eigestates of L that are ot shared with either of L - e.g., all liear combiatios of eigestates of differet compoet operators. 46 See, e.g., MA Eq. (3.). 47 The importace of this issue stems from the followig fact: it is easy (ad is hece left to the reader :-) to use Eqs. (5.78) to prove that operators of all L ad of L commute with the Hamiltoia of a particle i the spherically-symmetric potetial U(r), ad hece all their eigestates are the statioary states i such a field. 48 This is ust a particular eample of Ve diagrams (itroduced i the 88s by J. Ve) that show possible relatios (such as itersectios, uios, complemets, etc.) betwee various sets of obects, ad are a very useful tool i the geeral set theory. 49 Note that this particular result is cosistet with the classical picture of the agular mometum vector: eve whe is legth is fied, the vector may be orieted i various directios, correspodig to differet values of its Cartesia compoets. However, i the classical picture, all these compoet may be fied simultaeously, while i the quatum picture this is ot true. Chapter 5 Page 35 of 5

241 Let us focus o ust oe of these 3 oit sets of eigestates by traditio, of operators L ad L z. (This traditio is due to the caoical form of spherical coordiates, i which the polar agle is measured from ais z. Ideed, usig Eqs. (63), i the coordiate represetatio we get the followig epressio, L p yp z y i y i i. (5.8) y Writig the stadard eigeproblem for the operator i this represetatio, L z m Lz m, we see that it is satisfied by eigefuctios (75), with eigevalues L z = m - at was already coectured i Sec. 3.5.) More specifically, let us cosider a set of eigestates {l, m} correspodig to a certai degeerate eigevalue of operator L but all possible eigevalues of operator L z, i.e. all possible quatum umbers m. (At this poit, l is ust some parameter that determies the eigevalue of L ; it will be defied more eplicitly i a miute.) I order to aalyze this set, it is istrumetal to itroduce the so-called ladder (also called, respectively, raisig ad lowerig ) operators L L i (5.8) L y - ote a substatial similarity betwee this defiitio ad Eqs. (98). It is straightforward to use this defiitio ad the last of Eqs. (78) to calculate the followig commutators: Ladder operators ad mai commutatio relatios L, L L z, ad L z, L L ad use Eq. (79) to prove aother importat relatio: L, (5.83) z z Now let us rewrite the last of Eqs. (83) as L L L L. (5.84) L L L L L, (5.85) ad act by its both parts o the ket-vector l, m of the set specified above: L L z z l, m z L L l, m L l, m. (5.86) z Importat commutatio relatios Sice eigevalues of operator With that, Eq. (86) may be recast as L z are equal to m, i the first term of the right-had part we may write L z L z l, m m l, m. (5.87) L l, m m L l, m. (5.88) I a spectacular similarity with Eqs. ()-() for the harmoic oscillator, Eq. (88) meas that states L l, m are also the eigestates of operator L, correspodig to eigevalues (m ). Thus z the ladder operators act eactly as the creatio ad aihilatio operators i the oscillator, movig the system up or dow a ladder of eigestates (Fig. 9). The most sigificat differece is that ow the state Chapter 5 Page 36 of 5

242 Relatio betwee m ad l ladder must ed i both directios, because a ifiite icrease of m, with whatever sig, would cause the epectatio values of operator L Ly L Lz, (5.89) which correspods to a o-egative observable, to become egative. Hece there should be two states o both eds of the ladder, l, m ma ad l, m mi, for whom L l, m, L l, m. (5.9) ma Due to the symmetry of the whole problem with respect to the replacemet m -m, we should have m mi = - m ma. This m ma is eactly the quatum umber that is traditioally called l, so that mi l m l. (5.9) eigeket l,l L l, m l, m L l, m l, l L L L L L L L L eigevalue of Lz l m m m Fig Hierarchy (ladder diagram) of the commo eigestates of operators L ad L z. l Evidetly, this relatio of quatum umbers m ad l is compatible with the almost-classical image of various orietatios of the agular mometum vector of the same legth i various directios, with its z-compoet takig several (l + ) possible values m. I this simple picture, however, L would be equal to square of (L z ) ma, i.e. to (l) ; however, this is ot so. Ideed, applyig the operator equality (84) to the top state l, m ma l, l, we get Eigevalues of L L l, l L l z l, l L z l l, l. l, l L L l, l l l, l l l, l (5.9) Sice by our iitial assumptio, all eigevectors l, m correspod to the same eigevalue of operator L, this result meas that all these eigevalues are equal to l(l + ). Just as i case of the spi-½ vector operators, the deviatio of this result from l may be iterpreted as the result of uavoidable ucertaities ( fluctuatios ) of the - ad y-compoets of the agular mometum, that give a fiite positive cotributio to L eve if the agular mometum vector is aliged i the best possible way with the z-ais. Chapter 5 Page 37 of 5

243 Now let us compare our results with those of Sec Usig the epressio of Cartesia coordiates via the spherical oes eactly as was doe i Eq. (8), we get the followig epressios for the ladder operators (8) i the coordiate represetatio: i L e i cot a. (5.93) Now pluggig this equatio, together with Eq. (8), ito Eq. (84), we get L si. (5.94) si si But this is eactly the operator (besides its divisio by costat parameter mr) that stads i the lefthad part of Eq. (3.56). Hece that equatio, which was eplored by the brute-force (wavemechaical) approach i Sec. 3.6, may be uderstood as the eigeproblem for operator L i the coordiate represetatio, with eigefuctios Y m l (,) correspodig to eigekets {l, m}, ad eigevalues L = mre. As a remider, the mai result of that, rather ivolved aalysis was epressed by Eq. (3.63), which ow may be rewritte as Ll mr El l( l ), (5.95) i a full agreemet with what was obtaied i this sectio by much more efficiet meas based o the bra-ket formalism. I particular, it is fasciatig to see how easy are ow may operatios with eigevectors l, m, albeit wavefuctios of these states, spherical harmoics Y l m (,), have rather comple spatial behavior please have oe more look at Eq. (3.7) ad Fig Coordiate represetatio of agular mometum operators 5.7. Spi ad its additio to orbital agular mometum Surprisigly, the theory described i the last sectio is useful for much more tha orbital motio aalysis. I particular, it helps to geeralize the spi-½ results discussed i Chapter 4 to other values of spi s the parameter still has to be defied. For that, let us otice that the commutatio relatios that were derived, for s = ½, from the Pauli matri properties, may be rewritte i eactly the same form as Eqs. (78) ad (8) for the orbital mometum: S, S i S, S, S ' (5.96) " It has bee postulated (ad cofirmed by umerous eperimets) that these relatios hold true for ay quatum particle. Now ote that all the calculatios of the last sectio have bee based almost eclusively o such relatios the eceptio will be discussed immietly. Hece, we may repeat them for spi operators, ad get the relatios similar to Eq. (87) ad (9): '" Spi operators: commutatio relatios 5 The reader is challeged to use the commutatio relatios discussed above to prove oe more importat property of the commo eigestates of operators L z ad L : l, m r l', m', if either l' l, or m m', or both. This property is the basis of the selectio rules for dipole quatum trasitios, to be discussed later i the course, especially i Sec Chapter 5 Page 38 of 5

244 Spi operators: eigestates ad eigevalues Total agular mometum operator z s, m s s s s S m s m S,, s, m s( s ) s, m, s, s m s, (5.97) where m s is a quatum umber similar to the orbital umber m, ad the o-egative costat s is defied as the maimum value of m s. This parameter is eactly what is called particle s spi - i the arrow sese of the word. Now let us retur to the oly part of our orbital momet calculatios that has ot bee derived from the commutatio relatios. This was the fact, based o solutio (75) of the orbital motio problems, that quatum umbers m (the aalog of m s ) are iteger. For spi, we do ot have such a solutio, so that the spectrum of umbers m s (ad hece its limits s) should be foud from the more loose requiremet that the eigestate ladder, etedig from s to + s, has a iteger umber of steps. Hece, s has to be iteger, i.e. spi s of a quatum particle may be either iteger (as it is, for eample, for photos ad gluos), or half-iteger (e.g., for all quarks ad leptos icludig electros). 5 For s = ½, this picture yields all spi properties of electro that were derived i Chapter 4 from postulate (4.7). I particular, operators Ŝ ad Ŝ z have oly commo eigestates, with S z = m s = /, ad both with S = s(s +) = (3/4). Note that this aalogy with the agular mometum sheds a ew light o the symmetry properties of electros. Ideed, the fact that m i Eq. (75) is iteger was derived i Sec. 3.5 from the requiremet that makig a full circle aroud ais z, we should fid a similar value of wavefuctio m, which differs from the iitial oe by a icosequetial factor ep{im}. With the replacemet m m s = ½, such operatio would multiply the wavefuctio by ep{i}, i.e. reverse its sig. O course, spi caot be described by a usual wavefuctio, but this odd parity of electros (ad all other spi-½ particles) is clearly revealed i multiparticle systems see Chapter 8. Now we are sufficietly equipped to aalyze particles that have both the orbital mometum ad the spi. I classical mechaics, such a particle would be characterized by the total agular mometum vector J = L + S. Followig the correspodece priciple, we may make a assumptio that quatummechaical properties of this observable may be aalyzed usig the similarly defied vector operator: with Cartesia compoets etc, ad the magitude squared equal to J s J L S, (5.98) z L S, (5.99) y z J z z J J J. (5.) Let us eamie the properties of this vector operator. Sice its two compoets describe differet degrees of freedom of the particle (agai, you may say belog to differet Hilbert spaces ), they may be cosidered as completely commutig: L, S, L, S '. (5.) 5 As a remider, i the Stadard Model of particle physics, such hadros as mesos ad baryos (otably icludig protos ad eutros) are essetially composite particles, with the spi equal to the sum of its compoet quark spis. However, at o-relativistic eergies, protos ad eutros may be cosidered fudametal particles with s = ½. s Chapter 5 Page 39 of 5

245 These above equalities are sufficiet to derive the commutatio rules of the total agular mometum, ad, ot surprisigly, they tur out to be absolutely similar to those of its compoets: J, J i J, J, J '. (5.) " '" Now repeatig all argumets of the last sectio, we may derive the followig epressios for the commo eigestates of operators Ĵ ad Ĵ : J z, m m, m, J z, m ( ), m,, m, (5.3) where ad m are ew quatum umbers. Repeatig the argumets made for m s, we may coclude that ad m may be either iteger or half-iteger. Before we proceed, oe remark o otatio: it is very coveiet to use the same letter m for umberig eigestates of all mometum compoets participatig i Eq. (99), with correspodig idices (, l, ad s), i particular, to replace what we called m with m l. With this replacemet, the mai results of the last sectio may be summarized i the form similar to Eqs. (97) ad (3): Total mometum: commutatio relatios Total mometum: eigestates, ad eigevalues L l, m m l, m, L l, m l( l ) l, m z l l l l l, l, l m l l. (5.4) I order to uderstad which eigestates used is Eqs. (97), (3), ad (4) are compatible with each other, let us use Eqs. (98)-() to calculate the mutual commutators of the operators squared ad their z-compoets. The result is,,, L J S,, L, J, S. J (5.5) J (5.6) z This result may be preseted schematically o the followig Ve diagram (Fig. ), i which the crossed arrows idicate the oly o-commutig pairs of operators. z operators diagoal i the ucoupled represetatio L L z Ŝ Ŝ z Ĵ Ĵ z operators diagoal i the coupled represetatio Fig. 5.. Ve diagram for agular mometum operators, ad their mutually-commutig groups. This meas that ust as for each compoet agular mometum (J, L, ad S) cosidered separately we could select a group of commo eigestates for its magitude squared ad the z- compoet, we also may fid eigestates shared by two broader groups of operators, ecircled with colored lies i Fig.. The first group (withi the red circle), cosists of all operators but J. This meas that there are eigestates shared by 5 remaiig operators, ad they may be characterized by certai values of the correspodig quatum umbers: l, m l, s, m s, ad m. Actually, oly 4 of these Chapter 5 Page 4 of 5

246 Coupled ad ucoupled bases Defiitio of Clebsch- Jorda coefficiets umbers are idepedet, because due to Eq. (99) for these compatible operators, for each eigestate of the group, their magetic quatum umbers m have to satisfy the followig relatio: m m m. (5.7) Hece the commo eigestates of the operators of this group are fully defied by ust 4 quatum umbers, for eample, l, m l, s, ad m s. For some calculatios, especially those for systems whose Hamiltoias iclude oly operators of this group, it is coveiet to the use this set of eigestates as the basis; frequetly this is called the ucoupled represetatio. However, i some situatios we caot igore iteractios betwee the orbital ad spi degrees of freedom (i the commo argo, the spi-orbit couplig), which leads i particular to splittig (called the fie structure) of atomic eergy levels eve i the absece of eteral magetic field. I will discuss these effects i detail i the et chapter, ad ow will oly ote that they may be described by a separate term, proportioal to product L S, i the system s Hamiltoia. If this term is ot egligible, the ucoupled represetatio becomes icoveiet. Ideed, writig l s ) ( J L S L S L S, (5.8) ad lookig at Fig. agai, we see that the operator L S, describig the spi-orbit couplig, does ot commute with operators L z ad Ŝ z. This meas that statioary states of the system with such term i the Hamiltoia do ot belog to the ucoupled represetatio basis. O the other had, Eq. (8) shows that operator L S does commute with all 4 operators of aother group, ecircled with the blue lie i Fig.. Accordig to Eqs. (), (), ad (5), all operators of that group also commute to each other, so that they have commo eigestates that may be marked by the correspodig quatum umbers, l, s,, ad m. This group is the basis for the coupled represetatio of particle s state. Ecludig the quatum umbers l ad s, commo for both groups, from otatio, it is coveiet to deote the commo ket-vectors of each group as, respectively, m, m l, m s,, for the ucolpled represetatio's basis, for the coupled represetatio's basis. (5.9) As we will see i the et chapter, for solutio of some importat problems (e.g., the fie structure of atomic spectra ad the Zeema effect), we will eed the relatio betwee the kets, m ad the kets m l, m s. This relatio may be represeted as the usual liear superpositio,, m m, m m, m, m, (5.) ml, ms l whose bra-kets (c-umbers), essetially the elemets of the uitary matri of the trasformatio betwee two eigestate bases (9), are called the Clebsch-Gorda coefficiets. The best (though imperfect) classical iterpretatio of Eq. () I ca offer is as follows. If the legths of vectors L ad S (i quatum mechaics associated with umbers l ad s, respectively), ad also their scalar product LS, are all fied, the so is the legth of vector J = L + S (whose legth i quatum mechaics is described by quatum umber ). Hece, the classical image of a specific eigeket, m, i which l, s,, ad m are all fied, is a state i which L, S, J, ad J z are fied. s l s Chapter 5 Page 4 of 5

247 However, this fiatio still allows for a arbitrary rotatio of the pair of vectors L ad S (with a fied agle betwee them, ad hece fied LS ad J ) about the directio of vector J - see Fig.. z J z L z S z S L J z J z L z S z L S J Fig. 5.. Classical image of two quatum states with the same l, s,, ad m, but differet m l ad m s. Hece the compoets L z ad S z i these coditios are ot fied, ad i classical mechaics may take a cotiuum of values, two of which (with the largest ad smallest possible values of S z ) are show i Fig.. I quatum mechaics, these compoets are quatized, with their states represeted by eigekets m l, m s, so that a liear combiatio of such kets is ecessary to represet ket, m. This is eactly what Eq. () does. Some of properties of the Clebsch-Gorda coefficiets m l, m s, m may be readily established. For eample, the coefficiets do ot vaish oly if the ivolved magetic quatum umbers satisfy Eq. (7); let us prove this fact. 5 All matri elemets of the ull-operator J ( L S ) (5.) should equal zero i ay basis; i particular z z z, m J ( L S ) m, m. (5.) z z z l s Actig by operator Ĵ o the bra-vector, ad by the sum L S ) o the ket-vector, we get z ( z z ( m m ), m m, m, m (5.3) l s l s thus provig that m l, m s, m, m m l, m s * =, if m (m l + m s ). For the most importat case of spi-½ particles (s = ½, ad hece m s = ½), whose ucoupled represetatio basis icludes (l + ) states, restrictio (7) eables the represetatio of all ovaishig Clebsch-Gorda coefficiets o the simple diagram show i Fig.. Ideed, each coupled-represetatio eigeket, m, with m = m l + m s = m l ½, may be related with o-zero Clebsch-Gorda coefficiets to at most two ucoupled-represetatio eigestates m l, m s. Sice m l may oly take iteger values from l to +l, m may oly take semi-iteger values o the iterval [- l - ½, l + ½]. Hece, by the defiitio of as (m ) ma, its maimum value has to be l + ½, ad for m = l + ½, this is the oly possible value. This meas that the ucoupled state with m l = l ad m s = ½ should be idetical to the coupled-represetatio state with = l + ½ ad m = l + ½: l, m l ml m, ms. (5.4) 5 Oe may thik that Eq. (7) is a trivial corollary of Eq. (99). However, ow we should be a bit more careful, because i the Clebsch-Gorda coefficiets, these quatum umbers characterize differet groups of eigestates. Chapter 5 Page 4 of 5

248 m l m l / m s m l / l / l / l / / l / l... l / l m l l / / m l Fig. 5.. All possible sets of eigevalues m l, m s for a particle with fied l, ad s = ½. Each ucoupled-represetatio state is represeted by a dot, while each the coupled-represetatio state, by a sigle sloped lie coectig the dots. However, already for the et value, m = l ½, we eed to have two values of, so that two m l, m s kets is to be related to two, m kets by two Clebsch-Gorda coefficiets. Sice l chages i uit steps, these values of have to be l ½. This choice, l /, (5.5) evidetly satisfies all lower values of m (agai, with oly oe value, = l + ½, ecessary for the lowest m = -l - ½) see Fig.. Note that the total umber of the coupled-represetatio states is + l + = (l + ), i.e. the same as i the ucoupled represetatio. So, each sum (), for fied, m (ad fied commo parameter l), has at most terms, i.e. ivolves at most Clebsch-Gorda coefficiets. These coefficiets may be calculated i a few steps, all but the last oe rather simple eve for arbitrary spi s. First, the matri elemets of ladder operators L i the stadard z-basis (i.e. i the basis of kets m l ) may be calculated from Eq. (84). Net, the similarity of vector operators J ad S to operators L, epressed by Eqs. (97), (3), ad (4), may be used to argue that the matri elemets of operators S ad J, defied absolutely similarly to L, have similar matri elemets i the bases of kets m s ad m, respectively. After that, actig by operator J L S upo both parts of Eq. (), ad the ier-multiplyig the result by the bra vector m l, m s ad usig the above matri elemets, we get recurrece relatios for the Clebsch-Gorda coefficiets. Fially, these relatios may be recurretly applied to the adacet states i both represetatios, startig from ay of the two states commo for them for eample, from state with ket-vectors (4), correspodig to the top right poit i Fig.. Let me leave these straightforward but a bit tedious calculatios for reader s eercise ad ust cite the fial result of this procedure for s = ½: 53 Clebsch - Gorda coefficiets for s = ½ m m l l m m,, m m s s l l,, m m l m / l l m / l / /,. (5.6a) 53 For arbitrary spi s, the calculatios ad eve the fial epressios for the Clebsch-Gorda coefficiets are rather bulky. They may be foud, typically i a table form, mostly i special moographs see, e.g., A. R. Edmods, Agular Mometum i Quatum Mechaics, Priceto U. Press, 957. Chapter 5 Page 43 of 5

249 For applicatios, it may be more coveiet to use this result i the followig equivalet form: l, m l m / l / m l m, m s l m / l / m l m, m s. (5.6b) We will use this relatio i Sec. 6.4 for a aalysis of the aomalous Zeema effect, based o the perturbatio theory. Moreover, most of the agular mometum additio theory described above is immediately applicable to the additio of agular mometa i multiparticle systems, so we will revisit it i Chapter 8. To coclude this sectio, I have to ote that the Clebsch-Gorda coefficiets (for arbitrary s) participate also i the so-called Wiger-Eckart theorem that epresses matri elemets of certai spherical tesors, i the coupled-represetatio basis, m, via a reduced set of matri elemets. Ufortuately, a discussio of this theorem ad its applicatios would require a higher mathematical backgroud tha I ca epect from my readers, ad more time/space tha I ca afford Eercise problems 5.. Use the discussio of Sec. to fid a alterative solutio of Problem A two-level system is i a quatum state, described by ket-vector = +, with give (geerally, comple) c-umber coefficiets. Prove that we ca always select a 3-compoet vector a = {a, a y, a z } of real c-umbers, such that is a eigestate of operator a σ, where σ is the operator described, i z-basis, by the Pauli matri vector. Fid all possible values of a satisfyig this coditio, ad the secod eigestate of operator a σ, orthogoal to the give. Give a Bloch-sphere iterpretatio of your result A spi-½ particle is i a costat vertical field, so that its Hamiltoia H σ z, but its spi s iitial state is a eigestate of a differet Hamiltoia: 55 H ii a σ a σ a σ a σ. Use ay approach you like to calculate the time evolutio of the epectatio values of the spi compoets. Iterpret the results For ay periodic motio of a sigle particle i a cofiig potetial U(r), the virial theorem of o-relativistic classical mechaics 56 is reduced to the followig equality: y y z z 54 For the iterested reader I ca recommed, either Sec. 7.7 i E. Merzbacher, Quatum Mechaics, 3 rd ed., Wiley, 998, or Sec. 3. i J. Sakurai, Moder Quatum Mechaics, Addiso-Wesley, Cf. Problems 4., 4.3, See, e.g., CM Problem.. Chapter 5 Page 44 of 5

250 T r U, where T is particle s kietic eergy, ad the top bar meas averagig over the period of motio. Prove the quatum-mechaical versio of the theorem for a arbitrary statioary quatum state, i the absece of spi effects: T r U, where the agular brackets mea, as everywhere i these otes, the epectatio value of the variable iside them. Hit: Mimickig the proof of the classical virial theorem, cosider the time evolutio of operator G r p A costat force F is applied to a (otherwise free) D particle of mass m. Calculate the eigefuctios of the problem, usig (i) the coordiate represetatio, ad (ii) the mometum represetatio. Discuss the relatio betwee the results The mometum represetatio of a operator, defied i the Hilbert space of D orbital states of a particle, equals p -. Fid its coordiate represetatio * For a particle movig i a 3D periodic potetial, develop the bra-ket formalism for the q- represetatio, i which a comple amplitude similar to a q i Eq. (.34) (but geeralized to 3D ad all eergy bads) plays the role of the wavefuctio. I particular, calculate operators r ad v i this represetatio, ad use the result to prove Eq. (.37) for D motio i the low-field limit. Hit: Try to geeralize the aalysis of the mometum represetatio i Sec I the Heiseberg picture of quatum dyamics, fid the operator of velocity ad acceleratio, dr dv v ad a, dt dt of a electro movig i a arbitrary electromagetic field. Compare the results with the correspodig classical epressios Calculate, i the WKB approimatio, the trasmissio coefficiet T for tuelig of a D particle with eergy E < U through a saddle-shaped potetial pass where U ad a are real costats. U y, y) U, a ( Chapter 5 Page 45 of 5

251 5.. Calculate the so-called Gamow factor 57 for the alpha decay of atomic uclei, i.e. the epoetial factor i the trasparecy of the tuel barrier, resultig from the followig simple model of the particle s potetial eergy as a fuctio of its distace from the uclear ceter: U r U, ZZ'e, 4 r for r R, for R r, (where Ze = e > is the charge of the alpha-particle, Z e > is that of the ucleus after the decay, ad R is the ucleus radius), i the WKB approimatio. 5.. For a D harmoic oscillator with mass m ad frequecy, calculate: (i) all matri elemets 3 ', ad (ii) diagoal matri elemets 4, where are the Fock states. 5.. Calculate the sum (over all > ) of the so-called oscillator stregths, f m, E E of quatum trasitios betwee the th eergy level ad the groud state, for (i) a D harmoic oscillator, ad (ii) a D particle cofied i a arbitrary statioary potetial * Prove the so-called the Bethe sum rule, ik k E E ' e ' ' m (where k is a c-umber), valid for a D particle movig i a arbitrary time-idepedet potetial U(), ad discuss its relatio with the Thomas-Reiche-Kuh sum rule, whose derivatio was the subect of the previous problem. Hit: Calculate the epectatio value, i a statioary state, of the followig double commutator, D H e ik e ik,,, i two ways first, ust spellig out both commutators, ad, secod, usig the commutatio relatios betwee operators p ad e ik, ad compare the results Simplify the followig operators: (i) ep ia p ep ia, ad 57 Named after G. Gamow, who made this calculatio as early as i 98. Chapter 5 Page 46 of 5

252 (ii) ep iap ep ia p, where a is a c-umber Use the Heiseberg equatio of motio for a direct derivatio of time evolutio law (5.7) of the creatio ad aihilatio operators of a harmoic oscillator Calculate: (i) the epectatio value of eergy, ad (ii) the laws of time evolutio of epectatio values of the coordiate ad mometum for a D harmoic oscillator, provided that i the iitial momet (t = ) it was i state 5 6, where are ket-vectors of the statioary (Fock) states of the oscillator * Re-derive the Lodo dispersio force potetial betwee two 3D harmoic oscillators (already calculated i Problem 3.9), usig the laguage of mutually-iduced polarizatio The discussio of the Glauber state properties i Sec. 5 has used the followig geeral statemet: if A, B I, where  ad B are arbitrary operators, ad is a arbitrary c-umber, the A B ep A B I. ep Prove the statemet. Hit: Oe (of several) ways to prove the statemet is to epad operator f ( ) ep A B ep A ito the Taylor series i c-umber, ad the evaluate it at = A eteral force pulse F(t), of a fiite time duratio T, is eerted o a D harmoic oscillator, iitially i its groud state. Use the Heiseberg-picture equatios of motio to calculate the epectatio value of oscillator s eergy at the ed of the pulse. 5.. Calculate the eergy of the squeezed groud state s of a harmoic oscillator, defied by Eq. (7). 5.. Use Eqs. (5.78) of the lecture otes to prove that operators L ad Hamiltoia of a spiless particle placed i ay cetral potetial field. L commute with the 5.. Prove the followig relatios for the operators of the agular mometum: L L L L L L z z z L L L z. Chapter 5 Page 47 of 5

253 (Oe of them, Eq. (84), was already used i Sec. 6.) 5.3. Accordig to Eqs. (88) ad their discussio, actio of the ladder operators o the commo eigekets l, m of operators L ad L may be described as z ( m, ) l m L L l, m. Calculate coefficiets L (m), assumig that the eigestates are ormalized: l, ml, m = I the basis of commo eigestates of operators L z ad L, described by eigekets l, m: (i) calculate matri elemets l, m L l, m ad l, m L l, m ; (ii) spell out your results for diagoal matri elemets (with m = m ) ad their y-ais couterparts; ad (iii) calculate diagoal matri elemets l, m L L l m ad l, m L L l m. y, y, 5.5. For the state described by the commo eigeket l, m of operators L z ad L i a referece frame {, y, z}, calculate the epectatio values L z ad L z i the referece frame whose ais z forms agle with ais z Write dow the matrices of the followig agular mometum operators: L, L, L, ad, i the z-basis of states with l =. y z L 5.7. Fid the agular part of the orbital wavefuctio of a particle with a defiite value of L, equal to 3, ad the largest possible value of L. What is this value? 5.8. * A charged D particle is trapped i a soft i-plae potetial well U(, y) = m ( +y )/. Calculate its eergy spectrum i the presece of a additioal uiform magetic field B, ormal to the plae Calculate the spectrum of rotatioal eergies of a aially-symmetric, rigid molecule For the state with wavefuctio = Cye -r, with a real, positive, calculate: (i) the epectatio values of observables L, L y, L z ad L, ad (ii) the ormalizatio costat C A agular state of a spiless particle is described by the followig ket-vector: l 3, m l 3, m. Fid the epectatio values of the - ad y-compoets of its agular mometum. Is it sesitive to a possible phase shift betwee two compoet eigekets? Chapter 5 Page 48 of 5

254 5.3. * Simplify the followig double commutator: r, L, r Epress the commutators listed i Eq. (6), J, A. ' L z ' ad J, S z, via L ad Fid the operator T describig the state rotatio by agle about a certai ais, usig the similarity of this operatio with the shift of a Cartesia coordiate, discussed i Sec. 5. The use it to calculate the probabilities of measuremets of a beam of particles with z-polarized spi-½, by a Ster- Gerlach istrumet tured by agle withi the [z, ] plae (where y is the ais of particle propagatio see Fig. 4.) The rotatio ( agle traslatio ) operators T, aalyzed i the previous problem, ad the coordiate traslatio operator T X, discussed i Sec. 5.5 of the lecture otes, have a similar structure: C T ep i, where is a real c-umber, characterizig shift s magitude, ad Ĉ is a Hermitia operator that does ot eplicitly deped o time. (i) Prove that all such operators T are uitary. (ii) Prove that if the shift by, iduced by operator T, leaves the Hamiltoia of some system uchaged for ay, the the variable C, correspodig to the operator Ĉ, is a costat of motio. (iii) Discuss what does the last coclusio give for the particular operators T X ad T A particle is i a state with the orbital wavefuctio proportioal to the spherical harmoic Y (, ). Fid the agular depedece of the wavefuctios correspodig to the followig ket-vectors: (i) L, (ii) L, (iii) L, (iv) L, ad (v) L. y z For a state with defiite quatum umbers l ad, prove that observable LS also has a defiite value, ad calculate this value * Derive the geeral recurrece relatios for the Clebsh-Gorda coefficiets. Hit: Usig the similarity of commutatio relatios, discussed i Sec. 7, geeralize the solutio of Problem 9 to all agular mometum operators, ad apply them to Eq. (98) The byproduct of the solutio of the previous problem is the geeral relatio for the spi operators (valid for ay spi s), which may be rewritte as L Ŝ. 58 Note that the last task is ust a particular case of Problem 4.7 (see also Problem ). Chapter 5 Page 49 of 5

255 m s s m s m / S ms s s, provided that all other quatum umbers are fied. Use this result to spell out the matrices S, S y, S z, ad S of a particle with s =, i the z-basis - defied as the basis i which the matri S z is diagoal * For a particle with spi s, movig i a spherically-symmetric field, fid the rages of possible values of quatum umbers m ad, ecessary to describe, i the coupled represetatio basis: (i) all states with a defiite quatum umber l, ad (ii) a state with defiite value of ot oly l, but also m l ad m s. Give a iterpretatio of your results i terms of the classical geometric vector diagram (see Fig. ) A spi-½ particle moves i a cetrally-symmetric potetial U(r). Usig Eqs. (6) for the Clebsch-Gorda coefficiets, (i) write eplicit epressios for the ket vectors for states that would be simultaeously the eigestates of operators L J,, ad J z, via spi eigekets ad ; (ii) for each such state, fid all the possible values of observables L, L z, S, ad S z, the probability of each listed value, ad the epectatio value for each of the observables Takig ito accout electro s spi, fid the eergy spectrum of a electro, free to move withi a plae, besides beig placed ito a uiform magetic field B, ormal to the plae. Compare the result with the Ladau level picture discussed i Sec. 3.. Chapter 5 Page 5 of 5

256 Chapter 6. Perturbatio Theories This chapter discusses several perturbative approaches to problems of quatum mechaics, ad their simplest applicatios icludig the Stark effect, the fie structure of atomic levels, ad the Zeema effect. Moreover, the discussio of the perturbatio theory of trasitios to cotiuous spectrum ad the Golde Rule of quatum mechaics i the ed of this chapter will aturally brig us to the issue of ope quatum systems to be discussed i more detail i the et chapter. Weakly aharmoic oscillator 6.. Eigevalue/eigestate problems Ufortuately, oly a few problems of quatum mechaics may be solved eactly i the aalytical form. Actually, i the previous chapters we have solved a substatial fractio of such problems for a sigle particle, ad for multi-particle problems the eactly solvable cases are eve more rare. However, most practical problems of physics feature a certai small parameter, ad this smalless may be eploited by various approimate aalytical methods. Earlier i the course, we have eplored oe of them, the WKB approimatio, which is adequate for a particle movig through a slowly chagig potetial profile. Now I will discuss alterative approaches that are more suitable for other cases. The historic ame for these approaches is the perturbatio theory, but it is more fair to speak about several such theories, because they differ depedig o the type of the problem. The simplest perturbatio theories address eigeproblems for systems described by timeidepedet Hamiltoias of the type ` () () H H H, (6.a) () where the perturbatio operator Ĥ is small - i the sese its additio to the uperturbed operator () Ĥ results i a relatively small chage of eigevalues E of the system. A typical problem of this type is the D weakly aharmoic oscillator (Fig. ) described by Hamiltoia (a) with () p m, () 3 4 H H... (6.b) m with small coefficiets,,. I will use the aharmoic oscillator as our first particular eample, but let me start from describig the perturbative approach to the geeral time-idepedet Hamiltoia (a). I the bra-ket formalism, the eigeproblem for the perturbed system is ( H () H () ) E. (6.) Let the eigestates ad eigevalues of the uperturbed Hamiltoia, which satisfy equatio H () () () () E, (6.3) be kow. I this case, to solve problem () meas to fid, first, its perturbed eigevalues E ad, secod, coefficiets () of the epasio of perturbed state vectors i series over the uperturbed oes, () : K. Likharev

257 () () ' '. (6.4) ' U m ( ) () E () E U ( ) E E E U () H () Fig. 6.. The simplest problem for the perturbatio theory applicatio: a D weakly aharmoic oscillator. (Dashed lies characterize the uperturbed, harmoic oscillator.) Let us plug Eq. (4), with the summatio ide replaced with, ito both parts of Eq. (): " " H () () () () () () () " " H " " E ", (6.5) ( ) " ad the ier-multiply all terms by a arbitrary uperturbed ket-vector (). Assumig that the system of uperturbed eigestates is orthoormal, () () =, ad usig Eq. (3) i the first term of the left-had part, we get the followig system of liear equatios () () () () " H '" ' ( E E' ), (6.6) " where the matri elemets of the perturbatio are calculated i uperturbed bra-kets: " ( ) () () " () H '" ' H. (6.7) The liear equatio system (6) is still eact, ad is frequetly used for umerical calculatios. (Sice the matri coefficiets (7) typically decrease whe ad/or become very large, the sum i the left-had part of Eq. (6) may be typically trucated, still givig acceptable accuracy of the solutio.) For gettig aalytical results we eed to make more eplicit approimatios. I the simple perturbatio theory we are discussig ow, this is achieved by the epasio of both eigeeergies ad coefficiets ito the Taylor series i a certai small parameter of the problem: E () () () E E E..., (6.8) Perturbatio s matri elemets ' () () () () () () ' ' "..., (6.9) () where E ( k ) ( k ) k () '. (6.) Please ote its similarity with Eq. (.5) of the D bad theory. Ideed, the latter equatio is ot much more tha a particular form of Eq. (6) for D wave mechaics, ad a specific (periodic) potetial U() cosidered as perturbatio. Moreover, the approimate treatmet of the weak potetial limit i Sec..7 was essetially a particular case of the more geeral perturbatio theory we are discussig ow. Note that, by defiitio, () () =. Chapter 6 Page of 4

258 I order to eplore the st -order approimatio, which igores all terms O( ) ad higher, let us plug oly the two first terms of epasios (8) ad (9) ito the basic system of equatios (6): " H () '" () () () () () () () " " ' ' ( E E E' ). (6.) Now let us ope the paretheses, ad disregard all the remaiig terms O( ). The result is H () ' () () () () () E ' ( E E ), (6.) ' ' st - order correctio of eergies st - order result for vectors This equatio is valid for ay set of idices ad ; let us start from the case = ad immediately get a very simple (ad the most importat!) result: E ( ) () () () () H H. (6.3) For eample, let us see what does this result give for two first perturbatio terms i the weakly aharmoic oscillator (b) ( ) () 3 () () 4 () E. (6.4) As the reader should kow from the solutio of Problem 5.6, the first term is zero, while the secod oe yields 3 3 () 4 E. (6.5) 4 Naturally, there should be some cotributio from the (typically, larger) term proportioal to, so we eed to eplore the d approimatio of the perturbatio theory. However, before doig that, let us complete our discussio of its st order. For, Eq. () may be used to calculate the eigestates rather tha the eigevalues: () () () H ' ', for '. (6.6) () () E E' This meas that the eigeket s epasio (4), i the st order, may be represeted as () E ' H () () ' E () ' ' () C (). (6.7) Coefficiet C caot be foud from Eq. (), however, requirig the fial state to be ormalized, we see that other terms may provide oly correctios O( ), so i the st order we should take C =. The 3 The result for = may be readily calculated i the wave-mechaics style as well, usig Eq. (.69) for uperturbed groud state wavefuctio, ad the table itegral MA Eq. (6.9d): / ( ) 4 () * 4 4 () 4 () 4 3 d ep d, but for higher values of, such calculatios are much harder, because of more ivolved Eq. (.79) for (). Note also that at >>, Eq. (5) gives predictios which coicide with those of the classical theory of weakly oliear oscillatios see, e.g., CM Sec. 4., i particular, Eq. (4.49). Chapter 6 Page 3 of 4

259 most importat feature of Eq. (7) is its deomiator: the closer the (uperturbed) eigeeergies of two states, the larger is their mutual cotributio (hybridizatio), created by the perturbatio. This feature also affects the st approimatio s validity coditio that may be quatified usig Eq. (6): the magitudes of all the bra-kets it describes have to be much less the the uperturbed product () =, so that all elemets of the perturbatio matri have to be much less that the differece betwee the correspodig uperturbed eergies. For the aharmoic oscillator s eergy correctio (5), this requiremet is reduced to E () <<. Now we are ready for goig after the d secod order approimatio to Eq. (6). Let us focus o the case =, because as we already kow, oly this term will give us a correctio to eigeeergies. Moreover, we see that sice the left-had side of Eq. (6) already has the small factor H (), the bra-ket coefficiets i that part may be take from the st order result (6). As a result, we get () () () () () () H "H " E " H ". (6.8) () () E E " " " () Sice Ĥ represets a observable (eergy), ad hece has to be Hermitia, we may rewrite this epressio as () () () () E H ' H () ' () () () ( ' E E ' E E ) ' '. (6.9) This is the much celebrated d order perturbatio result that frequetly (i sufficietly symmetric problems) is the first ovaishig correctio to the state eergy for eample, from the cubic term (proportioal to ) i our weakly aharmoic oscillator problem (). I order to calculate the correspodig correctio, we may use aother result of Problem 5.6: ' / ( )( ) / 3 3 / 3( ) ( )( )( 3) /. ', 3 ', So, accordig to Eq. (9), we eed to calculate E () 6 ', ', 3 3/ ( )( ) 3 3( ) ( )( )( 3), 3 / 3/ / ', 3 ', ', '. ' ( ' ) (6.) (6.) The summatio is actually ot as cumbersome as may look, because all mied products are proportioal to differet Kroecker deltas ad hece vaish, so that we eed to sum up oly the squares of each term: d - order correctio for eergies () E ( )( ) ( ) 3 ( )( )( 3) 3 (6.) Chapter 6 Page 4 of 4

260 Please otice that all eergy level correctios are egative, regardless of the sig of. O the cotrary, the st order correctio E () (5) depeds o the sig of parameter, so that the et correctio, E () + E (), may be of ay sig. Results (7) ad (9) are clearly iapplicable to the degeerate case where, i the absece of perturbatio, several states correspod to the same eergy level, because of the divergece of their deomiators. 4 This divergece hits that the largest effect of the perturbatio i that case is the degeeracy liftig, e.g., splittig of the iitially degeerate eergy level E () (Fig. ), ad that for the aalysis of this case we ca, to the first approimatio, igore the effect of all other eergy levels. (A careful aalysis shows that this is ideed the case util the level splittig becomes comparable with the distace to other eergy levels.) () E () ( )... N () H H () H H E E... E N H ( ) () Fig. 5.. Liftig the eergy level degeeracy by a perturbatio (schematically). Limitig the summatio i Eq. (6) to the group of N degeerate states with equal E () E(), we reduce it to N " () () () " H ' ( E E '" () ). (6.3) where () ad () umber N states of the degeerate group. 5 Equatio (3) may be rewritte as N " () () () H E, where E E E. (6.4) () " '" " '" For each =,, N, this is a system of N liear, homogeous equatios (with N terms each) for ukow coefficiets (). I this problem, we readily recogize the problem of diagoalizatio of the perturbatio matri H () - cf. Sec. 4.4 ad i particular Eq. (4.). As i the geeral case, i the coditio of self-cosistecy of the system, we ca chage the otatio of the lower ide of E (), for eample to : Eergy levels of iitially degeerate system H () H E ()... H H () () E (6.5) 4 This is eactly the reaso why such perturbatio theories ru ito serious problems for systems with cotiuous spectrum, ad other approimate techiques (such as the WKB approimatio) are ofte ecessary. 5 Note that the choice of the basis is to some etet arbitrary, because due to the liearity of equatios of quatum mechaics, ay liear combiatio of states () is also a eigestate of the uperturbed Hamiltoia. However, for usig Eq. (4), these combiatios have to be orthoormal, as was suggested at the derivatio of Eq. (6). Chapter 6 Page 5 of 4

261 Accordig to the defiitio (4) of E (), the resultig N eergy levels E may be foud as E () + E (), where E () are the N roots of Eq. (5). If the perturbatio matri is diagoal, the result is etremely simple, E () () () E E H, (6.6) ad formally coicides with Eq. (3) for the o-degeerate case, but ow may give a differet result for each of N previously degeerate states. Let us see what does this theory give for several importat eamples. First of all, let us cosider a two-level system (or a system with two degeerate states with eergy far from all others levels), with a arbitrary perturbatio matri 6 () H H H H H. (6.7a) Sice that both the uperturbed Hamiltoia ad the operator of its perturbatio are Hermitia, the diagoal elemets of matri H () are real, ad its off-diagoal elemets are comple cougates of each other. As a result, we ca preset the matri i the same form as i Eq. (4.6): () a az a ia y H ai aσ a yσ y azσ z ai a σ. (6.7b) a ia y a az where scalar a ad the Cartesia compoets of vector a are real c-umber coefficiets. The correspodig characteristic equatio, a a a z E ia y a a a ia z y E has the solutio that is familiar to the reader from Chapters ad 4: E () E E a a a a a a, (6.8) / / H H H H y z HH.(6.9) Let us discuss physics of this simple result. Parameter a = (H + H )/ is evidetly the correctio to the average eergy of both states, that does ot give ay cotributio to the level splittig. The splittig, E = E + - E -, is a hyperbolic fuctio of coefficiet a z = (H H )/ that describes the direct cotributios (3) to the eigestates due to the perturbatio. A plot of this fuctio is the famous level-aticrossig diagram (Fig. 3) that has already bee discussed i Sec..5 i a particular cotet of the weak potetial limit of the D bad theory see Fig..9. Now we see that this is a geeral result for ay two-level system. The eamples of this behavior that we already kow iclude the coupled quatum wells (see Fig..9 ad its discussio), bad theory i the weak couplig limit (Sec..5), ad spi-½ systems discussed through Chapter 4 ad i Sec. 5.. By the way, from Sec. 4.4 we already kow the perturbed states i the middle of the aticrossig 6 For brevity, I am droppig the upper ide () i the matri elemets. Chapter 6 Page 6 of 4

262 diagram (at a z = ). For eample, if a y =, the our perturbatio Hamiltoia matri (7), besides the trivial term proportioal to a, is proportioal to, ad hece we ca use the result (4.4) to write: 7 where () ad () are system s states i the absece of the perturbatio. () (), (6.3) E () ( E a ) E a / a y a / a y E a z Fig Level-aticrossig diagram for a arbitrary two-level system. This aalysis shows that other results of our discussios of particular two-level systems i Sec..6 ad 4.6 are also geeral. For eample, if we put such ay two-level system ito a iitial state differet from oe of the eigestates, the probability of its fidig it i ay of states () or () will oscillate with frequecy E E E. (6.3) Hece, for a spi-½ particle i a z-orieted magetic field, the periodic oscillatios of the - ad y- compoets of spi vector, described by Eqs. (4.96) ad (4.), may be iterpreted ot oly as the torque-iduced precessio of spi withi the [, y] plae, but alteratively as the quatum oscillatios of the of the z-compoet of spi betwee states ad with eergies E ad E give by Eq. (4.67). Some other eamples of such oscillatios may be rather uepected. For eample, the ammoium molecule NH 3 (Fig. 4) has two symmetric states which differ by the iversio of the itroge atom relative to the plae of the three hydroge atoms, ad are coupled due to quatummechaical tuelig of the itroge atom through the plae of hydroge atoms. 8 Sice for this molecule, the level splittig E correspods to a eperimetally coveiet frequecy / 4 GHz, it played a importat historic role for the iitial developmet of first atomic frequecy stadards ad microwave quatum geerators (masers) i the 95s, 9 which paved the way toward the developmet of the laser techology. 7 Alteratively, if a =, the = (/)( () i () ). Note that besides a phase coefficiet, these states are similar i that they preset a coheret superpositio of the uperturbed states, with a 5/5 chace to fid the perturbed system i ay of those states. I that sese, the effects of perturbatio coefficiets a ad a y are similar. 8 Sice the hydroge atoms are much lighter, it is more fair to speak about their correlated tuelig aroud the (early immobile) itroge atom. 9 I particular, these molecules were used i the demostratio of the first maser by C. Towes group i 954. Chapter 6 Page 7 of 4

263 . m N 7.8 H H H Fig Ammoia molecule ad its iversio. 6.. The Stark effect Aother eample of the level degeeracy lifted by a perturbatio is the liear Stark effect atomic level splittig by a eteral electric field. Let us study this effect, i the liear approimatio, for a hydroge-like atom. Takig the directio of eteral electric field E (which is practically uiform o the atomic scale) for the z-ais, the perturbatio may be represeted by the followig Hamiltoia: () H qez qer cos. (6.3) (Sice we will work i the coordiate represetatio, we may skip the operator sig from this poit o.) As you (should :-) remember, eergy levels of a hydroge-like atom deped oly o the mai quatum umber - see Eq. (3.9); hece all states but the groud state = ( s i the spectroscopic omeclature) i which l = m =, have some degeeracy that grows rapidly with. This is why I will carry out the calculatios oly for the lowest degeerate level with =. Sice geerally l, here l may be equal either (oe s state, with m = ) or (three p states, with m =, ). Due to this 4-fold degeeracy, H () is a 44 matri with 6 elemets: l l m m m m H () H H H H 3 4 H H H H 3 4 H H H H H H H H m, l, m, m, l. m, (6.33) However, please do ot be scared. First, due to the Hermitia character of the operator, oly of the matri elemets (4 diagoal oes ad 6 off-diagoal elemets) may be substatially differet. Moreover, due to a high symmetry of the problem, there are a lot of zeros eve amog these elemets. Ideed, let us have a look at the agular compoets Y l m of the correspodig wavefuctios, described by Eqs. (3.74)-(3.75). For states with m =, the azimuthal parts of wavefuctios are proportioal to ep{i}; hece the off-diagoal elemets H 34 ad H 43 of matri (33), relatig these fuctios, are proportioal to Stark effect s perturbatio If there is ay doubt why, please revisit the discussio of Eq. (.47), i which we should ow take F = qe. Chapter 6 Page 8 of 4

264 dωy * i * () i H Y d e e. (6.34) The azimuthal-agle symmetry also kills the off-diagoal elemets H 3, H 4, H 3, H 4 (ad hece their comple cougates H 3, H 4, H 3, ad H 4 ), because they relate states with m = ad m, ad are proportioal to * Ω () i d Y H Y d e. (6.35) For the diagoal elemets H 33 ad H 44, correspodig to m =, the azimuthal-agle itegral does ot vaish, but sice the spherical fuctios deped o the polar agle as si, the matri elemets are proportioal to * dy H () Y si si cos si cos d cos d(cos ), (6.36) i.e. are equal to zero as ay limit-symmetric itegral of a odd fuctio. Fially, for states s ad p with m =, the diagoal elemets H ad H are also killed by the polar-agle itegratio: * () d Y H Y sid cos cos d(cos ), (6.37a) * () 3 3 d Y H Y sid cos cos d(cos ). (6.37b) Hece, the oly ovaishig matri elemets are two off-diagoal elemets H ad H relatig differet states with m =, because they are proportioal to * 3 d Y cosy si cos. 4 d d (6.38) 3 What remais is to use Eqs. (3.99) for the radial parts of these fuctios to fiish the calculatio of those two matri elemets: H qe H r drr, ( r) rr,( r) 3, (6.39) where the radial fuctios are give by Eqs. (3.99). Due to the structure of fuctio R, (r), the itegral falls ito a sum of two parts, both of the type we have already met. The fial result is H H 3qE, (6.4) r where r is the radius scale give by Eq. (3.83); for the hydroge atom it is ust the Bohr radius r B (.3). Thus, for our case the perturbatio matri (33) is reduced to See, e.g., MA Eq. (6.7d). Chapter 6 Page 9 of 4

265 H () 3qEr so that the coditio (5) of self-cosistecy is E 3qEr givig a very simple characteristic equatio with the roots 3qEr E E E E 3qEr E E, 3q r, (6.4) (6.4) E. (6.43) (), 3,4 r so that the degeeracy is oly partly lifted - see Fig. 5. (), E 3qE. (6.44) Liear Stark effect for = () E s p 3qEr 3qEr s p m m m Fig Liear Stark effect for level = of a hydroge-like atom. Geerally, i order to uderstad the ature of states correspodig to these levels, we should go back to Eq. (4) with each calculated value of E (), ad calculate the correspodig epasio coefficiets (), which describe the perturbed states. However, i our simple case the outcome of the procedure is clear i advace. Ideed, sice the states with m = are ot affected by the perturbatio (i the liear approimatio i electric field), their degeeracy is ot lifted, ad eergy uaffected see the middle level i Fig. 5. O the other had, the perturbatio matri coectig states s ad p, i.e. the top left part of the full matri (4), is proportioal to the Pauli matri, ad we already kow the result of its diagoalizatio see Eqs. (4.4). This meas that the upper ad lower split levels correspod to very simple liear combiatios of the previously degeerate states, both with m =. s p. (6.45) Chapter 6 Page of 4

266 Fially, let us estimate the magitude of the liear Stark effect for a hydroge atom. For a very high electric field of E = 3 6 V/m, q = e.6-9 C, ad r = r B.5 - m, we get a level splittig of 3qEr.8 - J.5 mev. This umber is much lower tha the uperturbed eergy of the level, E = -E H / -3.4 ev, so that the perturbatio result is quite valid. O the other had, the splittig is much larger tha the resolutio limit imposed by the atural liewidth (~ -7 E, see Chapter 9), so that the effect is quite observable eve i substatially lower electric fields Fie structure of atomic levels Now let us aalyze, for the simplest case of a hydroge-like atom, the so-called fie structure of atomic levels their degeeracy liftig eve i the absece of eteral fields. I the limit whe the effective speed v of electro motio is much smaller tha the speed of light c (as it is i the hydroge atom), the fie structure may be aalyzed as a sum of two small relativistic effects. To aalyze the first of these effects, let us epad the well-kow classical relativistic epressio 3 for the kietic eergy T = E mc of a free particle with the rest mass m, / 4 / p T m c p c mc mc, (6.46) m c ito the Taylor series with respect to the small ratio (p/mc) (v/c) : 4 4 p p p p T mc......, (6.47) 3 mc 8 mc m 8m c ad eglect all the terms besides the first (o-relativistic) oe ad the et term represetig the first ovaishig relativistic correctio of T. I accordace with the correspodece priciple, the quatum-mechaical problem i this approimatio may be described by the perturbative Hamiltoia (a), where the uperturbed (orelativistic) Hamiltoia of the problem, whose eigestates ad eigeeergies were discussed i Sec. 3.5, is p C H () U ( r), U ( r), (6.48) m r while the small kietic-relativistic perturbatio is Kieticrelativistic perturbatio 4 () p p H. (6.49a) 3 8m c Usig Eq. (48), we may rewrite the last formula as mc m This value approimately correspods to the threshold of electric breakdow i air, due to the impact ioizatio o the surface of typical metallic electrodes. (Reducig air pressure oly ehaces the ioizatio ad lowers the breakdow threshold.) As a result, eperimets with higher fields are rather difficult. 3 See, e.g., EM Sec. 9.3, i particular Eq. (9.78) - or ay udergraduate tet o special relativity. Chapter 6 Page of 4

267 H () () H U ( r), (6.49b) mc so that its matri elemets, participatig i the characteristic equatio (5) for a give degeerate eergy level (3.9), i.e. a give pricipal quatum umber, are H () U r H ( ( ) U ( r l'm' lm H () ) l'm' lm ), (6.5) mc where the bra- ad ket vectors describe the uperturbed eigestates whose eigefuctios (i the coordiate represetatio) are give by Eq. (3.9):,l,m = R,l (r)y m l (,). It is straightforward (ad hece left for the reader :-) to prove that all off-diagoal elemets of the set (5) are equal to. Thus we may use Eq. (6) for each set of quatum umbers{, l, m}: () () () () E, l, m E, l, m E lm H lm H U ( r) mc, l, m (6.5), E E E E U U C C 4 mc, l, l mc 4 r, l r, l where ide m has bee dropped, because the radial wavefuctios R,l (r), which affect the averages, do ot deped o that quatum umber. Now usig Eqs. (3.83), (3.9) ad the first two of Eqs. (3.), we fially get () mc 3 E 3 E, l. (6.5) 4 c l / 4 mc l / 4 Let us discuss this result. First of all, its last form cofirms that that correctio (5) is ideed much smaller tha the uperturbed eergy E (ad hece the perturbatio theory is solid) if the latter is much smaller tha the relativistic rest eergy mc of the particle. Net, sice i the Bohr problem l +, the first fractio i the paretheses of Eq. (5) is always larger tha, so that the relativistic correctio to kietic eergy is egative for all ad l. (This is already evidet from Eqs. (6.49), which show that the correctio Hamiltoia is a egatively defied form.) Fially, at a fied pricipal umber, the egative correctio s magitude decreases with the growth of l. This fact may be classically iterpreted usig Eq. (3.): the larger is l (at fied ), the smaller is particle s average distace from the ceter, ad hece the smaller is its effective velocity, the smaller is the magitude of the quatummechaical average of the egative relativistic correctio (49a) to the kietic eergy. Result (5) is coceptually valid for ay physics of iteractio U(r) = -C/r. However, if the iteractio is Coulombic, say betwee a electro with charge (-e) ad a ucleus of charge (+Ze), there is also aother relativistic correctio to eergy, due to the so-called spi-orbit iteractio. Its physics may be uderstood from the followig semi-qualitative, classical reasoig: from the the poit of view of a electro rotatig about the ucleus at costat distace r with velocity v, it is the ucleus, of charge Ze, that rotates about the electro with velocity (-v) ad hece time period T = r/v. From the poit of view of magetostatics, such circular motio of electric charge Q = Ze is equivalet to the Kieticrelativistic eergy correctio Chapter 6 Page of 4

268 Spiorbit perturbatio costat circular electric curret I = Q = (Ze)(v/r) which creates, at electro s locatio, i.e. i the ceter of the curret loop, a magetic field with magitude 4 Zev Zev B a I. (6.53) r r r 4r The field s directio is perpedicular to the apparet plae of the ucleus rotatio (i.e. that of the real rotatio of the electro), ad hece its vector may be readily epressed via the similarly directed vector L of electro s agular (orbital) mometum: Zev Ze Ze Ze B a m L L 3 evr, (6.54) 3 3 4r 4r m 4r m 4 r m c e where the last trasitio is due to the basic relatio betwee the SI uit costats: c. A more careful (but still classical) aalysis of the problem 5 brigs both good ad bad ews. The bad ews is that result (54) is wrog by a factor of eve for the circular motio, because the electro moves with acceleratio, ad the referece frame boud to its caot be cosidered iertial (as was implied i the above reasoig), so that the actual magetic field felt by the electro is Ze B L. (6.55) 3 8 r mec The good ews is that, so corrected, the result is valid (o the average) for ot oly circular but arbitrary (elliptic 6 ) orbital motio i the Coulomb field U(r). Hece from the discussio i Sec. 4. ad Sec. 4.4 we may epect that the quatum-mechaical descriptio of the iteractio betwee this apparet magetic field ad electro s spi momet (4.6) is give by the followig perturbatio Hamiltoia () μ e S Ze L Ze H B S L me r mec me c 4 r, (6.56a) where the small correctio to value g e = of electro s g-factor has bee igored, because Eq. (56) is already a small correctio. This epressio is cofirmed by the fully-relativistic Dirac theory, to be discussed i Sec. 9.7 below: it yields, for a arbitrary cetral potetial U(r), the followig Hamiltoia of the spi-orbit couplig: H () m c e e e du ( r) S L. (6.56b) r dr For the Coulomb potetial U(r) = -Ze /4 r, this formula is reduced to Eq. (56a). As we already kow from the discussio i Sec. 5.7, such Hamiltoia commutes with all operators diagoal i the coupled represetatio (iside the blue lie i Fig. 5.): L, Ŝ, Ĵ, ad Ĵ z. Hece, usig Eq. (5.8) to rewrite the spi-orbit Hamiltoia as 4 See, e.g., EM Sec. 5., i particular, Eq. (5.4). 5 See, e.g., R. Harr ad L. Curtis, Am. J. Phys. 55, 44 (987). 6 See, e.g., CM Sec Chapter 6 Page 3 of 4

269 H Ze ( ) J L S 3 mme c 4 r, (6.57) we may coclude that this operator is diagoal i the coupled represetatio with fied quatum umbers l, s,, ad m. As a result, i this represetatio, we may agai use Eq. (6) for each set {l,, m }: E (),, l mm e c Ze 4 3 r, l J L S, s, (6.58) where the idices irrelevat for each particular term have bee dropped. (As a remider, the spi quatum umber s is fied by particle s ature; for our case of a electro, s = ½.) Now usig the last of Eqs. (3.), ad similar epressios (5.9), (5.97), ad (5.3), we get a eplicit epressio for the spi-orbit correctios 7 E (),, l m c e Ze 4 r 3 ( ) l( l ) 3/ 4 E 3 l( l / )( l ) m c e ( ) l( l ) 3/ 4. (6.59) l( l / )( l ) The last form of its right-had part shows very clearly that this correctio has the same scale as the kietic correctio (5), 8 so that they should be cosidered together. I the first order of the perturbatio they may be ust added, givig a very simple formula for the et fie structure of level : () E 4 Efie 3. (6.6) mec / This simplicity, as well as the idepedece of the result of the orbital quatum umber l, will become less surprisig whe (i Sec. 9.7) we see that this formula follows i oe shot from the Dirac theory, i which the Bohr atom s eergy spectrum i umbered oly with ad, but ot l. Let us recall (see Sec. 5.7) that for a electro (s = ½), the quatum umber may take positive half-iteger values, from ½ to ½. With the accout of this fact, Eq. (6) shows that the fie structure of th Bohr s eergy level has sub-levels see Fig. 6. Spiorbit eergy correctio Fie structure of H-like atom s levels E l... l, l, l /... 5/ 3/ / Fig Fie structure of a hydroge-like atom s level. 7 The factor l i the deomiator does ot give a divergece at l =, because i this case = s = ½, ad the omiator turs ito as well. A careful aalysis of this case (which may be foud, e.g., i G. K. Woolgate, Elemetary Atomic Structure, d ed., Oford, 983), as well as the eact solutio of the Bohr atom problem withi the Dirac theory (Chapter 9) show that the fial result (6), which is idepedet of l, is valid eve i this case. 8 This is atural, because the magetic iteractio of charged particles is a essetially relativistic effect, of the same order (~v /c ) as the kietic correctio (49a) see, e.g., EM Sec. 5., i particular Eq. (5.3). Chapter 6 Page 4 of 4

270 Please ote that accordig to Eq. (5.3), each of these sub-levels is still ( + )-times degeerate i quatum umber m. This degeeracy is very atural, because i the absece of eteral field the system is still isotropic. Moreover, o each fie-structure level, besides the lowest ( = ½) ad the highest ( = ½) oes, each of the m -states is doubly-degeerate i the orbital quatum umber l = ½ - see the labels of l i Fig. 6. (Accordig to Eq. (5.5), each of these states, with fied ad m, may be represeted as a liear combiatio of two states with adacet values of l, ad hece differet electro spi orietatios, m s = ½, weighed with the Clebsch-Gorda coefficiets.) These details aside, oe may crudely say that the relativistic correctios make the total eigeeergy to grow with l, cotributig to the effect already metioed at our aalysis of the periodic table of elemets i Sec The relative scale of this icrease may be evaluated from the largest deviatio from the uperturbed eergy E, reached for the state with = ½ (ad hece l = ): () E ma E 3 Ze 3 3 Z. (6.6) E mec 4 c 4 4 where is the fie structure ( Sommerfeld ) costat, e c 37, (6.6) 4 that was already metioed i Sec These epressios show that the fie structure is ideed a relatively small correctio (~ ) for the hydroge atom, but it rapidly grows (as Z ) with the uclear charge (atomic umber), ad becomes rather substatial for the heaviest atoms with Z ~ The Zeema effect Now, we are ready to review the Zeema effect - the liftig of atomic level degeeracy by a eteral magetic field. Usig Eq. (3.6) (with q = -e) for the descriptio of electro s orbital motio i the field, ad Eq. (4.6) for the operator of electro s magetic momet due to its spi-½, we see that eve for a hydroge-like (i.e. sigle-electro) atom, eglectig the relativistic effects, the full Hamiltoia is rather bulky: H m Ze e p ea B S. 4 r e m e (6.63) There are several simplificatios we may make. First, let us assume that the eteral field is spatial-uiform o the atomic scale (which is a very good approimatio for most cases), so that we ca take the vector-potetial i a aially-symmetric gauge cf. Eq. (3.3): 9 See the Selected Physical Costats appedi for the more eact value of this costat. Its epressio i Gaussia uits, = e /c, makes eve more evidet the fact that is the ust fudametal costat ratio which characterizes the stregth (or rather the weakess :-) of electromagetic effects i quatum mechaics - that i particular makes the perturbative quatum electrodyamics possible. The alterative epressio = E H /m e c, where E H is the Hartree eergy (.9), the scale of all E, is also very revealig. It was discovered eperimetally i 896 by P. Zeema who, amazigly, was fired from the Uiversity of Leide for a uauthorized use of lab equipmet for this work ust to receive a Nobel Prize for it i a few years. Chapter 6 Page 5 of 4

271 A B r. (6.64) Secod, let us eglect the terms proportioal to B, which are small i practical magetic fields of the order of a few Tesla. The remaiig term i the effective kietic eergy, describig the iteractio with the magetic field, is liear i the mometum operator, so that we may repeat the stadard classical calculatio to reduce it to the product of B by the orbital magetic momet s compoet m z = - el z /m e - besides that both m z ad L z should be uderstood as operators ow. As a result, the Hamiltoia reduces to Eq. (a), () () () H H, where Ĥ is that of the atom at B =, ad () H e B ( L z S z ). (6.65) m e The form of the perturbatio immediately reveals the maor complicatio with the Zeema effect descriptio. Namely, i compariso with its cotributio (5.98) to the total agular mometum of the electro, its spi-/ produces a twice larger cotributio ito the magetic momet, so that the righthad part of Eq. (65) is ot proportioal to the total agular momet. As a result, the effect descriptio is simple oly i two limits. If the magetic field is so high that its effects are much stroger tha the relativistic (fiestructure) effects discussed i the last sectio, we may treat two terms i Eq. (48) as idepedet perturbatios of differet (orbital ad spi) degrees of freedom. Sice i the z-basis each of the perturbatio matrices is diagoal, we ca agai use Eq. (6): E E () eb m B( m B e eb, l, m L, l, m m S m m m l l ). z l s z s m e l s (6.66) This result describes splittig of each (l + )-degeerate eergy level, with certai ad l, ito (l +3) levels (Fig. 7), with the adacet level splittig of B B, equal to ~ -3 J ~ -4 ev/t. Note that all levels, besides the top ad bottom oe, remai doubly degeerate. This limit of the Zeema effect is sometimes called the Pasche-Back effect which simplicity was recogized oly i the 9s, due to the eed i very high magetic fields for its observatio..ml, ms / B B ml, ms / () E,l.BB m m l l, m, m ml, ms ml, m s s s / / / / Fig The Pasche-Back effect. Zeema effect s perturbatio Pasche- Back effect Despite its smalless, the quadratic term is ecessary for descriptio of the egative cotributio of the orbital motio to the magetic susceptibility m (the so-called orbital diamagetism, see EM Sec. 5.5), whose aalysis, usig Eq. (63), is left for reader s eercise. See, e.g., EM Sec. 5.4, i particular Eqs. (5.95) ad (5.). Chapter 6 Page 6 of 4

272 Aomalous Zeema effect for s = / I the opposite limit of low magetic field, the Zeema effect takes place o the backgroud of the fie structure splittig. As was discussed i Sec. 3, at B = each split sub-level has a (l + )-fold degeeracy correspodig to (l + ) differet values of the half-iteger quatum umber m, ragig from to +, ad values of iteger l = ½ - see Fig. 6. The magetic field lifts this degeeracy. 3 Ideed, i the coupled represetatio discussed i Sec. 5.7, perturbatio (48) is described by the matri with elemets H () eb m e eb m e eb, m Lz S z ', m ', m me m, m m m S z ', m '. ' J z S z ', m ' (6.67) Now pluggig ito the last term the Clebsh-Gorda epasios (5.6a) for the bra- ad ket-vectors, ad takig ito accout that operator Ŝ z gives o-zero bra-kets oly for m s = m s, matri (67) becomes diagoal, ad may agai use Eq. (6) to get E E () eb m eb m e e l m / l m / m l l m BBm, l l for where two sigs correspod to the two possible values of l = ½ - see Fig. 8. m, (6.68) m 3/ m 3 / m. / m. / () () E, E, l / / l / m /..m m 3/ m 3 / Fig Aomalous Zeema effect i a hydroge-like atom schematically. We see that the magetic field splits each sub-level of the fie structure, with a give l, ito + levels, with the distace betwee the levels depedig o l. I the ed of the 89s, whe the Zeema effect was first observed, there was o otio of spi at all, so that this puzzlig result was called the aomalous Zeema effect. (I this termiology, the ormal Zeema effect is the oe with o spi splittig, i.e. without the secod terms i the paretheses of Eqs. (66)-(68); it may be observed eperimetally i atoms with the et spi s =.) 3 I almost-hydroge-like, but more comple atoms (such as those of alkali metals), the degeeracy i l is lifted by electro-electro iteractio eve i the absece of the eteral magetic field. Chapter 6 Page 7 of 4

273 The strict quatum-mechaical aalysis of the aomalous Zeema effect for arbitrary s (which is importat for applicatios to multi-electro atoms) is ot that comple, but requires eplicit epressios for the correspodig Clebsch-Gorda coefficiets, which are rather bulky. Let me ust cite the uepectedly simple result of this aalysis: where g is the so-called Lade factor: 4 ΔE BBm g, (6.69) ( ) s( s ) l( l ) g. (6.7) ( ) For s = ½ (ad hece = l ½), this factor is reduced to the paretheses i the last form of Eq. (68). It is remarkable that Eqs. (69)-(7) may be readily derived usig very plausible classical argumets, similar to those used i Sec see Fig. 5. ad its discussio. As we have see above, i the absece of spi, the quatizatio of observable L z is a etesio of the classical torque-iduced precessio of the correspodig vector (say, L) about the magetic field directio, so that the iteractio eergy, proportioal to BL z = BL, remais costat (Fig. 9a). At the spi-orbit iteractio without eteral magetic field, the Hamiltoia icludes the operator of product SL, so that it has to be quatized, i.e. costat, together with J, L, ad S. Hece, this system s classical image is a rapid precessio of vectors S ad L about the directio of vector J = L + S, so that the spi-orbit iteractio eergy, proportioal to product LS, remais costat (Fig. 9b). O this backdrop, the aomalous Zeema effect i a relatively weak magetic field B = B z correspods to a slow precessio of vector J ( draggig the rapidly rotatig vectors L ad S with it) about ais z. Aomalous Zeema effect for arbitrary s z (a) z J (b) L z B L B L cost (L J ) z (S J ) z S L J S J L L S cost Fig Classical images of (a) the orbital agular mometum s quatizatio i eteral magetic field ad (b) the fie-structure level splittig. This picture allows us to coecture that what is importat for the slow precessio rate are oly the vectors L ad S averaged over the period of the much faster precessio about vector J - i other words, oly their compoets L J ad S J directed alog vector J. Classically, these compoets may be calculated as L J S L J, S J J J. J (6.7) J J The scalar products participatig i these epressios may be readily epressed via the squared legth of the vectors, usig the followig evidet formulas: 4 This formula is frequetly used with capital letters J, S, ad L, which deote the quatum umbers of the atom as a whole. Chapter 6 Page 8 of 4

274 S ( J L) J L L J, L ( J S) J S J S. (6.7) As a result, we get the followig time average: L z S z z L S J J L J S J J J z ( J J z L L J J S ) ( J J S J J S z L J J ) J z J S J L. (6.73) The last move is to smuggle i some quatum mechaics by usig, istead of vector legths squared, ad the z-compoet of J z, their eigevalues give by Eqs. (5.97), (5.3), ad (5.4). As a result, we immediately arrive at the eact result give by Eqs. (69)-(7). This coicidece ecourages thikig about quatum mechaics of agular mometa i classical terms of torque-iduced precessio, ad turs out to be very fruitful i more comple problems of atomic ad molecular physics. The high-field limit ad low-field limits of the Zeema effect, described respectively by Eqs. (66) ad (68), are separated by a medium field stregth rage i which the Zeema splittig is of the order of the fie-structure splittig aalyzed i Sec. 3. There is o time i this course for a quatitative aalysis of this crossover Time-depedet perturbatios () Now let us proceed to the case whe perturbatio Ĥ i Eq. (a) is a fuctio of time, while () Ĥ is time-idepedet. The adequate perturbative approach to this problem, ad its results, deped critically o the relatio betwee the characteristic frequecy (or the characteristic reciprocal time) of the perturbatio ad the distace betwee the iitial system s eergy levels: E E '. (6.74) I the easiest case whe all essetial frequecies of a perturbatio are very small i the sese of Eq. (74), we are dealig with the so-called adiabatic chage of parameters, that may be treated essetially as a time-idepedet perturbatio (see the previous sectios of this chapter). The most iterestig observatio here is that the adiabatic perturbatio does ot allow ay sigificat trasfer of system s probability from oe eigestate to aother. For eample, i the WKB limit of the orbital motio, the Bohr-Sommerfeld quatizatio rule (.), ad its multi-dimesioal geeralizatio, guaratee that itegral p dr, (6.75) C take alog the particle s classical traectory, is a adiabatic ivariat, i.e. does ot chage at a slow chage of system s parameters. (It is curious that classical mechaics also guaratees the ivariace of itegral (75), but its proof there 6 is much harder tha the quatum-mechaical derivatio of this fact, 5 For a more complete discussio of the Stark, Zeema, ad fie-structure effects i atoms, I ca recommed, for eample, either the moograph by G. Woolgate cited above, or the oe by I. Sobelma, Theory of Atomic Spectra, Alpha Sciece, 6. 6 See, e.g., CM Sec... Chapter 6 Page 9 of 4

275 carried out i Sec..4.) This is why eve if the perturbatio becomes large with time (while chagig sufficietly slowly), we ca epect the eigestate ad eigevalue classificatio to persist. Now let us proceed to the more importat (ad more comple) case whe both sides of Eq. (74) are comparable, ad use for its discussio the Schrödiger picture of quatum mechaics give by Eqs. (4.57) ad (4.58). Combiig these equatios, we get the Schrödiger equatio i the form () () H H ( t) ( t) i ( t). (6.76) t Very much i the spirit of our treatmet of the time-idepedet case i Sec., let us represet the timedepedet ket-vector of the system with its epasio, ( t) ( t), (6.77) over the full ad orthoormal set of the uperturbed, statioary ket-vectors defied by equatio H () E, (6.78) where bra-kets (t) are time-depedet coefficiets. Pluggig epasio (77), with replaced with, ito both sides of Eq. (76), ad the ier-multiplyig both its parts by bra-vector of aother uperturbed (ad hece time-idepedet) state of the system, we get a set of liear, ordiary differetial equatios for the epasio coefficiets: d () i ( t) E ( t) H ' ( t) ' ( t), (6.79) dt ' where the matri elemets of the perturbatio i the uperturbed state basis, defied similarly to Eq. (7), are ow fuctios of time: () H t H () ' ( ) ( t) '. (6.8) The set of differetial equatios (79), which are still eact, may be useful for umerical calculatios, because for virtually all practical problems the set of eigestates may be restricted with a acceptable error i the fial result. 7 However, Eq. (79) has a certai techical icoveiece, which becomes clear if we cosider its (evidet) solutio i the absece of perturbatio: 8 E ( t) () ep i t. (6.8) We see that the solutio oscillates very fast, ad its umerical modelig may preset a challege for eve fastest computers. These spurious oscillatios (whose frequecy, i particular, depeds of the eergy referece level) may be partly tamed by lookig for the geeral solutio of Eqs. (79) i a form ispired by Eq. (8): 7 Eve if the problem uder aalysis may be described by the wave-mechaics Schrödiger equatio (.5), a direct umerical itegratio of that partial differetial equatio is typically less coveiet tha that of the ordiary differetial equatios (79). 8 This is of course ust a more geeral form of Eq. (.6) of wave mechaics of time-idepedet systems. Chapter 6 Page of 4

276 Geeral equatios for probability amplitude evolutio Turig o siusoidal perturbatio E ( t) a ( t)ep i t. (6.8) Here a (t) are ew fuctios of time (essetially, the statioary states probability amplitudes), which may be used, i particular, to calculate the time-depedet level occupacies, i.e. the probabilities W to fid the perturbed system o the correspodig eergy levels of the uperturbed system: W ( t) ( t) a t. (6.83) Pluggig Eq. (65) ito Eq. (79), for these fuctios we readily get a slightly modified system of equatios: ( ) E E' () i' t i a a' H ' ( t)epi t a' H ' ( t) e, (6.84) ' ' where factors, defied by relatio ' ' E E (6.85) have the physical sese of frequecies of potetial quatum trasitios betwee the -th ad -th eergy levels of the uperturbed system. (The coditios whe such trasitios ideed take place will be discussed later i this chapter.) A advatage of Eq. (84) over Eq. (79) for umerical calculatios is the absece of ay depedece o the eergy referece selectio, ad lower frequecies of oscillatios of the right had part terms, especially whe the eergy levels of iterest are close to each other. I order to cotiue our aalytical treatmet, let us restrict ourselves to a particular but very importat case of a siusoidal perturbatio tured o at some momet - for eample, at t = : H (), ( t) Ae where the perturbatio amplitude operators  ad i for t, t it (6.86) A e, for t, Â, ad hece their matri elemets, * A ' A, A ', (6.87) ' A ' are time-idepedet. 9 I this case, for t >, Eq. (84) yields i( ' ) t * i( ' ) t i a a' A' e A ' e. (6.88) ' This is, geerally, still a comple system of coupled differetial equatios; however, it allows simple ad eplicit solutios i two very importat cases. First, let us assume that our system is iitially i oe eigestate (say, o the groud eergy level), ad that the occupacies W of all other levels stays very low all the time. (We will fid the correspodig coditio a posteriori - from the solutio.) With the correspodig assumptio 9 The otatio of the amplitude operators i Eq. (86) is ustified by the fact that the perturbatio Hamiltoia has to be self-adoit (Hermitia), ad hece each term i the right-had part of that relatio has to be a Hermitia cougate of its couterpart, which is evidetly true oly if the amplitude operators are also the Hermitia cougates of each other. Note, however, that each of the amplitude operators is geerally ot Hermitia. Chapter 6 Page of 4

277 a ' ; a, for ', (6.89) Eq. (88) may be readily itegrated, givig a A ' * i ( ) A ' t ' i( ) e e ' t, ' ' for '. (6.9) We see that the probability W (83) of fidig the system o each eergy level of the system oscillates i time, ad that our assumptio (89) is satisfied as soo as the ecitatio amplitude is ot too large, 3 A' '. (6.9) Epressio (9) also shows that this pheomeo has a clearly resoat character: the maimum occupacy W of a level grows ifiitely whe the correspodig detuig, 3 ' ', (6.9) teds to zero. I this limit, our iitial assumptio (89) may become a liability; i order to overcome it we may perform the followig trick - very similar to the oe we used for trasfer to the degeerate case i Sec.. Let us assume that for a certai level,,,, for all " ' (6.93) ' " "', - the coditio illustrated i Fig.. The, accordig to Eq. (9), we may igore the occupacy of all but two levels, ad, ad also the secod, o-resoat terms with frequecy + >> i Eqs. (88) writte for a ad a. 3 E E " ' Fig. 6.. Resoat ecitatio of oe of the higher eergy levels. E ' As a result, i this two-level approimatio (that is of course ot a approimatio at all for twolevel systems, such as spi-½ - see Sec. 5.), we get a simple system of two liear equatios: ia ia ' a ' a Ae A it, * it e, (6.94) 3 Strictly speakig, aother coditio is that the umber of resoat levels is also ot too high see Sec The otio of detuig is also very useful i the classical theory of oscillatios see, e.g., CM Chapter 4. 3 Such omissio of o-resoat terms is usually called the Rotatig Wave Approimatio (RWA); it is very istrumetal ot oly i quatum mechaics, but also i the classical theory of oscillatios - see, e.g., CM Secs Chapter 6 Page of 4

278 Rabi oscillatios (half-) frequecy where I have used shorthad otatio A A ad - ad will use it for a while - util other eergy levels become ivolved (i the begiig of the et sectio). This system of liear differetial equatios may be solved eactly by the itroductio of a ew variable (for oe of the levels oly!) Accordig to this formula, a b it b e, it a e. (6.95) it (6.96) a b ib e. Pluggig these relatios ito Eq. (94), we see that both equatios of the system loose their eplicit time depedece: * b ib a A, ia b A, i (6.97) ' ad ow may be readily solved by regular methods. For eample, we may differetiate the first equatio, ad the use the secod oe to elimiate variable a : ( b A A i b ) ib a' A A b. (6.98) i i From mathematics we kow that the resultig liear, secod-order differetial equatio, with time-idepedet coefficiets, has the followig geeral solutio, b t * t ' ( t) b e b e, (6.99) whose characteristic epoets may be readily foud by pluggig ay of the epoetial fuctios ito Eq. (98). I our case, both roots of the resultig characteristic equatio, A i, (6.) are purely imagiary: = i(/ ), where Δ Ω 4 A /. (6.) The coefficiets b are determied by iitial coditios. If, as before, the system was completely o level iitially, i.e. a () =, a () = b () = ; the Eq. (99) immediately yields b - = - b +, so that b ( t) ib e it / si t, a ( t) ib e it / si t, a () ib. (6.) Now the coefficiet b + may be readily foud from the compariso of the last equality i Eq. () with the first of Eqs. (94), take for t =, whe a =. This compariso yields ib + = A/i, ad hece so that the th level occupacy is A a ( t) e it / si t, (6.3) Chapter 6 Page 3 of 4

279 W a A si t A A / si t. (6.4) Rabi formula This is the famous Rabi formula. 33 It shows that a icrease of the perturbatio amplitude A leads ot oly to a icrease of the amplitude of the probability oscillatios, but also of their frequecy described by Eq. () see Fig...8 A 3 W t /( / ) Fig. 6.. Rabi oscillatios. Ultimately, at A >> (for eample, at the eact resoace, = ) Eqs. ()-() give = A/ ad (W ) ma =, i.e. describe a periodic, full repumpig of the system from oe level to aother ad back, with a frequecy proportioal to the perturbatio amplitude. This effect gives a very coveiet tool for maipulatig two-level-systems (qubits, i the quatum iformatio cotet). For eample, limitig the eteral ecitatio time to t = / (or a odd umber of such itervals) we may completely trasfer the system from oe eigestate (say, ) to the opposite oe (). 34 O the Bloch sphere (Fig. 5.), this trasfer correspods to the represetig poit s drive from the South Pole to the North Pole. Note, however, that accordig to Eq. (9), if the system has eergy levels other tha ad, they also become occupied to some etet. Sice the sum of occupacies should be, this meas that (W ) ma may approach oly if the ecitatio amplitude is very small, ad hece the state switchig time t = / = /A is very log. The ultimate limit i this sese is provided by the harmoic oscillator where all eergy levels are equidistat, ad probability repumpig betwee all of them occurs with the same rate. Hece, i that particular system, the implemetatio of the full Rabi oscillatios is impossible eve at the eact resoace. 35 I the opposite limit, whe the detuig is large i compariso with A/, though still small i the sese of Eq. (93), the frequecy of Rabi oscillatios is completely determied by the detuig, ad their amplitude is small: 33 It was derived i 95 by I. Rabi, i the cotet of his group s pioeerig eperimets with microwave ecitatio of quatum states, usig molecular beams i vacuum. 34 I the quatum iformatio sciece laguage, this is ust a logic operatio NOT performed o a sigle qubit. 35 We, of course, already kow what happes to the groud state of a oscillator at its eteral siusoidal (or ay other) ecitatio: it turs ito the Glauber state, i.e. a superpositio of all Fock states see Sec Chapter 6 Page 4 of 4

280 A t W ( t) 4 si, for A ( ). (6.5) However, I would ot like these quatitative details to obscure from the reader the most importat qualitative (OK, maybe semi-quatitative :-) coclusio of this sectio s aalysis: the resoat icrease of iterlevel trasitio itesity at. Usig the fudametal Kramer-Kroig dispersio relatios, 36 based essetially oly o very geeral causality argumets, it is easy to show (ad hece left for reader s eercise) that i a medium icorporatig may similar quatum systems (e.g., atoms or molecules), this icrease of quatum trasitios is accompaied by a sharp icrease of eteral field s absorptio. This effect has umerous practical applicatios icludig systems based o the electro paramagetic resoace (EPR) ad uclear magetic resoace (NMR) spectroscopies, which are broadly used i material sciece, chemistry, ad medicie. Ufortuately, I will ot have time to discuss the related techical issues (i particular, iterestig pulsig spectroscopy techiques) i detail, ad have to refer the reader to special literature Quatum-mechaical Golde Rule The last result of the past sectio, Eq. (5), may be used to derive oe of the most importat results of quatum mechaics its so-called Golde Rule. For that, let us cosider the case whe the perturbatio causes quatum trasitios from a discrete eergy level E ito a group of eigestates E with a dese (virtually cotiuous) spectrum see Fig. a. If, for all states of the group, the followig coditios are satisfied A', (6.6) ' the Eq. (5) coicides with the result that would follow from Eq. (9). This meas that we may apply Eq. (5), with idices ad duly restored, to ay level of our tight group. As a result, the total probability of havig our system trasferred from level to that group is ' W ( t) 4 W ( t) A ' ' si ' t. (6.7) (a) (b) E. E '. 5 5 ' t Fig. 6.. Derivig the Golde Rule: (a) the eergy level scheme, ad (b) the fuctio uder itegral (8). 36 See, e.g., EM Sec. 7.3, i particular, the correspodece betwee Eqs. (7.55) ad (7.56). 37 For itroductios see, e.g., J. Wertz ad J. Bolto, Electro Spi Resoace, d ed., Wiley, 7; J. Keeler, Uderstadig NMR Spectroscopy, d ed., Wiley,. Chapter 6 Page 5 of 4

281 Now comes the mai, absolutely beautiful trick: let us assume that the summatio over will be limited to a tight group of very similar states for which the matri elemets A are virtually similar (we will check the validity of this assumptio later o), so that we ca take it out of the sum (7) ad the replace the sum with the correspodig itegral: W ( t) 4 A ' ' si ' 4 A ' t d t t ' ' t si d( ' t), (6.8) where is the desity of eigestates o the eergy ais: d. (6.9) de This desity, as well as the matri elemet A, have to be evaluated at =, i.e. at eergy E = E +, ad are assumed to be costat withi the fiite state group. At fied E, the fuctio uder itegral (8) is eve ad decreases fast at t >> see Fig. b. Hece we may itroduce a dimesioless itegratio variable t, ad eted itegratio over this variable formally from - to +. The Eq. (8) is reduced to a table itegral, 38 ad yields 4 A' W ( t) t si 4 A d ' t t, (6.) where costat A'. (6.) is the called the trasitio rate. 39 This is oe of the most famous ad useful results of quatum mechaics, its Golde Rule (sometimes, rather ufairly, called the Fermi Golde Rule 4 ), which deserves much discussio. First of all, let us reproduce the reasoig already used i Sec..5 to show that the meaig of rate is much deeper tha Eq. () seems to imply. Ideed, due to the coservatio of the total probability, W + W =, we ca rewrite that equatio as W. (6.) ' t Evidetly, this result caot be true for t, otherwise probability W would become egative. The reaso for that apparet cotradictio is that result () was obtaied i the assumptio that iitially the system was completely o level : W () =. Now, if i the iitial momet the value of W is Desity of states Golde Rule of quatum mechaics 38 See, e.g., MA Eq. (6.). 39 I some tets, the desity of states i Eq. () is replaced with epressio (E E - ). Ideed, the itegratio of this epressio over ay fiite eergy iterval E gives the same result = (d/de )E = E as Eq. (). Such replacemet may be useful i some cases, but should be used with utmost care, ad for most applicatios the more eplicit form () is preferable. 4 Actually, this result was developed mostly by the same P. A. M. Dirac i 97; E. Fermi s role was ot much more tha advertisig it, uder the ame of Golde Rule No., i his lecture otes o uclear physics, which were published much later, i 95. (To be fair to Fermi, he has ever tried to pose as the Golde Rule s author.) Chapter 6 Page 6 of 4

282 Iitial state's occupacy decay Golde Rule s validity differet, result () has to be multiplied by that umber, due to the liear relatio (88) betwee da /dt ad a. Hece, istead of Eq. () we get a differetial equatio similar to Eq. (.59), W, (6.3) ' W ' which, for time-idepedet, has the evidet solutio, W ' Γt ( t) W () e, (6.4) ' describig a epoetial decay of the iitial state s occupacy, with time costat = /. I would ask the reader to thik agai about this fasciatig mathematical result: by summatio of periodic oscillatios (5) over may levels, we have got a epoetial evolutio (4) of the probability. The mai trick here is of course that the effective rage E of states E, givig the domiatig cotributio ito itegral (8), shriks with time: E ~ /t. 4 By the way, sice most of the decay takes place at times t ~ /, the rage of participatig fial eergies may be estimated as E ~. (6.5) This estimate is very istrumetal for the formulatio of coditios of validity of the Golde Rule (). First, we have assumed that the matri elemets of the perturbatio ad the desity of states do ot deped o eergy withi iterval (5). This gives the followig requiremet E E E ~, (6.6) ~ ' Secod, for the trasfer from sum (7) to itegral (8), we eed the umber of states withi that eergy iterval, N = E, to be much larger tha. Mergig Eq. (6) with Eq. (93) for all eergy levels, ot participatig i the resoat trasfer, we may summarize all coditios of the Golde Rule validity as Γ '". (6.7) (The reader may ask whether I have forgotte the coditio epressed by the first of Eqs. (6). However, for ~ E / ~, this coditio is ust A << (), so that pluggig it ito Eq. (),, (6.8) ad cacelig oe ad oe, we see that this requiremet coicides with the left relatio i Eq. (7) above.) Let us have a look at whether these coditios may be satisfied i practice, at least i some cases. For eample, let us cosider the optical ioizatio of a atom, with the released electro cofied i a volume of the order of cm 3 = -6 m 3. Accordig to Eq. (.8), with E of the order of the atomic ioizatio eergy E E m = ~ ev, the desity of electro states i that volume is of the order of 7 /ev. Thus coditios (7) provide a approimately 5-orders-of magitude rage for acceptable 4 Here we have ru agai, i a more geeral cotet, ito the eergy-time ucertaity relatio which was already discussed i the ed of Sec..5. Let me advise the reader to revisit that importat discussio. Chapter 6 Page 7 of 4

283 values of. This illustratio should give the reader a taste of why the Golde Rules is applicable to so may situatios. Fially, the physical picture of iitial state s decay (which will also be the key for our discussio of quatum mechaics of ope systems i the et chapter) is also very importat. Accordig to Eq. (4), the eteral ecitatio trasfers the system oto the cotiuous spectrum of levels, ad it ever comes back o the iitial level. However, it was derived from quatum mechaics of Hamiltoia systems, whose equatios are ivariat with respect to time reversal. This parado is a result of the geeralizatio (3) of the eact result (), that breaks the time reversal symmetry, but is absolutely adequate for the physics uder study. Some gut feelig of the physical sese of this irreversibility may be obtaied from the followig observatio. From our wave-mechaics eperiece, we kow that the distace betwee adacet orbital eergy levels teds to zero oly if the system size goes to ifiity. This meas that the assumptio of cotiuous eergy spectrum of fiite states essetially requires these states to be ifiitely eteded i space essetially beig free de Broglie waves. The Golde Rule approach correspods to the (physically ustified) assumptio that i a ifiitely large system the travelig waves ecited by a local source ad propagatig outward from it, would ever come back, ad eve if they do, the upredictable phase shifts itroduced by the ucotrollable perturbatios o their way would ever allow them to sum up i the way ecessary to brig the system back ito the iitial state. 4 Maybe the best illustratio of this iterpretatio is give by the followig problem - which is a toy model of the photoelectric effect that was briefly discussed i Sec..(iii). A D particle is iitially trapped i the groud state of a arrow quatum well, U ( ) W ( ). (6.9) Let us use the Golde Rule to fid rate of particle s ioizatio (i.e. its ecitatio ito a eteded, delocalized state) by a weak classical siusoidal force of amplitude F ad frequecy. As a remider, fidig the iitial, localized state ( ) of such particle was the task of Problem.4, ad its solutio was m ( ) ep, W / ', mw E. ' (6.) m Eteded states with cotiuous spectrum, for this problem eist oly at eergies E >, so that the ecitatio rate is differet from zero oly for frequecies mw t. (6.) 3 The weak siusoidal force may be described by the followig perturbatio Hamiltoia, () F ( ) cos it it H F t F t e e, for t, (6.) so that accordig to Eq. (86), that serves as the amplitude operator defiitio, i this case E ' 4 This situatio is very much similar to the etropy icrease i macroscopic systems, which is postulated i thermodyamics, ad ustified i statistical physics, eve though it is based o time-reversible laws of mechaics see, e.g., SM Sec.. ad Sec... Chapter 6 Page 8 of 4

284 F A A. (6.3) Now the matri elemets A that participate i Eq. () may be calculated i the coordiate represetatio: A * F * ( ) A( ) ' ( ) d ( ) ' ( d. (6.4) ' ) Sice, accordig to Eq. (), the iitial is a symmetric fuctio of, a ovaishig cotributio to this itegral is give oly by asymmetric fuctios (), proportioal to sik, with waveumber k related to the fial eergy by the well-familiar equality (.77): k m E. (6.5) As we kow from Sec..5 (see i particular Eq. (.4) ad its discussio), such asymmetric fuctios, with () =, are ot affected by the zero-cetered delta-fuctioal potetial (9), ad their desity is the same as i a completely free space, ad we ca use Eq. (.94). (Actually, sice that relatio was derived for travelig waves, it is more prudet to repeat the calculatio that has led to that result, cofiig the waves o a artificial segmet [-l/, +l/] - so log, k l, l, (6.6) that it does ot affect the iitial localized state ad the ecitatio process. The the cofiemet requiremet (l/) = immediately yields the coditio k l/ =, so that Eq. (.94) is ideed valid, but oly for positive values of k, because sik with k k does ot give a idepedet stadigwave eigestate.) Hece the fiite state desity is d d de l k / /. (6.7) de dk dk m k It may look troublig that the desity of states depeds o artificial segmet s legth l, but the same l also participates i the fial wavefuctio ormalizatio factor, 43 ad hece the matri elemet (4): A ' F l / l l si k e / lm si k, (6.8) l F d. i l / l ( e ik ) d l e ( ik ) d. (6.9) These two itegrals may be readily worked out by parts. Takig ito accout that, accordig to coditio (6), their upper limits may be eteded to, the result is 43 The ormalizatio to ifiite volume, usig Eq. (5.55), is also possible, but less coveiet i such problems. Chapter 6 Page 9 of 4

285 / k A ' F, (6.3) l ( k ) so that fially we get a epressio for the rate, which is idepedet of the artificially itroduced l: / 3 k lm 8F mk A' F. (6.3) 3 4 l ( k ) k ( k ) Note that due to the above defiitios of k ad, the epressio i paretheses i the deomiator of the last formula does ot deped o the quatum well parameter W, ad is a fuctio of oly the ecitatio frequecy (ad particle s mass): ( k ) E E'. (6.3) m As a result, Eq. (3) may be recast simply as 3 F W k. (6.33) 4 What is still hidde here is that k, defied by to Eq. (5) with E = E +, is a fuctio of frequecy, chagig as / at >> t (so that drops as -7/ at ), ad as ( - t ) / whe approaches the red boudary t E / = mw / 3 of the ioizatio effect, so that ( - t ) / i that limit as well. We see that our toy model does describe this mai feature of the photoelectric effect, whose eplaatio by Eistei was essetially the startig poit of quatum mechaics - see Sec... The (very similar) aalysis of this effect i a more realistic model, the hydroge atom s ioizatio, is left for reader s eercise Golde Rule for step-like perturbatios Now let us reuse some of our results for a perturbatio beig tured o at t =, but after that time-idepedet: (), t, H ( t) (6.34) () H cost, t. A superficial compariso of this equatio ad our former Eq. (69) seems to idicate that we may use all our previous results, takig =. However, that coclusio does ot take ito accout the fact that aalyzig both the two-level approimatio ad the Golde Rule for cotiuous spectrum, we have eglected the secod (o-resoat) term i Eq. (9). This why it is more prudet to use the geeral Eq. (86), () i' t i a a' H ' e, (6.35) ' i which the matri elemet of the perturbatio is ow time-idepedet. We see that it is formally equivalet to Eq. (88) with oly the first (resoat) term kept, if we make the followig replacemets: Step-like perturbatio Chapter 6 Page 3 of 4

286 A () H, ' ' '. (6.36) As a saity check, let us revisit a two-level system such as two quatum wells coupled by tuelig see Fig. 3a. It is coveiet to iclude the eergy differece E - E betwee the two levels ito the uperturbed Hamiltoia, so that perturbatio (34) describes oly the localized state couplig due to tuelig through the eergy barrier separatig the wells. (The turig o of the couplig, described by Eq. (34), may be achieved, for eample, by a rapid lowerig of the barrier at t =.) The, after replacemets (36), we are gettig a aalog of Eq. (4): W a H () ' si t, (6.37) where frequecy of the periodic probability repumpig betwee levels ad is ow described, istead of Eq. (4), by relatio / () H ' ' 4 ' 4 ' ' () () ( E E ) H H /. (6.38) But these are eactly the quatum oscillatios that have already bee discussed i Sec..6 ow derived for a arbitrary quatum wells ad tuel barrier shape. (a) (b) l co l co l co l l co ' ' Fig Quatum-well implemetatio of couplig of a discrete-eergy state to (a) aother discrete-eergy state, ad (b) a state cotiuum, due to tuelig through a potetial barrier. The similarity of Eqs. (4) ad (37) shows that the Rabi oscillatios ad the usual quatum oscillatios have essetially the same physical ature, besides that i the former case the eteral rf sigal quatum bridges over the state eergy differece. We may also compare result (38) with our aalysis of a two-level system, with a similar time-idepedet perturbatio, i Sec.. Accordig to Eq. (9), its eigeeergies differ by E ( H H ) 4H /. E H (6.39) But this is eactly the result give by Eq. (38), provided that we cosider (H - H ) as the differece (E E ) of uperturbed state eergies rather tha as a perturbatio, as we certaily have a right to do. Now let us cosider the effect of perturbatio (34) i the case whe it creates couplig betwee the iitial (discrete) eergy level ad a dese group of states with a quasi-cotiuum spectrum, i the same eergy rage. Figure 3b shows a eample of such a system: a quatum well separated by a Chapter 6 Page 3 of 4

287 peetrable tuel barrier from a eteded regio with a quasi-cotiuous eergy spectrum. Makig replacemets (36) i Eq. (), we may preset the Golde Rule for this case as () H ', (6.4) where states ad ow have the same eergy. 44 It is very iformative to compare this result with Eq. (38) for a symmetric (E = E ) double quatum well usig the same tuel barrier see Fig. 3. For the latter case, Eq. (38) yields () H '. (6.4) co Here I have used ide co (from cofiemet ) to emphasize that this matri elemet is rather differet from the oe participatig i Eq. (4). Ideed, i the latter case, the matri elemet, H () ' H () * ' H ' () d, (6.4) has to be calculated for two similar wavefuctios ad cofied to spatial itervals of the same scale l co, while i Eq. (4), wavefuctios are eteded to a much larger distace l >> l co see Fig. 3. As Eq. (9) tells us, i the D model we are cosiderig ow, this meas a additioal factor small factor of the order of (l co /l) /. Now usig Eq. (8) as a crude but suitable model for the fiitestate wavefuctios, we arrive at the followig estimate: () co () co ' co ~ H ' ~ co l l k () ' co l l lm H ~ H, (6.43) E E where E ~ /ml co is the scale of the differeces betwee eigeeergies of the particle i a uperturbed quatum well. Sice the coditio of validity of the perturbative formula (4) is << E, we see that 45 ~.. (6.44) E m Hece the rate of (irreversible) quatum tuelig ito cotiuum is always much lower that the frequecy of (reversible) quatum oscillatios betwee states separated with the same potetial barrier at least for the case whe both are much lower tha E /, so that the perturbatio theory is valid. A hadwavig iterpretatio of this result is that the cofied particle woders beyod the barrier ad back may times before fially decidig to perform a irreversible trasitio ito ucofied cotiuum. 46 ' ' 44 The coditio of its validity is agai give by Eq. (7), but with i the upper limit. 45 It is straightforward to show that i this form, the estimate is valid for a similar problem of ay spatial dimesioality, ot ust the D case we have aalyzed. 46 This qualitative picture may be verified, for eample, usig the eperimetally observable effects of dispersive electromagetic eviromet o electro tuelig - see P. Delsig et al., Phys. Rev. Lett. 63, 8 (989). Chapter 6 Page 3 of 4

288 Let me coclude this sectio (ad this chapter) with the applicatio of Eq. (4) to a importat case, which will provide us with a smooth trasitio to the et chapter s topics. Cosider a composite system cosistig of two parts, a ad b, with the eergy spectra sketched i Fig. 4. system a system b ' a b a () H iteractio A( a) B( b) ' b Fig Eergy relaatio i system a due to its couplig with system b (which serves as the eviromet of a). Let the systems be completely idepedet iitially. The idepedece meas that i the absece of perturbatio, the total Hamiltoia of the system at t < may be preseted as a sum () H H ( a) H ( b), (6.45) a where argumets a ad b symbolize the o-overlappig sets of variables of the two systems. The eigekets of the system may be aturally factored as 47 b a b, (6.46) while its eigeeergies separate ito a sum, ust as the Hamiltoia (45) does: () H H H H H a E E E E. a a b a b b b b a a a b a b b b a (6.47) Aalysis of such a composite system is much easier whe the iteractio of its compoets may be preseted as a product of two Hermitia operators, each depedig oly o the degrees of freedom of oly oe compoet system: () H A( a) B( b). (6.48) A typical eample of such a biliear iteractio Hamiltoia is the electric-dipole iteractio betwee a atomic-scale electro system (with a size of the order of the Bohr radius r B ~ - m) ad the electromagetic field at optical frequecies ~ 6 s -, with wavelegth = c/ ~ -6 m >> r B : 48 () H d E, with d q k r k, (6.49) where the dipole electric momet d depeds oly o positios r k of charged particles (umbered with ide k), while that of electric field E is a fuctio of oly the electromagetic field s degrees of freedom see Chapter 9 below. k 47 Sig is used to deote the formatio of a oit ket-vector from kets of idepedet systems ( belogig to differet Hilbert spaces ). Evidetly, the order of operads i such a product may be chaged at will. 48 See, e.g., EM Sec. 3., i particular Eq. (3.6), i which letter p is used for the electric dipole momet. Chapter 6 Page 33 of 4

289 Returig to the geeral situatio show i Fig. 4, if the compoet system a was iitially i a ecited state a, iteractio (48) may brig it to aother discrete state a of a lower eergy - for eample, the groud state. I the process of this trasitio, the released eergy, i the form of eergy quatum is picked up by system b: E ' E, (6.5) a a E '. (6.5) b E b (I typical applicatios, though ot always, the iitial state b of that system is its groud state.) If the fiite state b of the system is iside a state group with quasi-cotiuous eergy spectrum (Fig. 4), the process has the epoetial character (4) 49 ad may be iterpreted as the effect of eergy relaatio of system a, with the released eergy quatum absorbed by system b. Note that sice the quasicotiuous spectrum essetially requires a system of large spatial size, such model is very coveiet for descriptio of the eviromet of system a. (I physics, the eviromet typically meas all the Uiverse less the system uder cosideratio.) The relaatio rate may be described by the Golde Rule. Sice perturbatio (48) does ot deped o time eplicitly, ad the total eergy of the composite system does ot chage, we may use Eq. (4) that, with the accout of Eqs. (46) ad (48), takes the form A ' B ', where A ' a A ' a, B ' b B ' b, (6.5) with beig the desity of states of the fiite states of system b, at the relevat eergy E b = E b + = E b + (E a E a ). I particular, Eq. (5), with the dipole Hamiltoia (49), eables a very simple calculatio of the atural liewidth of atomic electric dipole trasitios. However, such calculatio has to be postpoed util Chapter 9 i which we will discuss the electromagetic field quatizatio - i.e., the eact ature of states b ad b for this problem. Istead, I will proceed to a discussio of the effects of iteractio of quatum systems with their eviromet, toward which the situatio show i Fig. 4 provides a clear path. Golde Rule for coupled systems 6.8. Eercise problems 6.. Use Eq. (3) to prove the Hellma-Feyma theorem: 5 E H, where is a arbitrary c-umber parameter, ad the use this theorem to prove the first of Eqs. (3.). 6.. Aalyze the relatio betwee Eq. (5) ad the results of classical aalysis 5 of a similar aharmoic ( oliear ) oscillator. 49 The process is evidetly spotaeous, i.e. does ot require ay eteral aget, ad starts as soo as either the iteractio (7) has bee tured o, or (if it is always o) as soo as system a is placed ito the ecited state a. 5 As a remider, its proof for the particular case of wave mechaics was the subect of Problem.4. Chapter 6 Page 34 of 4

290 6.3. A weak additioal force F is applied to a D particle that was placed ito a hard-wall quatum well with, for a, U, otherwise. Calculate, sketch, ad discuss the first-order perturbatio of its groud-state wavefuctio A D quatum particle is cofied i a square-shaped quatum well with ifiitely high walls ad slightly skewed floor: y, for L ad y L, U, otherwise. I the first order i the small parameter, fid eergies of the groud state ad the lowest ecited state of the system. Formulate the coditios of validity of your result. Hit: To save reader s time o a straightforward but logish itegratio by parts, I ca offer the followig itegral: 8 si( ) si() d Calculate the lowest-order relativistic correctio to the groud-state eergy of a D harmoic oscillator A D particle of mass m is localized at a arrow potetial well which may be approimated with a delta-fuctio: U W, with W. Calculate the chage of its groud state eergy by a additioal weak, time-idepedet force F, i the first ovaishig approimatio of the perturbatio theory. Discuss the limits of validity of this result, takig ito accout that at F, the localized state of the particle is metastable Use the perturbatio theory results to calculate the eigevalues of the observable L, i the limit l m >>, by purely wave-mechaical meas. Hit: Try the followig substitutio: () = f()/si / I the first ovaishig order of the perturbatio theory, calculate the shift of the groudstate eergy of a electrically charged spherical rotator (i.e. a particle of mass m, free to move over a spherical surface of radius R) due to a weak, uiform, time-idepedet electric field E Use the perturbatio theory to evaluate the effect of a costat electric field E o the groud state eergy E g of a hydroge atom. I particular: 5 See, e.g., CM Sec. 4.. Chapter 6 Page 35 of 4

291 (i) calculate the st -order shift of E g, (ii) brig the epressio for the d -order shift (eglectig the eteded uperturbed states with E > ) to the simplest possible aalytical form, (iii) fid the lower ad upper bouds o the result, ad (iv) discuss the simplest maifestatios of the shift (called the quadratic Stark effect). 6.. * A particle of mass m, with electric charge q, is i its groud s-state with kow eergy E g <, beig localized by a very short-rage, spherically-symmetric potetial well. Calculate its electric polarizability. 6.. I the first ovaishig order of the perturbatio theory, calculate the correctio to eergies of the groud state ad all lowest ecited states of a hydroge-like atom/io, due to electro s peetratio ito its ucleus, modelig it as a spiless, uiformly charged sphere of radius R << r B /Z. 6.. A spi-½ particle is placed ito a magetic field B B z z B, with B B z. Calculate its eergy levels: (i) eactly, ad (ii) i the first ovaishig order of the perturbatio theory i small B. Compare the results of the two approaches Use the perturbatio theory to aalyze the orbital diamagetism. Namely, calculate the magetic susceptibility m of a dilute gas due to the orbital motio of a sigle electro cofied iside each gas particle. Hit: You may like to use the well-kow formula for the magetic eergy u per uit volume of a liear medium: u B /, where B is the applied magetic field, ad is the magetic permeability, related to the susceptibility m. 5 as 6.4. * Aalyze the statistics of the spacig S E + - E - betwee eergy levels of a two-level system, assumig that all elemets H of its Hamiltoia matri (6.7) are idepedet radom umbers, with equal ad costat probability desities withi the eergy iterval of iterest. Compare the result with that for a purely diagoal matri, with the similar probability distributio of the diagoal elemets Discuss how to calculate the eergy level degeeracy liftig i the secod order of the perturbatio theory, assumig that it is ot lifted i the first order. Carry out such a calculatio for a plae rotator of mass m ad radius R, carryig electric charge q, ad placed ito a weak, uiform, costat electric field E. 5 See, e.g., EM Sec. 5.5, i particular Eqs. (5.7) ad (5.). Chapter 6 Page 36 of 4

292 6.6. Use the sigle-particle approimatio to fid the comple dielectric costat () of a dilute gas of similar atoms, due to their iduced electric polarizatio by a weak eteral ac field, for a field frequecy very close to oe of quatum trasitio frequecies defied by Eq. (85). Hit: I the sigle-particle approimatio, atom s respose to a eteral field is described as that of Z similar, o-iteractig electros movig i a effective static attractig potetial geerally iduced ot oly by the uclei but also by other electros Use the solutio of the previous problem to geeralize the epressio for the Lodo dispersio force betwee two electroeutral molecules (whose calculatio i the harmoic oscillator model was the subect of Problems 3.8 ad 5.) to the sigle-particle model with a arbitrary eergy spectrum Use the solutio of the previous problem to calculate the potetial eergy of iteractio of two hydroge atoms, both i their groud state, separated by distace r >> r B I a certai quatum system, distaces betwee three lowest eergy levels are slightly differet - see Fig. o the right ( <<, ). Fid the time ecessary to populate the first ecited level almost completely (with a give precisio << ), usig the Rabi oscillatio effect, if at t = the system is completely i its groud state. Hit: Assume that all matri elemets of the perturbatio Hamiltoia are kow, ad are all proportioal to the eteral rf field amplitude. E E E 6.. A weak eteral force pulse F(t), of a fiite time duratio, is applied to a D harmoic oscillator that iitially was i its groud state. (i) Calculate, i the lowest ovaishig order of the perturbatio theory, the probability that the pulse drives the oscillator ito a ecited state. (ii) Formulate the coditio of validity of the result, ad compare it with the eact solutio of the problem. (iii) Spell out the perturbative result for a Gaussia-shaped waveform, F t F ep t, / ad aalyze its depedece o the effective duratio of the pulse. 6.. A charged plae rotator, iitially i its groud state, is placed ito a spatially-uiform, but time-depedet eteral field E(t), applied at t =. (i) Calculate, i the lowest ovaishig order i field s stregth, the probability that the pulse drives the rotator oscillator ito its th ecited state. (ii) Spell out ad aalyze your results for a rotatig field. (iii) Same for a ac field with fied polarizatio. Chapter 6 Page 37 of 4

293 6.. (i) Develop the geeral theory of ecitatios of the higher levels of a discrete-spectrum system, iitially i the groud state, by a weak time-depedet perturbatio, up to the d order. (ii) Apply the theory to the system aalyzed i the previous problem (a plae rotator drive by a time-depedet electric field) to fid out what ecitatios, forbidde i the st order of the perturbatio theory, are allowed i its d order A heavy, relativistic particle, with the electric charge q = Ze, passes by a hydroge atom, iitially i its groud state, with the impact parameter (shortest distace) b withi the limits r B << b << r B /, where /37 is the fie structure costat. Calculate the probability of atom s trasitio to its lowest ecited states * A particle of mass m is iitially i the localized groud state, with the kow eergy E g <, of a very small, spherically-symmetric potetial well. Calculate the rate of its delocalizatio ( ioizatio ) by a applied force F(t) = F F cost, with a time-idepedet orietatio F, ad discuss its depedece o frequecy * Calculate the rate of ioizatio of a hydroge atom, iitially i its groud state, by a classical, liearly polarized electromagetic wave with electric field s amplitude E, ad frequecy withi the rage r m e B where r B is the Bohr radius. Recast your result i terms of the cross-sectio of this electromagetic wave absorptio process. Discuss semi-quatitatively what chages would be ecessary i the theory if either of the above coditios had bee violated For the system of two weakly coupled quatum wells (see Fig. 3a), write the system of differetial equatios for the probability amplitudes a defied by Eq. (.99), ad i particular prove Eqs. (.) - which were ust guessed i Sec * Use the quatum-mechaical Golde Rule to derive the geeral epressio for the electric curret I through a weak tuel uctio betwee two coductors, biased with dc voltage V, treatig the coductio electros as a Fermi gas, i which the electro-electro iteractio is limited to the Pauli eclusio priciple. Simplify the result i the low-voltage limit. Hit: The electric curret flowig through a weak tuel uctio is so low that its perturbatio of the electro states iside each coductor is egligible * Geeralize the result of the previous problem to the case whe a weak tuel uctio is biased with voltage V ( t) V Acost, with geerally comparable with e V ad ea * Use the quatum-mechaical Golde Rule to derive the Ladau-Zeer formula (.66). c r B, Chapter 6 Page 38 of 4

294 6.3. Calculate, i the d order of the perturbatio theory, the rate of trasitios betwee differet states of a cotiuous group (of virtually the same eergy E ), iduced by a moochromatic perturbatio of frequecy, with comparable to the distaces betwee other, discrete levels of the system. Chapter 6 Page 39 of 4

295 This page is itetioally left blak Chapter 6 Page 4 of 4

296 Chapter 7. Ope Quatum Systems This chapter discusses the effects of iteractio of a quatum system with its eviromet, ad i particular, with the istrumets used for measuremets. Some part of this material is o the fie lie betwee quatum mechaics ad (quatum) statistical physics. Here I will oly cover those aspects of this field which are of key importace for the basic goals of this course, i particular for the discussio of quatum measuremets i the ed of the chapter. 7.. Ope systems ad the desity matri All the way util the very ed of the previous chapter, we have discussed quatum systems isolated from their eviromet. Ideed, from the very begiig we have assumed that we are dealig with the statistical esembles of systems as similar to each other as oly allowed by laws of quatum mechaics. Each member of such a esemble, called pure or coheret, may be described by the same quatum state - i the wave mechaics case, by the same wavefuctio. Eve our discussio of the Golde Rule i the ed of the last chapter, i particular its form i which oe compoet system (i Fig. 6.3, system b) may be used as a model of the eviromet of aother compoet (a), was still based o the assumptio of a pure iitial state (6.46) of the system. Sice the iteractio of two compoet systems was described by a certai Hamiltoia (the oe give by Eq. (6.45) for eample), for the state of the system as a whole at arbitrary istat we might write, (7.) with a uique correspodece betwee eigestates states a ad b. However, i may importat cases our kowledge of quatum system s state is icomplete. This is especially uavoidable whe a relatively simple quatum system s of our iterest (say, a electro or a atom) is i a cotact with eviromet e here uderstood i a most geeral sese, say, as all the whole Uiverse less system s see Fig.. The there is virtually o chace of makig two or more eperimets with eactly the same composite system, because it would imply a repeated preparatio of the whole eviromet (icludig the eperimeter :-) i a certai quatum state - a rather challegig task, to put it mildly. I this case, it makes much more sese to cosider a statistical esemble of aother kid, with radom quatum states of the eviromet, though possibly with kow macroscopic parameters (e.g., temperature, pressure, etc.). I classical physics, such mied esembles are the subect of statistical (classical) mechaics. 3 Let us see how they may be described i quatum mechaics. For the begiig, we eed to assume a b For a broader discussio of statistical mechaics ad physical kietics, icludig those of quatum systems, the reader is referred to the SM part of this lecture ote series. Most of the mied esemble aalysis i this chapter will pertai also to the cases whe the systems of iterest are ot i a cotact with the eviromet curretly, ad our kowledge about them is icomplete by some other reaso for eample, if they had bee i such a cotact at some time betwee their perfect preparatio (i a certai quatum state) ad the observatio, or if such a perfect preparatio is impossible (or impracticable, or udesirable :-). 3 See, e.g., SM Sec... K. Likharev

297 agai that the couplig betwee the system of iterest ad its eviromet is weak i the sese accepted i the perturbatio theory. 4 I this case we ca still use the bra- ad ket-vectors of uperturbed states, that deped o differet sets of variables (agai, belogig to differet Hilbert spaces ). The the most geeral quatum state of the whole Uiverse, still assumed to be pure, 5 may be described as the followig liear superpositio: s e. (7.), k k The oly differece betwee the descriptio of such a etagled state ad the superpositio of separable states, described by Eq. (), is that coefficiets k i the right-had part of Eq. () are umbered with two idices: ide listig the quatum states of system s, ad k umberig the (eormously large) set of quatum states of the eviromet. So, i a mied esemble a certai state s of the system of iterest may coeist with differet states of its eviromet. 6 Of course, the eormity of the Hilbert space of the eviromet, i.e. the umber of k-compoets i sum (), strips us of ay opportuity to make direct calculatios usig that sum. For eample, accordig to the basic Eq. (4.5), i order to fid the epectatio value of a arbitrary observable A i state (), we would eed to calculate k Assumed quatum state of Uiverse A A * e s A s e. (7.3),' k,k' k 'k' k ' k' Eve if we assume that {s} ad {e} are sets of the basis states of, respectively, the system ad the eviromet, ad that each is full ad orthoormal, Eq. (3) still icludes a double sum over the eormous basis state set of the eviromet! weak iteractio The Uiverse system of iterest (s) eviromet (e) Fig. 7.. Quatum system ad its eviromet (VERY schematically :-). However, let us cosider a limited but the most importat subset of operators those of itrisic observables, which deped oly o the degrees of freedom of the system of iterest (s). These operators commute do ot act o eviromet s degrees of freedom, ad hece i Eq. (3) we may move the eviromet bra-vector e k over all the way to ket-vector e k. Assumig, agai, that the set of evirometal eigestates is full ad orthoormal, Eq. (3) is ow reduced to 4 I the opposite case, the very partitio of the Uiverse ito the system ad the eviromet is impossible. 5 Whether this assumptio is true is a iterestig issue, still beig debated (more by philosophers tha by physicists), but it is widely believed that its solutio is ot critical for the validity of the results of this approach. I Sec. 6, I will offer a strog argumet for this opiio - albeit ot its strict proof. 6 Actually, such coeistece has bee implied (but well hidde :-) i the derivatio of the quatum-mechaical Golde Rule, which i all fairess, also belogs to the ope systems class. Chapter 7 Page of 58

298 A A s ' ek ek' A' α α s,' k,k' * k 'k' ' k α * k α 'k. (7.4) Epectatio value of itrisic observable Desity matri: defiitio Statistical operator: defiitio This is already some relief, because we have oly a sigle sum over k, but the mai trick 7 is still ahead. After the summatio over k, the secod sum i the last form of Eq. (4) is some fuctio w of idices ad, so that, accordig to Eq. (4.96), this relatio may be preseted as where matri w, with elemets A A ' w ' Tr (Aw), (7.5) ' * k 'k i.e. ' w ', w, (7.6) k is called the desity matri of the system. Most importatly, Eq. (5) shows that the kowledge of this matri allows the calculatio of the epectatio value of ay itrisic observable A (ad, accordig to Eqs. (.33)-(.34), its r.m.s. fluctuatio as well if ecessary), eve for the very geeral statistical esemble of states (). This is why let us have a very good look at the desity matri. First of all, as we kow very well by ow that the epasio coefficiets i superpositios of the type () may be always epressed as bra-kets; i our curret case, we may write k k k * 'k e s. (7.7) k Pluggig this epressio ito Eq. (6), we get * w ' k 'k s ek ek s ' k k s w s '. (7.8) We see that from the poit of our system (i.e. i its Hilbert space whose basis states may be umbered by idices oly), the desity matri is ideed ust the matri of some costruct, 8 w e k e k, (7.9) k that is called the statistical (or desity ) operator. As evidet from its defiitio (9), i cotrast to the desity matri this operator does ot deped o the choice of a particular basis s ust as all previous operators cosidered i this course, but i cotrast to them, the statistical operator does deped o composite system s state, icludig the state of system s as well. However, i the -space it is mathematically still a operator whose matri elemets obey all formulas of the bra-ket formalism. I particular, due to its defiitio (6), the desity operator is Hermitia: * * * w w, (7.) ' k k 'k k 'k k ' 7 First suggested i 97 by J. vo Neuma. 8 Of course the bra-kets i this epressio are ot c-umbers, because state is defied i a larger Hilbert space (of the eviromet plus the system of iterest) tha the basis states e k (of the eviromet oly). Chapter 7 Page 3 of 58

299 so that accordig to the geeral aalysis of Sec. 4.3, there should be a certai basis {w} i which the matri of this operator is diagoal: w ' i w w. (7.) Sice ay operator, i ay basis may be preseted i form (4.59), i basis {w} we may write ' w w w w. (7.) This epressio remids, but is ot equivalet to Eq. (4.44) for the idetity operator, that has bee used so may times i this course, ad i the basis w has the form I w w. (7.3) I order to comprehed the meaig of coefficiets w participatig i Eq. (), let us use Eq. (5) to calculate the epectatio value of ay observable A whose eigestates coicide with those of the special basis set {w}: A Tr (Aw) A w. (7.4) A ' w ' ' where A is ust the epectatio value of observable A i state w. Hece, i order to comply with the geeral Eq. (.37), real c-umbers w must have the physical sese of probabilities W of fidig the system i state. As the result, we ca rewrite Eq. () i the form w w W w. (7.5) I oe ultimate case whe oly oe of probabilities (say, W ) is differet from zero, W, (7.6) the system is evidetly i a coheret (pure) state w. Ideed, it is fully described by oe ket-vector w, ad we ca use the geeral rule (4.86) to preset it i aother (arbitrary) basis {s} as a coheret superpositio w " * U s ' U ' ' " s ' ' ', (7.7) where U is the uitary matri of trasform from basis {w} to basis {s}. Accordig to Eqs. () ad (6), i such a pure state the desity matri is diagoal i the {w} basis,, (7.8a) w ' i w, " ', " but ot i a arbitrary basis. Ideed, usig the geeral rule (4.9), we get Statistical operator i diagoalizig basis Epectatio value of w -compatible variable w ' i s * U w U U U U U. (7.8b) l, l' l ll' i w l'' " "' " "' To make this result more trasparet, let us deote matri elemets U = w s (that, for fied, deped o ust oe ide ) by ; the Chapter 7 Page 4 of 58

300 Desity matri i a pure state w ' i s ', (7.9) so that N elemets of the whole NN matri is determied by ust oe strig of N c-umbers. For eample, for a two-level system (N = ), * * w i. * * s (7.) We see that the off-diagoal terms are, colloquially, as large as the diagoal oes, i the followig sese: w. w ww (7.) Sice the diagoal terms have the sese of probabilities W, to fid the system i the correspodig state, we may preset Eq. () i the form / i ( ) w W WW e. (7.) / i ( ) W W e W The physical sese of the (real) costat is the phase shift betwee the coefficiets i the liear superpositio (7) that presets the pure state w i basis s,. Now let us cosider a differet statistical esemble of two-level systems, that icludes member states idetical i all aspects (icludig similar probabilities W, i the same basis s, ), besides that the phase shifts are radom, with the phase probability uiformly distributed over the trigoometric circle. The the esemble averagig is equivalet to averagig over from to, so that it kills the offdiagoal terms of the desity matri (), ad the matri becomes diagoal. For a system with a timeidepedet Hamiltoia, such averagig is especially plausible i the basis of statioary states of the system, i which phase is ust the differece of itegratio costats i Eq. (4.58), ad radomess is aturally produced by mior fluctuatios of the eergy differece E E. (I Sec. 3 we will study the dyamics of such dephasig process.) The mied statistical esemble of systems with the desity matri diagoal i the statioary state basis is called the classical miture, ad presets the limit opposite to the pure (coheret) state. After that eample, the reader should ot be much shocked by the mai claim 9 of statistical mechaics that ay large esemble of similar systems i thermodyamic (or thermal ) equilibrium is eactly such a classical miture. Moreover, for systems i the thermal equilibrium with a much larger eviromet with fied temperature T (such eviromet is usually called a heat bath or a thermostat) statistical physics gives a very simple epressio, called the Gibbs distributio, for probabilities W : * Gibbs distributio W E ep. (7.3a) Z kbt 9 This is essetially a alterative formulatio of the basic postulate of statistical physics, called the microcaoical distributio - see, e.g., SM Sec... See. e.g., SM Sec..4. The Boltzma costat k B is oly eeded if temperature is measured i o-eergy uits, say i kelvis. Chapter 7 Page 5 of 58

301 where E is the eigeeergy of the correspodig statioary state, ad Z is the ormalizatio coefficiet called the statistical sum E Z ep. (7.3b) kbt A detailed aalysis of classical ad quatum esembles i thermodyamic equilibrium is the focus of statistical physics courses (such as my SM) rather tha this course of quatum mechaics. However, I would still like to attract reader s attetio to the key fact that, i cotrast with the similarlylookig Boltzma distributio for sigle particles, the Gibbs distributio is absolutely geeral ad is ot limited to classical statistics. I particular, for quatum gases of idistiguishable particles, it is absolutely compatible with quatum statistics (such as the Bose-Eistei or Fermi-Dirac distributios) of the compoet particles. For eample, if we use Eq. (3) to calculate the average eergy of a D harmoic oscillator of frequecy i thermal equilibrium, we easily get W ep ep, kbt kbt (7.4) Z ep / ep, k BT k BT (7.5) E WE coth. kbt ep / kbt (7.6a) A alterative way to preset the last result is to write E, with ep B / k T, (7.6b) ad to iterpret it as the fact that i additio to the so-called zero-poit eergy / of the groud state, the oscillator (o the average) has thermally-iduced ecitatios, with eergy each. I the harmoic oscillator, whose eergy levels are equidistat, such a laguage is completely appropriate, because the trasfer from ay level to oe ust above it adds the same amout of eergy,, to the system. The above epressio for is actually the Bose-Eistei distributio (for the particular case of zero chemical potetial); 3 we see that it does ot oly cotradict the Gibbs distributio (for the total eergy of the system), but immediately follows from it. 4 Harmoic oscillator i thermal equilibrium See, e.g., SM Sec..8. See, e.g., SM Sec..5 - but mid a differet eergy referece level, E =, used i Eqs. (.68)-(.69), affectig the epressio for Z. Actually, the calculatio is so straightforward (ust the summatio of a geometric progressio for the eumeratio of Z) that it is highly recommeded to the reader as a simple eercise. 3 See, e.g., SM Sec Because of the fudametal importace of Eq. (6) for may fields of physics, let me remid the reader of its mai properties. At low temperatures, k B T <<, there are virtually o thermal ecitatios,, ad the average eergy of the oscillator is domiated by that of its groud state. I the opposite limit of high temperatures, k B T / >>, ad E approaches the classical value k B T (followig, for eample, from the equipartitio theorem, which assigs eergy k B T/ to each quadratic cotributio to system s eergy i the D oscillator case, to oe potetial ad oe kietic eergy term). Chapter 7 Page 6 of 58

302 Desity matri i coordiate represetatio 7.. Coordiate represetatio ad the Wiger fuctio For may applicatios of the desity matri to wave mechaics, its coordiate represetatio is coveiet. (I will oly discuss it for D case; the geeralizatio to multi-dimesio case is straightforward.) Followig Eq. (4.47), it is atural to defie the followig fuctio of two argumets (frequetly also called the desity matri): w(, ' ) w '. (7.7) Isertig, ito the right-had part of this defiitio, two closure coditios (4.44) for a arbitrary (full ad orthoormal) basis {s}, ad the usig Eq. (5.9), we get 5 w(, ' ) s s w s ' s ' ', ', ' * ( ) w ( ' ). (7.8) ' i s I the special basis {w}, i which the desity matri is diagoal, this epressio is reduced to * w (, ' ) ( ) W ( ' ). (7.9) Let us discuss the properties of this fuctio. At coicidig argumets, =, this is ust the probability desity: 6 * w (, ) ( ) W ( ) w ( ) W w( ). (7.3) However, the desity matri gives more iformatio about the system tha ust the probability desity. As the simplest eample, let us cosider a pure quatum state, with W =,, so that () = (), ad * * w(, ' ) ( ) ( ' ) ( ) ( ' ). (7.3) ' ' We see that the desity matri carries the iformatio ot oly about the modulus but also the phase of the wavefuctio. (Of course oe may argue rather covicigly that i this ultimate limit the desitymatri descriptio is redudat, because all this iformatio is cotaied i the wavefuctio itself.) How may be the desity matri iterpreted? I the simple case (3), we ca write ' w(, ' ) w(, ' ) w * * * (, ' ) ( ) ( ) ( ' ) ( ' ) w( ) w( ' ), (7.3) so that the modulus squared of the desity matri may is ust as the oit probability desity to fid the system at poit ad poit. For eample, for a simple wave packet with the spatial etet, w(, ) is appreciable oly if the both poits are ot farther tha from the packet ceter, ad hece from each other. The iterpretatio becomes more comple if we deal with a icoheret miture of several wavefuctios, for eample the classical miture describig the thermodyamic equilibrium. I this case, we ca use Eq. (3) to rewrite Eq. (9) as follows: 5 For ow, I will focus o a fied time istat (say, t = ), ad hece write () istead of (, t). 6 This fact is the historic origi of desity matri ame. Chapter 7 Page 7 of 58

303 * E * w (, ' ) ( ) W ( ' ) ( )ep ( ' ). (7.33) Z k BT As the simplest eample, let us see what is the desity matri of a free (D) particle i the thermal equilibrium. As we kow very well, i this case, the set of eergies E p = p /m of statioary states (moochromatic waves) forms a cotiuum, so that we eed to replace sum (33) by a itegral, takig delta-ormalized travelig wavefuctios (5.59) as eigestates: w(, ' ) Z ip p ep ep mk B ip' ep dp. (7.34) T This is a usual Gaussia itegral, ad may be worked out, as we have doe repeatedly i Chapter ad beyod, by complemetig the epoet to the full square of mometum plus a costat. The statistical sum Z may be also readily calculated, 7 mk T /, Z (7.35) However, for what follows it is more useful to write the result for product wz (the so-called uormalized desity matri): / mk BT mkbt ( ' ) w(, ' ) Z ep. (7.36) This is a very iterestig result: the desity matri depeds oly o the differece of its argumets, droppig to zero fast as the distace betwee poits ad eceeds the followig characteristic scale (called the correlatio legth) / c '. (7.37) / mk T This legth may be iterpreted i the followig way. It is straightforward to use Eq. (3) to verify that the average eergy E p = p /m of a particle i the thermal equilibrium, i.e. i the classical miture (33), equals k B T/ this is ust oe more maifestatio of the equipartitio theorem. Hece the average mometum magitude may be estimated as B B Free particle i thermal equilibrium Free particle s correlatio legth p c / m E / mk T /, p (7.38) B so that c is of the order of the miimal legth allowed by the Heiseberg-like ucertaity relatio : c. (7.39) p c 7 Due to the delta-ormalizatio of the eigefuctio, the desity matri for the free particle (ad ay system with cotiuous eigevalue spectrum) is ormalized as w (, ' ) Zd' w(, ' ) Zd. Chapter 7 Page 8 of 58

304 Harmoic oscillator i thermal equilibrium Notice that with the growth of temperature, the correlatio legth (37) goes to zero, ad the desity matri (36) teds to the -fuctio: w(, ' ) Z T ( ' ). (7.4) Sice i this limit the average kietic eergy of the particle is larger tha its potetial eergy i ay fied potetial profile, Eq. (4) is the geeral property of the desity matri (33). Let us discuss the followig curious feature of Eq. (36): if we replace k B T with /i(t - t ), ad with, the u-ormalized desity matri wz for a free particle turs ito the particle s propagator see Eq. (.49). This is ot ust a occasioal coicidece. Ideed, i Chapter we saw that the propagator of a system with a arbitrary statioary Hamiltoia may be epressed via the statioary eigefuctio as E * G (, t;, t ) ( )ep i t t ( ). (7.4) Comparig this epressio with Eq. (33), we see that the replacemets i( t t ) k T, B ', (7.4) tur the pure-state propagator G ito the u-ormalized desity matri wz of the same system i thermodyamic equilibrium. This importat fact, rooted i the formal similarity of the Gibbs distributio (3) with the Schrödiger equatio s solutio (.67), eables a theoretical techique of the so-called thermodyamic Gree s fuctios, which is especially productive i codesed matter physics. 8 For our purposes, we ca use Eq. (4) to recycle some of wave mechaics results, i particular the followig formula for the harmoic oscillator s propagator G(, t;, t m ) i si[ ( t t )] / m ep cos[ ( t t )] i si[ ( t t )]. (7.43) that may be readily proved to satisfy the Schrödiger equatio for Hamiltoia (5.95), with the appropriate iitial coditio, G(, t ;, t ) = ( ). Makig substitutio (4), we immediately get m w(, ') Z sih[ / k B ] T / m ep ' cosh[ / k T ] ' B. sih[ / k B T ] (7.44) As a saity check, at very low temperatures, k B T <<, both hyperbolic fuctios, participatig i this epressio, are very large ad early equal, ad Eq. (44) yields w ' Z (, ) T m / 4 m ep ep k BT m / 4 m ' ep. (7.45) 8 I will have o time to discuss this techique, ad have to refer the iterested reader to special literature. Probably, the most famous tet of that field is A. Abrikosov, L. Gor kov, ad I. Dzyaloshiski, Methods of Quatum Field Theory i Statistical Physics, Pretice-Hall, 963. (Later repritigs are available from Dover.) Chapter 7 Page 9 of 58

305 I each of the square brackets we ca readily recogize the groud state s wavefuctio (.69), while the middle epoet is ust the statistical sum (4) i the low-temperature limit whe it is domiated by the groud-level cotributio: Z T ep. (7.46) k BT As a result, Z i both parts of Eq. (45) may be cacelled, ad the desity matri i this limit is described by Eq. (3), with the groud state as the oly state of the system. This is atural whe temperature is too low for the ecitatio of ay other state. Returig to arbitrary temperatures, Eq. (44) i coicidig argumets gives the followig epressio for the probability desity: 9 m m w(, ) Z w( ) Z ep tah. sih[ / ] (7.47) k BT k BT This is ust a Gaussia fuctio of, with the followig variace: coth. (7.48) m k T I order to compare this result with our earlier oes, it is useful to recast it as m U coth. (7.49) 4 k T Comparig this epressio with Eq. (6), we see that the average value of potetial eergy is eactly oe half of the total eergy - the other half beig the average kietic eergy. This is what we could epect, because accordig to Eqs. (5.9)-(5.3), such relatio holds for each Fock state ad hece should also hold for their classical miture. Ufortuately, besides the trivial case (3) of coicidig argumets, it is hard to give a straightforward iterpretatio of the desity fuctio i terms of system measuremets. This is a fudametal difficulty that has bee well eplored i terms of the Wiger fuctio (sometimes called the Wiger-Ville distributio ) defied as / B ~ ~ ~ X X ipx ~ W ( X, P) w X, X ep dx. (7.5) B Wiger fuctio: defiitio 9 I have to cofess that this otatio is imperfect, because from the poit of view of rigorous mathematics, w(, ) ad w() are differet fuctios, ad so are w(p, p ) ad w(p) used below. I the perfect world, I would use differet letters for them all, but I desperately wat to stay with w for all the probability desities, ad there are ot so may good differet fots for this letter. Let me hope that the differece betwee these fuctios is clear from their argumets, ad from the cotet. It was itroduced i 93 by E. Wiger o the basis of a geeral (Weyl-Wiger) trasform suggested by H. Weyl i 97, ad re-derived i 948 by J. Ville o a differet mathematical basis. Chapter 7 Page of 58

306 From the mathematical stadpoit, this is ust the Fourier epasio of the desity matri i oe of two ew coordiates (Fig. ) defied by relatios ~ ~ X X X, ' X. (7.5) Physically, the ew argumet X = ( + )/ may be uderstood as the average positio of the ~ particle durig the time iterval (t t ), while X ' as the distace passed by the particle durig that time iterval, so that P may be iterpreted as the characteristic mometum of a particle durig that motio. As a result, the Wiger fuctio is a costruct iteded to characterize the system spread simultaeously i the coordiate ad mometum space - for D systems, o the phase plae [X, P] that we cosidered before see Fig Let us see how fruitful these itetios are. ' ~ X X Fig. 7.. Coordiates X ad X ~ employed i the Weyl- Wiger trasform (5). They differ from the coordiates obtaied by the rotatio of the referece frame by agle / oly by coefficiets, describig scale stretchig. First of all, we may write the Fourier trasform reciprocal to Eq. (5): ~ ~ ~ X X ipx w X X, W ( X, P)ep dp. (7.5) ~ For the particular case X, this relatio yields w ( X ) w( X, X ) W ( X, P) dp. (7.53) Hece the itegral of the Wiger fuctio over mometum P gives the probability desity to fid the system at poit X. Actually, the fuctio has the same property for itegratio over X. To prove that, we should first itroduce the mometum represetatio of the desity matri, i the full aalogy with its coordiate represetatio (7): w( p, p' ) p w p'. (7.54) Isertig, as usual, two idetity operators, i the form give by Eq. (5.), ito the right had part of this equality, we ca get the followig relatio betwee the mometum ad coordiate represetatios: w( p, p' ) p w p' dd' p w ' ' p' ip ip'' dd' ep w(, ' ) ep.(7.55) Chapter 7 Page of 58

307 This is of course othig else tha the uitary trasform of a operator from the -basis to p-basis, ad is similar to the first form of Eq. (5.67). For coicidig argumets, p = p, Eq. (55) is reduced to ip( ' ) w( p) w( p, p) dd'w(, ' )ep. (7.56) Usig Eq. (9) ad the Eq. (5.6), this fuctio may be preseted as * ip( ' ) * w( p) W dd' ( ) ( )ep W p p, (7.57) ad hece iterpreted as the probability desity of the particle s mometum at poit p. Now, i variables (5), Eq. (56) has the form ~ ~ ~ X X ipx ~ w( p), ep. w X X dx dx (7.58) Comparig this equality with defiitio (5) of the Wiger fuctio, we see that w ( P) W ( X, P) dx. (7.59) Thus, accordig to Eqs. (53) ad (59), the itegrals of the Wiger fuctio over either the coordiate or mometum give the probability desities to fid them at certai values of these variables. This is of course the mai requiremet to ay cadidate oit probability desity, (X,P), to fid a classical represetatio poit of a stochastic system o the phase plae [X, P]. Let us look how does the Wiger fuctio look for the simplest systems i the thermodyamic equilibrium. For a free D particle, we ca use Eq. (34), igorig for simplicity the ormalizatio issues: ~ mk TX ~ B ipx ~ W ( X, P) ep ep dx. (7.6) The usual Gaussia itegratio yields: P W ( X, P) cost ep. (7.6) mk BT We see that the fuctio is idepedet of X (as it should be for this traslatioal-ivariat system), ad coicides with the Gibbs distributio (3). We could get the same result directly from classical statistics. This is atural, because as we kow from Sec.., the free motio is essetially ot quatized at least i terms of its eergy ad mometum. Now let us cosider a substatially quatum system, the harmoic oscillator. Pluggig Eq. (44) ito Eq. (5), for that system i thermal equilibrium it is easy to show (ad hece is left for reader s eercise) that the Wiger fuctio is also Gaussia, but ow i both its argumets: Thermal equilibrium: free particle Note that the last lie of Eq. (5.67) is ivalid for the desity operator ŵ, because it is ot local! Such desity, which would epress the probability dw to fid the system i a small area of the phase plae as dw = (X, P)dXdP, is the basic otio of (D) classical statistics see, e.g., SM Sec... Chapter 7 Page of 58

308 Thermal equilibrium: harmoic oscillator m X P W ( X, P) cost ep C, (7.6) m though coefficiet C is ow differet from /k B T, ad teds to that limit oly at high temperatures, k B T >>. Moreover, for the Glauber state it also gives a very plausible result a Gaussia distributio similar to Eq. (6), but shifted to the cetral poit of the state see Sec Ufortuately, for some other possible states of the harmoic oscillator, e.g., ay pure Fock state with >, the Wiger fuctio takes egative values i some regios of the [X, P] plae - Fig Fig The Wiger fuctio of several Fock states of a harmoic oscillator: (a) =, (b) = ; (c) = 5. Adapted from The same is true for most other quatum systems. Ideed, this fact could be predicted ust by lookig at defiitio (5) applied to a pure quatum state, i which the desity fuctio may be factored see Eq. (3): 3 Please ote that i otatios of that sectio, argumets {X, P} of the Wiger fuctio should be replaced with {, p}, ad capital letters saved for the Cartesia coordiates of the cetral poit (5.33), i.e. the classical comple amplitude of the oscillatios. 4 Spectacular eperimetal measuremets of this fuctio (for = ad = ) were carried out recetly by E. Bimbard et al., Phys. Rev. Lett., 336 (4). Chapter 7 Page 3 of 58

309 ~ ~ ~ X * X ipx ~ W ( X, P) ( X ) ( X )ep dx. (7.63) Chagig argumet P (say, at fied X), we are essetially chagig the spatial frequecy (waveumber) of the wavefuctio product s Fourier compoet we are calculatig, ad we kow that Fourier images typically chage sig as the frequecy is chaged. Hece the wavefuctios should have some high-symmetry properties to avoid this effect. Ideed, the Gaussia fuctios (describig, for eample, the Glauber states, ad as the particular case, the groud state of the harmoic oscillator) have such a symmetry, but may other fuctios do ot. Hece the Wiger fuctio caot be used i the role of classical probability desity (X, P), otherwise we would get a egative probability for measuremet i certai itervals dxdp the otio hard to iterpret. However, the Wiger fuctio is still used for a semi-quatitative iterpretatio of states of ope quatum systems. Wiger fuctio: D pure quatum state 7.3. Ope system dyamics: Dephasig So far we have discussed the desity matri as somethig give. Now let us discuss the evolutio of the matri i time, startig from the simplest case whe the system is i state (5) with time-idepedet probabilities W. I the Schrödiger picture we ca rewrite Eq. (5) as w ( t) w ( t) W w ( t). (7.64) Differetiatig this equatio by parts, ad usig Eqs. (4.57)-(4.58), with the accout of the Hermitia ature of the Hamiltoia operator, we get iw i H w ( t) W w ( t) w ( t) W w ( t) H w ( t) W w ( t) w ( t) W w ( t) H w ( t) W w ( t) w ( t) W w ( t) H. Now usig Eq. (64) agai (twice), we get the so-called vo Neuma equatio 5 (7.65) i w H, w. (7.66) This equatio is similar i structure to Eq. (4.99) describig the time evolutio of the Heisebergpicture operators: i A A, H., (7.67) besides the operator order i the commutator, i.e., the sig of the right-had part. This is quite atural, because Eq. (66) belogs to the Schrödiger picture, while Eq. (67) to the Heiseberg picture of the quatum dyamics. vo Neuma equatio 5 I may tets, it is called the Liouville equatio, due to the philosophical proimity to the classical Liouville theorem for the distributio fuctio (X, P) or its multi-dimesioal aalog see, e.g., SM Sec. 6., i particular Eq. (6.5). Chapter 7 Page 4 of 58

310 System s iteractio with eviromet I the geeral case whe a system, iitially out of equilibrium, comes ito a cotact with the eviromet, probabilities W chage, ad dyamics is described by equatios more comple tha Eq. (66). However, we still ca use this equatio to discuss, usig a simple model, the secod (after the eergy relaatio) maor effect of the eviromet, dephasig (also called decoherece ). 6 Let us cosider the followig model of a system iteractig (weakly!) with eviromet: 7 H H H H. (7.68) Let us cosider the simplest, two-level system, takig its Hamiltoia i the simplest form, s e s a z z it H, (7.69) (as we kow from Sec. 4.6, such Hamiltoia is sufficiet to avoid the eergy level degeeracy), ad a factorable (biliear) iteractio - cf. Eq. (6.48) ad its discussio: it H z f. (7.7) Here f is a Hermitia operator depedig oly o the set {} of evirometal degrees of freedom ( coordiates ). These coordiates belog to the Hilbert space differet from that of the two-level system, ad hece operators f ad Ĥ e (that describes the eviromet) commute with z - ad ay other itrisic operator of the two-level system. Of course, ay realistic Ĥ e is very comple, so that it may be surprisig how much we will be able to achieve without specifyig it. Before we proceed to solutio, let me remid the reader of the importat two-level systems that may be described by this model. The first eample is a electro i a eteral magetic field of a fied directio (take for ais z), which icludes both a average compoet B z ad a radom (fluctuatig) compoet B ~ z. As it follows from the discussio i Chapter 4, it may be described by Hamiltoia (68)- (7) with a z B f B ~. (7.7) B z, B The secod importat eample is a particle i a double-quatum-well potetial (Fig. 4), with a barrier betwee them sufficietly high to be impeetrable, ad a additioal force F(t) eerted by the eviromet. If the force is sufficietly weak, we ca eglect its effects o the shape of quatum wells ad hece o the localized wavefuctios L,R, so that the force effect is reduced to the variatio of the differece E L E R = F(t) betwee well eigeeergies. As a result, it may described by Eqs. (68)- (7) with ~ F / ; f F /. (7.7) a z z 6 Aother eample whe W are costat i time, ad hece Eq. (66) is valid, is the thermodyamic equilibrium. However, i this case the statistical operator is diagoal i the statioary state basis ad hece commutes with the Hamiltoia. Hece the right-had part of Eq. (66) vaishes, ad it shows that the desity matri does ot evolve i time at all as it should. 7 Though this model works very well i may cases (see the eamples give below), it is ot adequate for a particle iteractig with the eviromet of similar particles. I this case the methods discussed i the et chapter are more relevat. Chapter 7 Page 5 of 58

311 L R U(, t) U s ( ) F( t) F ( t) Fig Dephasig i a double quatum well system. Returig to the geeral model (68)-(7), let us start its aalysis from writig the usual equatio of motio for the Heiseberg operator : 8 z, H ( a f),, i (7.73) z z so that operator z does ot evolve i time. What does this mea for the observables? For a arbitrary desity matri of the two-level system, w w w, (7.74) w w we ca readily calculate the trace of operator z (sice operator traces are basis idepedet, we ca do this i ay basis, i particular i the usual z-basis): w w Trσ zw Trσ zw Tr w w W W w w. (7.75) Hece, accordig to Eq. (5), z may be cosidered the operator for observable W W, so that i the case (73), the differece W W does ot deped o time, ad sice the sum of the probabilities is also fied, W + W =, both of them are costat. (The physics of this result is especially clear for the model show i Fig. 4: sice the potetial barrier separatig the quatum wells is so high that tuelig through it is egligible, the iteractio with eviromet caot move the system from well ito aother oe. It may look like othig iterestig may happe i such situatio, but i a miute we will see this is ot true.) Hece, we may use the vo Neuma equatio (66) for the desity matri evolutio (i the Schrödiger picture). I the usual z-basis: w iw i w w w H, w a f σ, w z w w w w w a f a f. z w w z w z w z z z w (7.76) 8 This ca be doe because we may cosider the whole system, icludig the eviromet, as a Hamiltoia oe see Eq. (68). Chapter 7 Page 6 of 58

312 Correlatio fuctio of classical variable This meas that though the diagoal elemets, i.e., the probabilities of the states, do ot evolve i time (as we already kow), the off-diagoal coefficiets do chage; for eample, i w a f w, (7.77) ( z ) with a similar but comple-cougate equatio for w. The solutio of the liear differetial equatio (77) is straightforward, ad yields w a () ep i tepi t z ( t) w ' f( t' ) dt. (7.78) The first epoet is a determiistic c-umber factor, while i the secod oe f ( t) f ( t) is still a operator i the Hilbert space of the eviromet, ad, from the poit of view of the system of our iterest, a radom fuctio of time. Let us start from the limit whe the eviromet behaves classically. 9 I this case, the operator i Eq. (78) may be cosidered as a classical radom fuctio of time f(t), provided that we average the result over the esemble of may fuctios f(t) describig may (macroscopically similar) eperimets. For a small time iterval t = dt, we ca use the Taylor epasio of the epoet, trucatig it after the quadratic term: epi dt f ( t' ) dt' i i dt dt f ( t' ) dt' f ( t' ) dt' i dt dt' dt dt f ( t' ) dt' i dt" dt f ( t' ) f ( t" ) f ( t" ) dt" dt dt' dt dt"k f ( t' t" ). (7.79) Here we have used the fact that the first average is equal to zero (it is evidet from Eqs. (69)-(7) that if f had ay average compoet, it could be icluded ito parameter a), while the secod average, called the correlatio fuctio, i a statistically- (i.e. macroscopically-) statioary state of eviromet may oly deped o the time differece t t : f ( t' ) f ( t" ) K ( t' t" ) ( ). (7.8) f K f If this differece is much larger tha some time scale c, called the correlatio time of the radom force, the values f(t ) ad f(t ) are completely idepedet (ucorrelated), as illustrated i Fig. 5a, so that the correlatio fuctio has to ted to zero. O the other had, at =, i.e. t = t, the correlatio fuctio is ust the variace of f: K f () f, (7.8) ad has to be positive. As a result, the fuctio looks (qualitatively) like the sketch i Fig. 5b. 9 This assumptio is ot i ay cotradictio with the quatum treatmet of the two-level system, because a typical eviromet has very dese eergy spectrum, so that the distaces betwee them may be readily bridged by thermal ecitatios of eergies ~ k B T << a z, ofte makig its essetially classical. Chapter 7 Page 7 of 58

313 f (t) t' t" (a) f ( t' ) f ( t" ) (b) t t' t" Fig (a) Typical radom process ad (b) its correlatio fuctio schematically. c Hece, if we are oly iterested i time differeces much loger tha c, we may approimate K f () with a delta-fuctio. Let us take it i the followig coveiet form K f ( ) D ( ), (7.8) where D is a positive costat called the phase diffusio coefficiet. The origi of this term stems from the very similar effect of diffusio of atoms or small solid particles i real space the so-called (the Browia motio. 3 Ideed, if a small classical particle moves i a highly viscous medium, its velocity is approimately proportioal to the eteral force. Hece, if the radom hits of a D particle by the molecules may be described by a force which obeys a law similar to Eq. (8), the velocity (alog ay Cartesia coordiate) is also delta-correlated: v( t), v( t' ) v( t" ) D ( t' t" ). (7.83) Now we ca itegrate the kiematic equatio v, to calculate particle s deviatio from the iitial positio, Phase diffusio coefficiet ad its the variace: ( t) () t t ( t) () v( t' ) dt', (7.84) v( t' ) dt' v( t" ) dt" dt' dt" v( t' ) v( t" ) dt' dt" D ( t' t" ) Dt. (7.85) t t t t t This is the famous law of diffusio, showig that the r.m.s. deviatio of the particle from the iitial poit grows with time as (Dt) /, where costat D is called the diffusio coefficiet. Returig to the diffusio of the quatum-mechaical phase, usig Eq. (8), the last double itegral i Eq. (79) yields D φ dt, so that az w ( dt) w () ep i dt Applyig this formula to sequetial time itervals, D dt. (7.86) 3 The theory of this pheomeo, first observed eperimetally by biologist R. Brow i the early 8s, was pioeered by A. Eistei i 95 (see i particular Eq. (6) below) ad developed i detail by M. Smoluchowski i 96-97, ad A. Fokker i 93. Chapter 7 Page 8 of 58

314 az D dt w () ep i dt D, az w (dt) w ( dt) ep i dt dt (7.87) Two-level system s dephasig etc., for a fiite time t = Ndt, i the limit N ad dt (at fied t) we get, 3 az w( t) w()ep i tlim N D t. (7.88a) N By the defiitio of the atural logarithm base e, 3 this limit is ust ep{-d t}, so that, fially: a a t w( t) w()ep i tep D t w()ep i tep. (7.88b) T So, due to couplig to eviromet, the off-diagoal elemets of the desity matri decay with the characteristic dephasig time T = /D, providig a atural evolutio from the desity matri () of a pure state, to the diagoal matri, W w, (7.89) W with the same probabilities W,, describig a fully dephased (icoheret) classical miture. Our simple model offers a very clear look at the ature of decoherece: force f(t), eerted by the eviromet, shakes the eergy differece betwee two eigestates of the system ad hece the istat velocities (a z - f)/ of their mutual phase shift φ(t) cf. Eq. (4). Due to radomess of the force, φ(t) performs a radom walk aroud the trigoometric circle, so that evetually, averagig of its trigoometric fuctios ep{±iφ} over the possible states of eviromet yields zero, killig the offdiagoal elemets of the desity matri. Our aalysis, however, has left ope two importat issues: (i) Is it approach valid for a quatum descriptio of a typical eviromet? (ii) If yes, what is D? N 7.4. Fluctuatio-dissipatio theorem Similar questios may be asked about a more geeral situatio, whe the Hamiltoia Ĥ s of the system of iterest (s), i the composite Hamiltoia (68), is ot specified at all, but the iteractio betwee that system its eviromet still has the biliear form similar to Eqs. (7) ad (6.3): it H F{ }, (7.9) 3 This result is valid oly if approimatio (8) may be applied at time iterval dt which, i tur, should be much smaller tha T, i.e. if the dephasig time is much loger that the eviromet s correlatio time c. This requiremet is usually well satisfied, because i most eviromets, c very short. For eample, i the origial Browia motio eperimets with few-m ik particles i water, it is of the order of the average iterval betwee sequetial molecular impacts, of the order of - s. 3 See, e.g., MA Eq. (.a). Chapter 7 Page 9 of 58

315 where is some observable of the subsystem s (say, a geeralized coordiate or a geeralized mometum). It may look icredible that i this very geeral situatio oe may make a very simple ad powerful statemet about the statistical properties of the geeralized eteral force F, uder oly two (iterrelated) coditios which are satisfied i a huge umber of cases of iterest: (i) the couplig of system s of iterest to eviromet e is weak - i the sese of the perturbatio theory (see Chapter 6), ad (ii) the eviromet may be cosidered as stayig i thermodyamic equilibrium, with certai temperature T, regardless of the process i the system of iterest. 33 This famous statemet is called the fluctuatio-dissipatio theorem (FDT). 34 Due to the importace of this fudametal result, let me derive it. 35 Sice by writig Eq. (68) we treat the whole system (s + e) as a Hamiltoia oe, 36 we may use the Heiseberg equatio (4.99) to write F, H F, because, as was discussed i the last sectio, operator i F, (7.9) H e F commutes with operators Ĥ s ad. Geerally, very little may be doe with this equatio, because the time evolutio of the eviromet s Hamiltoia depeds, i tur, o that of the force. This is where the perturbatio theory becomes idispesable. Let us decompose the eteral force s operator ito the followig sum: ~ ~ F, (7.9) F F( t), with F( t) where (util further otice) sig meas the statistical averagig over the eviromet aloe. 37 From the poit of view of system s, the first term of the sum (still a operator!) describes the average respose 33 The most frequet eample of violatio of these coditios is eviromet s overheatig by the eergy flow from the subsystem. I leave it to the reader to estimate the overheatig of a stadard physical laboratory room by a typical dissipative quatum process the emissio of a optical photo by a atom. (Hit: etremely small.) 34 The FDT was first derived by H. Calle ad T. Welto i 95, o the backgroud of a earlier derivatio of its classical limit by H. Nyquist i 98, ad the pioeerig 95 work by A. Eistei see below. 35 The FDT may be proved i several ways which are differet from, ad shorter tha the oe give i this sectio see, e.g., either SM Secs. 5.5 ad 5.6 (based o H. Nyquist s argumets), or the origial paper by H. Calle ad T. Welto, Phys. Rev. 83, 34 (95) - woderful i its clarity. The loger approach I describe here, besides givig the importat Kubo formula (9) as a byproduct, is a very useful eercise i the operator maipulatio ad the perturbatio theory i its itegral form, differet from the differetial form used i Chapter 6. If the reader is ot iterested i this eercise, he or she may skip the derivatio ad ump directly to the result epressed by Eq. (34), which uses the otios defied by Eqs. (4) ad (3). 36 We ca always do that if the local eviromet is large eough, so that the processes i our subsystem would ot deped o the type of boudary betwee it ad the eteral eviromet; i particular we may assume the total system closed, i.e. Hamiltoia. 37 For usual ( ergodic ) eviromets, without itrisic log-term memories, this statistical averagig over a esemble of eviromets is equivalet to averagig over relatively short times - much loger tha the correlatio time c of the eviromet, but still much shorter tha the characteristic time of evolutio of the system uder aalysis, such as the dephasig time T ad the eergy relaatio time T both still to be calculated. As was already metioed, i most practical eviromets, c is very short. Thus, for relatively massive (iertial) systems of iterest the separatio of the averagig ito two steps is well ustified. Chapter 7 Page of 58

316 of the eviromet to the system dyamics (possibly, icludig such irreversible effects as frictio), ad has to be calculated with accout of their iteractio as will do later i this sectio. O the other had, the secod term i Eq. (9) presets fluctuatios of the eviromet, which eist eve i the absece of system s. Hece, i the first ovaishig approimatio i the iteractio stregth, the fluctuatio part may be calculated igorig the iteractio, i.e. treatig the eviromet as beig i the thermodyamic equilibrium: 38 ~ ~ i F F, H eq e. (7.93) Sice i this approimatio the eviromet s Hamiltoia does ot have a eplicit depedece of time, the solutio of this equatio may be writte combiig Eqs. (4.75) ad (4.9): i i F t ep H tf e eq ep H e eq t. (7.94) Let us use this relatio to calculate the correlatio fuctio of fluctuatios, defied similarly to Eq. (8), but payig close attetio to the order of the time argumets (very soo we will see why): ~ F ~ i i i t Ft' ep H t F ep H t ep H t' F ep H t', e e i e e (7.95) where the thermal equilibrium of eviromet is implied. We are at will to calculate this epectatio value i ay basis, ad the best choice is evidet, because i the eviromet s statioary state basis, its Hamiltoia, the epoets i Eq. (95), ad the desity operator of the eviromet are all represeted by diagoal matrices. Usig Eq. (5), the correlatio fuctio becomes ~ F ~ t Ft' Tr w ep H t F ep H t ep H t' F, ', ' i ep w H etf W W i ep E F ' i e tf epi ' E i i ep ep H et H et' F i ep E E' i ' e ( t t' ), i i tep E ' t' F ' ~ where E E e i ep Et' i ep H et' i ep H et' E '. " (7.96) Here W are the Gibbs distributio probabilities, give by Eq. (3) with eviromet s temperature T, ad F are the Schrödiger-picture matri elemets of the iteractio force operator. We see that correlator (96) is a fuctio of the differece t t oly (as it should be for fluctuatios i a macroscopically statioary system), but may deped o the order of the operads. This is why let us deote this particular correlatio fuctio by upper ide +, 38 Here we assume that for the equilibrium, Eq. (9) has zero average, because if this is ot so, this average part of force may be always icluded ito the Hamiltoia of subsystem s. Chapter 7 Page of 58

317 ~ E K ~ ~ F Ft Ft' W F' epi, where E ~ E E', (7.97), ' ad its couterpart by upper ide - : ~ ~ ~ E K F K F Ft' Ft W F' ep i. (7.98), ' So, i cotrast with classical processes, i quatum mechaics the correlatio fuctio of fluctuatios F ~ is ot ecessarily time-symmetric: ~ ~ ~ ~ ~ E K F K F K F K F Ft Ft' Ft' Ft iw F' si, (7.99) so that F t gives a good eample of a Heiseberg-picture operator whose values, take i differet momets of time, geerally do ot commute the opportuity already metioed i Sec Now let us retur to the force decompositio (9), ad calculate the first (average) compoet of the force. I order to do that, let us write the formal solutio of Eq. (9) as follows: F ( t) i t F t', H et', ' dt'. (7.) I the right-had part of this relatio, we caot treat the Hamiltoia of the eviromet as a uperturbed (equilibrium) oe, because the result would have zero statistical average. Hece, we should make oe more step i our perturbative treatmet, ad take ito accout (i the first ovaishig approimatio) the effect of our system of iterest (s) o the eviromet. To do this, let us write the (so far, eact) Heiseberg equatio of motio for the eviromet s Hamiltoia, H, H H, F i H, (7.) ad its formal solutio, similar to Eq. (), but for a arbitrary time t rather tha t: H e ( t' ) i e e t ( t" ) H e e t", F t" dt". (7.) Pluggig this equality ito the right-had part of Eq. (), ad averagig the result (agai, over the eviromet oly!), we get F ( t) t dt' t' dt" t" F t', H t", F t" e. (7.3) As we will see immietly, this epressio gives a ovaishig result eve if the right-hadpart averagig is carried over the uperturbed (thermal-equilibrium) eviromet, so that uless we are iterested i higher-order correctios, there is o eed to refie the result ay further. This fact eables us to calculate the average i the right-had part of Eq. (3) absolutely similarly to that i Eq. (96), usig Eq. (94): 39 A good saity check here is that at =, the differece (99) betwee K F () ad K F (-) vaishes. Correlatio fuctios of a operator Chapter 7 Page of 58

318 F t', H t" F e eq, t" Trw Ft', H eft" Tr w Ft' H Ft" Ft' Ft" H H Ft" Ft' Ft" H Ft', ' W, ' F ' ~ W E F e e e t' E F t" F t' F t" E E F t" F t' F t" E F t" ' ' ' ~ E epi ' ' t' t" c.c.. ' e ' ' ' ' (7.4) Now, if we try to itegrate each term of this sum, as Eq. (3) seems to require, we will see that the lower-limit substitutio (at t, t - ) is ucertai, because the epoets oscillate without decay. This techical difficulty may be overcome by the followig reasoig. As illustrated by the eample cosidered i the previous sectio, couplig to a disordered eviromet makes the memory horizo of the subsystem of our iterest (s) fiite: its curret state does ot deped o its history beyod certai time scale i that eample, the dephasig time T. (Actually, this is true for virtually all real physical systems, i cotrast to the idealized models such as a dissipatio-free pedulum that swigs for ever ad ever with the same amplitude.) As a result, the fuctios uder itegrals of Eq. (3), i.e. the sum (4), should self-average at a certai fiite time. Oe simple techique for epressig this fact mathematically is ust droppig the lower-limit substitutio; this would give the correct result for Eq. (3). However, a better (mathematically more acceptable) trick is to first multiply the fuctio uder each itegral by, respectively, ep{(t t )} ad ep{(t t )}, where is a very small positive costat, the carry out the itegratio, ad after that take the limit. The physical ustificatio of this procedure may be provided by sayig that system s behavior should ot be affected if its iteractio with the eviromet was ot kept costat but was tured o gradually say, epoetially with a ifiitesimal rate. With this modificatio, Eq. (3) becomes t t' ~ ~ E F( t) W lim E F' dt' dt" t" epi t' t" t" t c.c.. (7.5), ' This double itegratio is over the area shaded i Fig. 6, so that the order of itegratio may be chaged to the opposite oe as where t t, ad t t. t t' t t t dt' dt"... dt" dt'... dt" d'..., (7.6) t" t" t t' t" t t' Fig D itegratio area i Eqs. (5) ad (6). Chapter 7 Page 3 of 58

319 whose kerel, As a result, Eq. (5) may be rewritte as a sigle itegral, G F t t G( t t" ) ( t" ) dt" G( ) ( t ) d, ~ E si e W E F lim ep i ' lim, ' ~, ' W ' F ' ~ E, ' W c.c. d' ~ E F' si, (7.7) (7.8) does ot deped o the particular law of evolutio of the subsystem (s) uder study, i.e. provides a geeral characterizatio of its couplig to the eviromet. I Eq. (7) we may readily recogize the most geeral form of the liear respose of a system (i our case, the eviromet), takig ito accout the causality priciple, where G() is the respose fuctio (also called the temporal Gree s fuctio ) of the eviromet. 4 Comparig Eq. (8) with Eq. (99), we get a woderfully simple uiversal relatio, 4 ~ ~ F( ), F() ig( ). (7.9) that emphasizes oce agai the quatum ature of the correlatio fuctio s time asymmetry. However, the relatio betwee G() ad the force ati-commutator, ~ ~ ~ ~ ~ ~ F ( t ), F( t) F( t ) F( t) F( t) F( t ) K F K F, (7.) Esemble average of eviromet s respose Fluctuatio commutator via Gree s fuctio is much more importat because of the followig reaso. Relatios (97)-(98) show that the so-called symmetrized correlatio fuctio, ~ K ~ ~ F K E F K F F ( ), F() lim W F' cos e, ' ~ (7.) E W F' cos,, ' Symmetrized correlatio fuctio that is evidetly a eve fuctio of time differece, looks very similar to the respose fuctio (8), oly with aother trigoometric fuctio uder the sum. This similarity my be used to obtai a eact algebraic relatio betwee the Fourier images of these two fuctios of. Ideed, fuctio () may be represeted as the Fourier trasform 4 4 For a more detailed discussio of this fuctio ad the causality priciple, see, e.g., CM Sec This relatio, called the Kubo (or Gree-Kubo ) formula, after the works by M. Gree (954) ad R. Kubo (957), does ot come up i the easier derivatios of the FDT, discussed i the begiig of this sectio. 4 Due to their practical importace, ad certai mathematical issues with their ustificatio for radom fuctios, Eqs. ()-(3) have their ow grad ame, the Wieer-Khichi theorem, though the math rigor aside, they are ust a straightforward corollary of the Fourier itegral trasform (5) see, e.g., SM Sec Chapter 7 Page 4 of 58

320 K with the reciprocal trasform F i ( ) S ( ) e d S ( )cos d, (7.) F F Symmetrized spectral desity S F i ( ) K F ( ) e d K F ( )cos d. (7.3) via the symmetrized spectral desity of variable F, defied as S F, F, ) ( ' ) F Fω' Fω' F (7.4) F ( ω' where F (also a operator rather tha a c-umber!) is defied as F ~ it ~ it F( t) e dt, so that F( t) F e dt. (7.5) The physical meaig of fuctio S F () becomes evidet if we write Eq. () for the particular case = : ~ K F ( ) F S F ( ) d S F ( ) d. (7.6) This formula implies that if we pass fuctio F(t) through a liear filter cuttig from its frequecy spectrum a arrow bad dω of real (positive) frequecies, the variace F f of the filtered sigal F f (t) would be equal to S F (ω)dω hece the ame spectral desity. 43 Let us use Eqs. () ad (3) to calculate the spectral desity for our model: S F, ' W, ', ' F W W ' F F lim ' ' lim lim i ~ E i cos e e d ~ E i epi c.c. e e d ~ E / i E / ~. (7.7) Now it is a coveiet time to recall that each of the two summatios here is over the eigeeergy spectrum of the eviromet whose spectrum is virtually cotiuous because of its large size, so that we may trasform each sum ito a itegral ust as this was doe i Sec. 6.6: E de d..., (7.8) 43 A alterative popular measure of spectral desity is S F () F f /d = 4S F (), where = / is the cyclic frequecy (measured i Hz). Chapter 7 Page 5 of 58

321 Chapter 7 Page 6 of 58 where (E) is the desity of eviromet s states at a give eergy. This trasformatio yields. / ~ / ~ ) ( ) ( lim E i E i F E de E E W de S ' ' ' F (7.9) Sice the square bracket depeds oly o a specific liear combiatio of two eergies,, ~ E E ' E it is coveiet to itroduce also aother, liearly-idepedet combiatio of the eergies, for eample, the average eergy / E E ' E, so that the state eergies may be preseted as ~, ~ E E E E E E '. (7.) With this otatio, Eq. (9) becomes. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ lim E i de F E E E E E E W E i de F E E E E E E W de S ' ' F (7.) Due to the smalless of parameter (which should be much less tha all real eergies, icludig k B T,, E, ad E ), each of the iteral itegrals is domiated by a ifiitesimal viciity of oe poit, E ~, i which the spectral desity, matri elemets, ad the Gibbs probabilities do ot chage cosiderably, ad may be take out of the itegrals, so that they may be worked out eplicitly: 44, ~ ~ ~ ~ ~ ~ lim ~ ~ ~ ~ lim de F W F W de E E i F W de E E i F W de E i de F W E i de F W de S F (7.) where idices mark fuctio values at the special poits E ~, i.e. E = E. The physics of these poits becomes simple if we iterpret state, that is the argumet of the equilibrium Gibbs distributio fuctio W, as the iitial state of the eviromet, ad as its fiite state. The the topsig poit correspods to E = E -, i.e. to the emissio of oe eergy quatum of the observatio frequecy by the eviromet ito subsystem s of iterest, while the bottom-sig poit E = E +, correspods to the absorptio of such quatum by the eviromet. As Eq. () shows, both processes give similar positive cotributios ito force fluctuatios. 44 Usig, e.g., MA Eq. (6.5a). (The imagiary parts of the itegrals vaish, because itegratio i ifiite limits may be always re-cetered to fiite poits.) A mathematically elighteed reader may have oticed that the itegrals might be take without the itroductio of small, usig the Cauchy theorem see MA Eq. (5.).

322 Chapter 7 Page 7 of 58 The situatio is differet for the Fourier image of the respose fuctio G(), 45 d e G i ) ( ) (, (7.3) that is frequetly called either the geeralized susceptibility or the respose fuctio - i our case, of the eviromet. Its physical meaig is that the comple fuctio () = () + i () relates the Fourier amplitudes of the geeralized coordiate ad geeralized force: 46 F ) (. (7.4) The physics of its imagiary part () is especially clear. Ideed, if both F ad represet a siusoidal classical process, say, i.e. cos ) (, e e t t t i t i (7.5) The, i accordace with the correspodece priciple, Eq. (4) should hold for the c-umber comple amplitudes F ad, eablig us to calculate the time depedece of force,. si cos ) ( t " t χ' e " i χ' e " i χ' e e e e e F e F t F t i t i t i t i t i t i t i t i (7.6) We see that () scales the part of the force that is /-shifted from the coordiate oscillatios, i.e. is i phase with those of velocity, ad hece characterizes the time-average power flow from the system ito the eviromet, i.e. the eergy dissipatio rate: 47 " t t " t χ' t t F P si si cos ) ( ) (. (7.7) Let us calculate this fuctio from Eqs. (8) ad (3), ust as we have doe for the spectral desity of fluctuatios: i E i E F W d e E i i F W d e G " ' ' e i i ~ ~ Im lim c.c. ~ ep Im lim ) ( Im ) ( ', ', 45 Itegratio i Eq. may be eteded to the whole time ais, - < < +, if we complemet defiitio (7) of G() for > with its defiitio as G( ) = for <, i correspodece with the causality priciple. 46 I order to prove this relatio, it is sufficiet to plug epressio t i s e, or ay sum of such epoets, ito Eqs. (7) ad the use defiitio (3). This simple eercise is highly recommeded to the reader. 47 The epressio F P = Fv used for the istat power flow is evidet if is the usual Cartesia coordiate of a mechaical system. Accordig to aalytical mechaics (see, e.g., CM Chapters ad ), it is valid for ay geeralized coordiate geeralized force pair which forms the iteractio Hamiltoia (9). Geeralized susceptibility

323 Chapter 7 Page 8 of 58. ~ ~ lim ', E E F W ' (7.8) Makig the trasfer (8) from the double sum to the double itegral, ad the the itegratio variable trasfer (), we get. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ lim ) ( de E F E E E E E E W de E F E E E E E E W de " ' ' (7.9) Now usig the same argumet about the smalless of parameter as above, we may take the spectral desities, matri elemets of force, ad the Gibbs probabilities out of the itegrals, ad work out the itegrals, gettig a result very similar to Eq. ():. ) ( de F W F W " (7.3) I order to relate these results, it is sufficiet to otice that accordig to Eq. (3), the Gibbs probabilities W are related by coefficiets depedet o oly the temperature T ad observatio frequecy : T k E W T k E Z E W E E W W B B ep / ep ~, (7.3) so that both the spectral desity ad the dissipative part of susceptibility may epressed via the same itegral over eviromet eergies:, cosh B de F F E W T k S F (7.3), sih B de F F E W T k " (7.33) ad hece are uiversally related as T k " S F B )coth ( ) (. (7.34) This is the Calle-Welto s fluctuatio-dissipatio theorem (FDT). It reveals the fudametal, itimate relatio betwee dissipatio ad fluctuatios iduced by eviromet ( o dissipatio without fluctuatios ) hece the ame. 48 I the classical limit, << k B T, the FDT is reduced to 48 A curious feature of the FDT is that Eq. (34) icludes the eactly same fuctio of temperature as the average eergy (6) of a quatum oscillator of frequecy, though, as the reader could witess, the otio of the Fluctuatio- dissipatio theorem

324 Drag coefficiet Nyquist formula k BT k BT Im ( ) S F ( ) " ( ). (7.35) I most systems of iterest the last fractio teds to a fiite (positive) costat i a substatial rage of relatively low frequecies. Ideed, epadig Eq. (3) i the Taylor series i small, we get i..., with G d, ad G d. (7.36) Sice the temporal Gree s fuctio is real by defiitio, the Taylor epasio of () Im() starts with the liear term i, where is a certai real coefficiet, ad uless =, is domiated by this term at small. (The physical sese of costat becomes clear if we cosider a eviromet that provides viscous frictio with the simple law F,. (7.37) For the Fourier images of coordiate ad force this gives the relatio F = i, so that accordig to Eq. (4), " ( ) Im i, i.e.. (7.38) Hece, eve i the geeral case, coefficiet describes a effective low-speed drag (kiematic frictio) provided by the eviromet.) I this case Eq. (34) turs ito the Nyquist formula: 49 k T SF ( ) B, i.e. Ff kbtd 4. (7.39) Accordig to Eq. (), if such a costat spectral desity 5 persisted at all frequecies, it would correspod to a delta-correlated process F(t), with K ( ) S () ( ) k BT ( ), (7.4) similar to already discussed above see Eq. (8). F F oscillator was by o meas used i its derivatio. As will see i the et sectio, this fact leads to rather iterestig cosequeces ad eve coceptual opportuities. 49 Actually, the 98 work by H. Nyquist was about electroic oise i resistors, ust discovered eperimetally by his Bell Labs colleague J. Johso. For a Ohmic resistor, as a dissipative eviromet of the electric circuit it is coected with, Eq. (37) is ust the Ohm s law, ad may be recast as either V = -R(dQ/dt) = RI, or I = - G(d/dt) = GV. Thus for voltage V i a ope circuit, correspods to resistace R, while for curret I i the short circuit, to coductace G = /R. I this case, the fluctuatios described by Eq. (39) are referred to as the Johso-Nyquist oise. (Because of this importat applicatio, ay model leadig to Eqs. (36)-(37) is frequetly referred to as Ohmic dissipatio, eve if the physical ature of variables ad F is quite differet.) Aother ote: the Nyquist formula (39) should ot be cofused with the Nyquist-Shao theorem describig the miimum samplig rate of a aalog sigal. 5 A radom process whose properties may be reasoably approimated by costat spectral desity is frequetly called the white oise, because the it is a radom miture of all possible siusoidal compoets with equal weights, remidig atural white light s compositio. Chapter 7 Page 9 of 58

325 Sice i the classical limit the right-had part of Eq. (9) is egligible, ad the correlatio fuctio may be cosidered a eve fuctio of time, the symmetrized fuctio uder the itegral i Eq. (3) may be rewritte ust as F()F(). I the limit of low observatio frequecies (i the sese that is much smaller tha ot oly the quatum frotier k B T/, but also the frequecy scale of fuctio ()/), Eq. (38) may be used to recast Eq. (35) i the form 5 " lim F F d. (7.4) k T To coclude this sectio, let me retur for a miute to the questios formulated i our earlier discussio of dephasig i the two-level model. I that problem, the dephasig time scale is T = /D. Hece the classical approach to the eviromet, used i Sec. 3, is adequate if D << k B T. Net, we may idetify operators f ad z participatig i Eq. (7) with, respectively, operators F ad of the geeral Eq. (9). The the compariso of Eqs. (8), (88) ad (4) yields 4k BT D, (7.4) T so that, for the model described by Eq. (37) with temperature-idepedet drag coefficiet, the dephasig rate is proportioal to temperature. B Dephasig time via dissipatio 7.5. The Heiseberg-Lagevi approach The fluctuatio-dissipatio theorem opes a very simple ad efficiet way for aalysis of the system of iterest (s i Fig. ). It is to write its Heiseberg equatios (4.99) of motio for relevat operators, which would ow iclude the evirometal force operator, ad eplore these equatios usig the Fourier trasform ad the Wieer-Khichi theorem ()-(3). Such approach to classical equatios of motio is commoly associated with the ame of Lagevi, 5 so that its etesio to dyamics of Heiseberg-picture operators is frequetly referred to as the Heiseberg-Lagevi (or quatum Lagevi ) approach to ope system aalysis. 53 Perhaps the best way to describe this method is to demostrate how it works for the very importat case of a D harmoic oscillator, so that the geeralized coordiate of Sec. 4 is ust the oscillator s coordiate. For the sake of simplicity, let us assume that the eviromet provides the simple Ohmic dissipatio described by Eq. (37) - which is a good approimatio i may cases. As we already kow from Chapter 5, the Heiseberg equatios of motio for operators of coordiate ad mometum of the oscillator, i the presece of eteral force, are 5 I some fields (especially i physical kietics ad chemical physics), this particular limit of the Nyquist formula, is called the Gree-Kubo (or ust Kubo ) formula. As was discussed above, these ames may be more reasoably associated with Eq. (9). 5 After P. Lagevi, whose 98 work was the first systematic developmet of Eistei s ideas (95) of the Browia motio theory i the radom force laguage, as a alterative to M. Smoluchowski s approach usig the probability desity laguage see Sec. 6 below. 53 Perhaps the largest credit for this etesio belogs to M. La whose work, i the early 96s, was motivated mostly by quatum electroics applicatios see, e.g., his moograph M. La, Fluctuatio ad Coheret Pheomea i Classical ad Quatum Physics, Gordo ad Breach, 968, ad refereces therei. Chapter 7 Page 3 of 58

326 p, p m F, m (7.43) so that usig Eqs. (9) ad (37), we get p ~, p m F. t (7.44) m Combiig Eqs. (44), we may write their system as a sige differetial equatio m ~ m F, (7.45) that is absolutely similar to the classical equatio of motio. 54 (I the view of Eqs. (5.4) ad (5.48), whose corollary the Ehrefest theorem (5.49) is, this should be by o meas surprisig.) For the Fourier images of the operators, defied similarly to Eq. (5), Eq. (45) gives the followig relatio, F, (7.46) m i that should be also well kow to the reader from the classical theory of forced oscillatios. However, sice the Fourier compoets are still Heiseberg-picture operators, ad their values for differet do ot commute, we have to tread carefully. The best way to proceed is to write a copy of Eq. (46) for frequecy (- ), ad the combie these equatios to form a symmetrical combiatio similar that used i Eq. (4). The result is ' t ' F ' '. F F F (7.47) m i Sice the spectral desity defiitio similar to Eq. (4) is valid for ay observable, i particular for, Eq. (47) allows us to relate the symmetrized spectral desities of coordiate ad force: SF ( ) SF ( ) S ( ). (7.48) m i m Now usig a aalog of Eq. (6) for, we ca calculate coordiate s variace: S F ( ) K () S ( ) d m d, (7.49) where ow, i cotrast to the otatio used i Sec. 4, sig meas the averagig over the usual statistical esemble of may systems of iterest i our curret case, of may harmoic oscillators. If the couplig to eviromet is so weak that drag coefficiet η is small (i the sese that the oscillator s dimesioless Q-factor 48 is large, Q mω /η >> ), this itegral is domiated by the resoace peak i a arrow viciity, -, of its resoace frequecy, ad we ca take the relatively smooth fuctio S F () out of the itegral, thus reducig it to a table itegral: See, e.g., CM Sec See, e.g., MA Eq. (6.5a). Chapter 7 Page 3 of 58

327 S S F F ( ) ( ) m S ( ) F m S ( ) F (m / ) d d d S m m F ( ). (7.5) With the accout of the FDT (34) ad Eq. (38), this gives coth coth. (7.5) m k BT m k BT But this is eactly Eq. (48) that was obtaied from the Gibbs distributio, without ay eplicit accout of the eviromet - though keepig it i mid by usig the otio of the thermally-equilibrium esemble. 56 (Notice that the drag coefficiet, which characterizes the oscillator-to-eviromet iteractio stregth, has cacelled!) Does this mea that we have toiled i vai? By o meas. First of all, the FDT result has a importat coceptual value. For eample, let us cosider the low-temperature limit k B T <<, whe Eq. (5) is reduced to. (7.5) m Let us ask a aïve questio: What eactly is the origi of this coordiate ucertaity? From the poit of view of the usual quatum mechaics of closed (Hamiltoia) systems, there is o doubt: this ovaishig variace of coordiate is the result of the fial spatial etesio of the groud-state wavefuctio, reflectig the Heiseberg s ucertaity relatio (that i tur results from the fact that the operators of coordiate ad mometum do ot commute) see Eq. (.7). However, from the poit of view of the Heiseberg-Lagevi equatio (45), variace (5) is a ualieable part of the oscillator s respose to the fluctuatio force F ~ t eerted by the eviromet at frequecies. Though it is impossible to refute the former, absolutely legitimate poit of view, i may applicatios it is much easier to subscribe to the latter stadpoit, ad treat the coordiate ucertaity as the result of the socalled quatum oise of the eviromet. This otio has received umerous cofirmatios i eperimets that did ot iclude ay oscillators with the eigefrequecies close to the oise measuremet frequecy. 57 The advatage of the Heiseberg-Lagevi approach is that for ay > it is possible to calculate the (eperimetally measurable!) distributio S (), i.e. decompose the fluctuatios ito spectral compoets. This procedure is ot restricted to the limit of small (large Q factors); for ay dampig we may ust plug the FDT (34) ito Eq. (49) ad itegrate. As a eample, let us have a look at the so-called quatum diffusio. A free D particle may be cosidered as the particular case of a D harmoic oscillator with =, so that combiig Eqs. (34) ad (49), we get 56 By the way, the simplest way to calculate S F (), i.e. to derive the FDT, is to require that Eqs. (48) ad (5) give the same result for a oscillator with ay eigefrequecy. This is eactly the approach used by H. Nyquist (for the classical case) see also SM Sec See, for eample, R. Koch et al., Phys. Lev. B 6, 74 (98).. Chapter 7 Page 3 of 58

328 Quatum diffusio S F ( ) d coth d. (7.53) ( m ) k T ( m ) This itegral has two divergeces. The first oe, of the type d/ at the lower limit, is ust a classical effect: accordig to Eq. (85), particle s displacemet variace grows with time, so it caot have a fiite time-idepedet value that Eq. (53) tries to calculate. However, we still ca use that result to sigle out the quatum oise effect o diffusio - say, by comparig it with a similar but purely classical case. These effects are promiet at high frequecies, especially if the quatum oise overcomes the thermal oise before the dyamic cut-off, i.e. if k BT. (7.54) m I this case there is a broad rage of frequecies where the quatum oise gives a substatial cotributio to the itegral: / m / m d d l ~. Q (7.55) mkbt k T / k T / B Formally, this cotributio diverges at either m or T, but this logarithmic (i.e. etremely weak) divergece is readily queched by a almost ay chage of the eviromet model at very high frequecies, where the Ohmic approimatio give by Eq. (36) becomes urealistic. The Heiseberg-Lagevi approach is etremely simple ad powerful, 58 but is has its limitatios. The mai oe is that if the equatios of motio for the Heiseberg operators are ot liear, there is o liear relatio, such as Eq. (46), betwee the Fourier images of the geeralized force ad geeralized coordiate, ad as the result there is o simple relatio, such as Eq. (48), betwee their spectral desities. I other words, if the Heiseberg equatio of motio are oliear, there is o regular simple way to use them to calculate statistical properties of the observables. For eample, let us retur to the dephasig problem described by Eqs. (68)-(7), ad assume that the geeralized force is characterized by relatios similar to (93) ad (34). Now writig the Heiseberg equatios of motio for the two remaiig spi operators, ad usig the commutatio relatios betwee them, we get ~ a f ( t) a f ~ ( t) y, y. (7.56) These equatios do ot provide a liear relatio betwee the Pauli operators ad the fluctuatio force, so eve if we kow spectral properties of the latter from the FDT, this does ot help too much - uless we retur to the approimate, classical approach described i Sec. 3 above. 59 B B 58 Its atural geeralizatios eable aalyses of fluctuatios i arbitrary liear systems, i.e. the systems described by liear differetial (or itegro-differetial) equatios of motio, icludig those with may degrees of freedom, ad distributed systems (cotiua). 59 For some calculatios, this problem may be avoided by liearizatio: if we are oly iterested i small fluctuatios, the Heiseberg equatios of motio may be liearized about their epectatio values (see, e.g., CM Sec. 4.), ad the liear equatios for variatios solved either as has bee show above, or (if the epectatio values evolve i time) by their Fourier epasios. Chapter 7 Page 33 of 58

329 7.6. Desity matri approach The mai alterative approach, that is essetially a geeralizatio of that used i Sec., is to etract the fial results from the dyamics of the desity matri of our subsystem s of iterest (which, from this poit o, will be called w s ). I will discuss this approach i detail, 6 cuttig ust a few techical corers, i each case referrig the reader to special literature. We already kow that the desity matri allows the calculatio of the epectatio value of ay observable of system s see Eq. (5). However, our iitial recipe (6) for the desity matri calculatio, which requires the kowledge of the eact state () of the whole Uiverse, is ot too practicable, while the vo Neuma equatio (66) for the desity matri evolutio is limited to cases i which probabilities W of the system states are fied thus ecludig such importat effects as the eergy relaatio. However, such effects may be aalyzed usig a differet assumptio that the system of iterest iteracts oly with some local eviromet (say, with the lab room) that is i the thermallyequilibrium state described by a diagoal desity matri see Eqs. (5) ad (3). This calculatio is facilitated by the followig observatio. Let us umber the basis states of the full local system (the system of our iterest plus its local eviromet) by ide l, ad apply Eq. (5) to write A Tr Aw All wl ' l l, l' l A l' l' w ' l, (7.57) where ŵ is the statistical operator of this full composite system. At weak iteractio betwee the system s ad local eviromet e, their variables reside i differet Hilbert spaces, so that we ca write l l,l' s e k. (7.58) ad if observable A depeds oly o the coordiates of system s, Eq. (57) yields where A,' k,k',' k,k' s e k A s ' s kk' ŵ s is defied as A s e k' ' s ' e k ' w s e k ' w s e k s ' w s, ' A ' s e k ' k e k w e k s (7.59) Tr ( Aw ), e w e Tr w. (7.6) k k Sice Eq. (59) is similar to Eq. (5), ŵ s may serve as the statistical operator defied i the Hilbert space of the system of our iterest. The huge advatage of Eqs. (59)-(6) is that they are valid for a arbitrary state of the local eviromet, icludig the case whe it is i the thermodyamic equilibrium. By the way, the similarity of Eqs. (5) ad (59) may serve as the strog argumet, promised i Sec., for the validity of the former relatio eve if the Uiverse as a whole is ot i a pure state. (The argumet is, however, imperfect, because the latter relatio has bee derived from the former oe.) k k s 6 As i Sec. 4, the reader ot iterested i the derivatio of the basic equatio (8) for the desity matri evolutio may immediately ump to the discussio of this equatio ad its applicatios. Chapter 7 Page 34 of 58

330 Now, sice at a sufficietly large size of the local eviromet e, the composite system (s + e) may be cosidered Hamiltoia, with fied probabilities of its states, for the descriptio of time evolutio of its statistical operator ŵ (agai, i cotrast to that, ŵ s, of the system of our iterest) we may use the vo Neuma equatio (66). Partitioig its right-had part i accordace with Eq. (68), we get: H, w H, w H, w i w. (7.6) s The et step is to use the perturbatio theory to solve this equatio i the lowest order i Ĥ it that yields ovaishig results due to the iteractio. For that, Eq. (6) is ot very coveiet, because its right-had part cotais two other terms, which are much larger tha the iteractio Hamiltoia. To mitigate this techical difficulty, the iteractio picture (which was discussed i the ed of Sec. 4.6), is very hady - though ot absolutely ecessary. As a remider, i that picture (whose etities will be marked with ide I, with the umarked operators assumed to be i the Schrödiger picture), both the operators ad the state vectors (ad hece the desity matri) deped o time. However, the time evolutio of the operator of ay observable A is described by Eq. (67) with the uperturbed part of the Hamiltoia oly see Eq. (4.4). I our curret case (68), this meas i A A, H. (7.6) where the uperturbed Hamiltoia cosists of two idepedet parts: H I I e it H s H e. (7.63) O the other had, the state vector evolutio is govered by the iteractio evolutio operator û I that obeys Eqs. (4.5). Sice this equatio, usig the iteractio-picture Hamiltoia (4.6), H u I, (7.64) H itu is absolutely similar to the ordiary Schrödiger equatio usig the full Hamiltoia, we may repeat all argumets give i the begiig of Sec. 3 to coclude that the dyamics of the desity matri i the iteractio picture of a Hamiltoia system is govered by the followig aalog of the vo Neuma equatio (66): i w H, w. (7.65) I Sice this equatio is similar i structure (with the opposite sig) to the Heiseberg equatio (66), we may use solutio Eq. (4.9) of the latter equatio to write its aalog: 6 t u t, w () u t. I I w I I I. (7.66) It is also straightforward to verify that i this picture, the epectatio value of ay observable A may be foud from the epressio similar to the basic Eq. (5): 6 Notice the opposite order of the uitary operators, which results from the already metioed sig differece. Note also that we could write a similar epressio i the Schrödiger picture: w t uw () u, where û is the full time-evolutio operator. Chapter 7 Page 35 of 58

331 A Tr A I w I, (7.67) so that the iteractio ad Schrödiger pictures give the same fial results. I the most frequet case of biliear iteractio (9), 6 Eq. (6) is readily simplified, i differet ways, for the both operators participatig i the product. I particular, for A, it yields I, H, H, H i. (7.68) I Sice operator of coordiate is defied i the Hilbert space of system s, it commutes with the Hamiltoia of the eviromet, so that we fially get I I I s s I e i, H. (7.69) O the other had, takig A F, we should take ito accout that the last operator is defied i the Hilbert space of the eviromet, ad commutes with the Hamiltoia of the uperturbed system s. As a result, we get This meas that with our time-idepedet uperturbed Hamiltoias I I e i F F, H. (7.7) Ĥ s ad Ĥ e, the time evolutio of the iteractio-picture operators is rather simple. I particular, the aalogy betwee Eq. (7) ad Eq. (93) allows us to immediately write the followig aalog of Eq. (94): i i F I t ep H tf e ep H et, (7.7) so that i the statioary (eigestate) basis of the eviromet, i i E E F ' I ( t) ep EtF' () ep E t F' i t ' ' () ep, (7.7) ad similarly (but i the basis of the eigestates of system s) for operator. As a result, Eq. (64) may be also factored: H I i i t u t, H u t, ep H H t F ep H H it s e s e t i ep i ep i ep i H t H t H t F()ep H t s s e e I Now, as i Sec. 4, we may rewrite Eq. (65) i the itegral form: w I i t t H t', w t' I I t F I. t (7.73) dt' ; (7.74) 6 A similar aalysis of a more geeral case, whe the iteractio with eviromet may be represeted as a sum of products of the type (9), may be foud i a moograph by K. Blum, Desity Matri Theory ad Applicatios, 3 rd ed., Spriger,. Chapter 7 Page 36 of 58

332 pluggig this result, for time t, ito the right-had part of Eq. (74) agai, we get w I t t I I I dt' ( t) F ( t), ( t' ) F ( t' ), w I ( t' ) t H t, H t', w t' dt', (7.75) where, for the otatio brevity, from this poit o I will strip operators ad F of their ide I. (Their time depedece idicates the iteractio picture clearly eough.) So far, this equatio is eact (ad caot be solved aalytically), but this is the right time to otice that eve if we take the desity matri i its right-had part equal to its uperturbed value (correspodig to o iteractio betwee system s ad its thermally-equilibrium eviromet e), w I t' w, with s t' we e we e' W ', (7.76) where e are the statioary states of the eviromet ad W are the Gibbs probabilities (3), Eq. (75) would still provide some ovaishig time evolutio of the desity operator. This is eactly the first ovaishig perturbatio we have bee lookig for. Now usig Eq. (6), we fid the equatio of evolutio of the desity operator of our system of iterest: t t Tr ( t) F ( t), ( t' ) F ( t' ), w ( t' ) w w s s e dt', (7.77) where the trace is over the statioary states of the eviromet. I order to spell out the right-had part of Eq. (77), ote agai that the coordiate ad force operators commute with each other (but ot with themselves at differet time momets!) ad hece may be swapped, so that we may write Tr...,...,... t t' w s t' Tr F t F t' we t ws t' t' Tr F t wef t' t' w Tr Tr s t' t F t' we F t ws t' t' t we F t' Ft t t' w s t' F' t F' t' W t ws t' t' F' t W ' F' t', ', ' t' w s t F' t' W' F' t ws t' t W F' t' F' t., ', ' (7.78) Sice the summatio o both idices ad i this epressio is over the same eergy level set (of all eigestates of the eviromet), we may swap the idices i ay of the sums. Doig that i the terms with factors W, we tur them ito W, so that this factor becomes commo: Tr Now usig Eq. (7), we get...,...,... W t t' w t' F t F t' t w t' t' F t F t', ' t' w t F t' F t w t' t F t' F t. s ' s ' ' ' s s ' ' ' ' (7.79) Chapter 7 Page 37 of 58

333 Tr...,...,... W F' t t' ws t', ' W F ', ' ~ Et t' cos t' w t t, t', w t' s s ~ ~ Et t' E t t' epi t w s t' t' ep i ~ ~ Et t' E t t' epi w s t' t ep i ~ Et t' iw F' si s,' t, t', w t', (7.8) where {, } meas the aticommutator see Eq. (4.34). Comparig the two double sums participatig i this epressio with Eqs. (8) ad (), we see that they are othig else tha, respectively, the symmetrized correlatio fuctio ad the Gree s fuctio (multiplied by /) of the time-differece argumet = t t. As the result, Eq. (77) takes a very simple form: w s t t K F t t' ( t), ( t' ), w s ( t' ) dt' Gt t' ( t), ( t' ), w s ( t' ) i t dt'. (7.8) Let me hope that the reader eoys this beautiful result as much as I do, ad that it is a sufficiet itellectual award for his or her effort of followig its derivatio. It gives a self-sufficiet equatio for time evolutio of the desity matri of the system of our iterest (s), with the effects of its eviromet represeted oly by two real algebraic fuctios of τ oe (K F ) describig eviromet s fluctuatios ad aother oe (G) represetig its the average respose to system s dyamics. Ad most spectacularly, these are eactly the same fuctios as participate i the Heiseberg-Lagevi approach to the problem, ad hece related to each other by the fluctuatio-dissipatio theorem (34). After a short celebratio, let us ackowledge that Eq. (8) is still a itegro-differetial equatio that eeds to be solved together with Eq. (69). Such equatios do ot allow eplicit aalytical solutios ecept for very simple (ad ot very iterestig) cases. For most applicatios, further simplificatios should be made. Oe of them is based o the fact (which was already discussed i Sec. 3) that both evirometal fuctios participatig i Eq. (8) ted to zero whe their argumet becomes larger that certai eviromet correlatio time c, which is frequetly much shorter that the time scales T of the evolutio of the desity matri elemets. Moreover, the characteristic time scale of the coordiate operator evolutio may be also short o the scale of T. I this limit, all argumets t of the desity operator givig substatial cotributios to the right-had part of Eq. (7) are so close to t that it does ot matter whether its argumet is t or ust t. This simplificatio (t t) is kow as the Markov approimatio. 63 However, this approimatio aloe is still isufficiet for fidig the geeral solutio of Eq. (8). Substatial further progress is possible i two importat cases. The most importat of them is whe the itrisic Hamiltoia Ĥ s of our system of iterest is time-idepedet ad has a very discrete eigeeergy spectrum E, 64 with well-separated levels: Desity matri time evolutio 63 Named after A. Markov (856-9; i older literature, Markoff ), because the result of this approimatio is a particular case of the Markov process whose future developmet is completely determied by its preset state. 64 Rather reluctatly, I will use this stadard otatio, E, for the eigeeergies of our system of iterest (s), i hope that the reader would ot cofuse these discrete eergy levels with the quasi-cotiuous eergy levels of its eviromet, participatig i particular i Eqs. (8) ad (). As a remider, by this stage of our calculatios the eviromet levels have disappeared, leavig behid their trace fuctios K F () ad G(). Chapter 7 Page 38 of 58

334 E E'. (7.8) T Let us see what does this coditio yield for Eq. (8) rewritte for the matri elemets i the statioary state basis (from this poit o, I will drop ide s for brevity): w ' t K F t t' ( t), ( t' ), w ' dt' Gt t' ( t), ( t' ), w ' i t ' dt' ; (7.83) after spellig out the commutators, it icludes 4 operator products, which differ oly by the operator order. Let us have a good look at the first product, ( t) ( t' ) w ' ( t) ( t' ) w, (7.84) m,m' where idices m ad m ru over the same set of eigeeergies of the system s of our iterest as idices ad. Accordig to Eq. (69) with a time-idepedet H s, matri elemets (i the statioary state basis) oscillate i time as ep{i t}, so that m mm' m'' ( t) ( t' ) w epi t t' ' m, m' m mm' w, (7.85) where the coordiate matri elemets are i the Schrödiger picture ow, ad I have used the atural otatio (6.85) for the quatum trasitio frequecies: ' ' m mm' m'' E E. (7.86) Accordig to coditio (8), frequecies with are much higher tha the speed of evolutio of the desity matri elemets (i the iteractio picture!) i both the left-had ad right-had parts of Eq. (83). As we already kow from Sec. 6.5, this meas that i the right-had part of Eq. (83) we may keep oly the terms that do ot oscillate with frequecies, because they would give egligible cotributio to the desity matri dyamics. 65 For that, i the double sum (85) we may keep oly the terms proportioal to differece (t t ), because they will give (after itegratio over t ) a slowly chagig cotributio to the right-had part. 66 These terms should have m + mm =, i.e. (E E m ) + (E m E m ) E E m =. For a o-degeerate eergy spectrum, this requiremet meas m = ; as a result, the double sum is reduced to a sigle oe: ( t) ( t' ) w epi t t' w epi t t' ' Aother product, w ( t' ) ( t) absolutely similarly, givig ' m m m m ' m m m w '. (7.87), that appears i the right-had part of Eq. (83), may be simplified w ( t' ) ( t) epi t' t ' m 'm 'm w '. (7.88) 65 This is essetially the same rotatig-wave approimatio (RWA) that is so istrumetal i other fields of ot oly quatum mechaics, but classical physics as well see, e.g., CM Secs As was already discussed i Sec. 4, the lower-limit substitutio (t = - ) i itegrals (74) gives zero, due to the fiite-time memory of the system, epressed by the decay of the correlatio ad respose fuctios at large values of the time delay = t t. Chapter 7 Page 39 of 58

335 These epressios hold true whether ad are equal or ot. The situatio is differet for two other products i the right-had part of Eq. (83), with w sadwiched betwee ad. For eample, ( t) w ( t' ) ( t) w ( t' ) w epi t t' ' m, m' m mm' m'' m, m' m mm' m''. (7.89) For this term, the same requiremet of havig a fast oscillatig fuctio of (t t ) oly yields a differet coditio: m + m =, i.e. E E E E. (7.9) m Here the double sum reductio is possible oly if we make a additioal assumptio that all iterlevel eergy distaces are uique, i.e. our system of iterest has o equidistat levels (such as i the harmoic oscillator). For diagoal elemets ( = ), the RWA requiremet is reduced to m = m, givig sums over all diagoal elemets of the desity matri: ( t) w ( t' ) epi t t' m m m' ' m w mm m m''. (7.9) (Aother similar term ( t' ) w ( t), is ust a comple cougate of Eq. (9).) However, for offdiagoal matri elemets ( ), the situatio is differet: Eq. (9) may be satisfied oly if m = ad also m =, so that the double sum is reduced to ust oe, o-oscillatig term: ( t) w ( t' ) ' w' '', for '. (7.9) The secod similar term, ( t' ) w ( t), is eactly the same, so that i oe of the itegrals of Eq. (83), these terms add up, while i the secod oe, they cacel. This is why the fial equatios of evolutio look differetly for diagoal ad off-diagoal elemets of the desity matri. For the former case ( = ), Eq. (83) is reduced to the so-called master equatio 67 relatig diagoal elemets w of the desity matri, i.e. the eergy level occupacies W : 68 W m m K i G F W W epi ep i W W epi ep i d, m m m m m m (7.93) where t t. Chagig the summatio ide otatio from m to, we may rewrite the master equatio i its caoical form where coefficiets Γ ' ' ' W' ' W W, (7.94) ' K F cos G si dt', ' ' (7.95) Master equatios ad iterlevel trasitio rates 67 The master equatios, first itroduced to quatum mechaics i 98 by W. Pauli, are sometimes called the Pauli master equatios, or kietic equatios, or rate equatios. 68 As Eq. (93) shows, the term with m = would vaish, ad thus may be legitimately ecluded from the sum. Chapter 7 Page 4 of 58

336 are called the iterlevel trasitio rates. 69 Equatio (94) has a very clear physical meaig of the level occupacy dyamics (i.e. the balace of probability flows W) due to the quatum trasitios betwee the eergy levels (Fig. 6), i our curret case caused by the iteractio betwee the system of our iterest ad its eviromet. 7 higher levels E W ' ' E ' lower levels W ' Fig Probability flows betwee the eergy levels, described by the master equatio (86). Trasitio rates via geeralized susceptibility Detailed balace equatios The Fourier trasforms (3) ad (3) eable us to epress two itegrals i Eq. (95) via, respectively, the symmetrized spectral desity S F () of eviromet force fluctuatios ad the imagiary part () of the geeralized susceptibility, both at frequecy =. After that we may use the fluctuatio-dissipatio theorem (34) to eclude the former fuctio, gettig fially ' ' ' " ' coth kbt ' ep " ( E E / k T ' ' ) B. (7.96) Note that sice the imagiary part of the geeralized susceptibility is a odd fuctio of frequecy, Eq. (96) is i compliace with the Gibbs distributio for arbitrary temperature. Ideed, accordig to this equatio, the ratio of up ad dow rates for each pair of levels equals ' ' ep{( E E ' ' ) / k B / T} ep{( E ' E ' ) / k B E E ep T} kbt '. (7.97) O the other had, accordig to the Gibbs distributio (3), i thermal equilibrium the level populatios should be i the same proportio, satisfyig the so-called detailed balace equatios, W ' W' ', (7.98) for each pair {, }, so that all right-had parts of all Eqs. (94) could vaish as they should. Thus, the statioary solutio of the master equatios ideed describes the thermal equilibrium. The closed system of master equatios (94), sometimes complemeted by additioal righthad-part terms that describe iterlevel trasitios due to other factors (e.g., by a eteral ac force with a frequecy close to oe of ), is the key startig poit for practical aalyses of may quatum 69 As Eq. (93) shows, the result for is described by Eq. (95) as well, provided that idices ad are swapped i all compoets of its right-had part, icludig the swap = -. 7 It is straightforward to show that at relatively low temperatures (k B T << E - E ), Eq. (96) gives the same result as the Golde Rate formula (6.34) see Eercise. (The low temperature limit is ecessary to esure that the iitial occupacy of the ecited level is egligible, as was assumed at the derivatio of Eq. (6.34).) Chapter 7 Page 4 of 58

337 systems icludig quatum geerators (masers ad lasers). It is importat to remember that it is strictly valid oly i the rotatig-wave approimatio, i.e. if Eq. (8) is well satisfied for all ad. For a particular (but very importat) case of a two-level system (say, with E > E ) i the lowtemperature limit k B T << = E E, rate >> defies the characteristic time T / of the eergy relaatio process that brigs the diagoal elemets of the desity matri to their thermally-equilibrium values (3). For the Ohmic dissipatio described by Eq. (38), Eq. (96) yields a simple epressio kbt. (7.99) T Of course, time T should ot be cofused with the characteristic time T of relaatio of the offdiagoal elemets, i.e. dephasig, which was already discussed i Sec. 3. By the way, let us see what do Eqs. (83) say about the dephasig rate. Takig ito accout our itermediate results (87)-(9), ad mergig the o-oscillatig compoets (with m = ad m = ) of sums Eq. (87) ad (88) with the terms (9), that also do ot oscillate i time, we get the followig equatio: 7 w ' K i G F epi ep i epi ep i d w, for '. m m m m m m m' m' 'm 'm 'm 'm ' '' (7.) I cotrast with Eq. (94), the right-had part of this equatio icludes both a real ad a imagiary part, ad hece it may be preseted as ' ' ' w ' w / T i, (7.) where both factors /T ad are real. 7 As should be clear from Eq. (), the secod term i the right-had part of this equatio causes slow oscillatios of the matri elemets w, that, after returig to the Schrödiger picture, add ust small correctios 73 to the uperturbed frequecies (83) of their oscillatios, ad are hece are ot importat for most applicatios. More importat is the first term, Eergy relaatio time 7 Because of the reaso eplaied above, this (relatively :-) simple result is ot valid for systems with equidistat eergy spectra, most importatly, for the harmoic oscillator (while Eq. (7.94) is). For the oscillator, with its simple matri elemets, it is straightforward to repeat the above calculatios, startig from (7.83), to obtai a equatio similar to Eq. (7.), but with two other terms, proportioal to w,, i its right-had part. Sice for the harmoic oscillator the Heiseberg-Lagevi approach allows obtaiig most results i a much simpler way, I will skip the derivatio of this equatio ad the discussio of its solutios. The iterested reader may fid such a discussio, for eample, i a paper by B. Zeldovich et al., Sov. Phys. JETP 8, 38 (969). 7 Sometimes Eq. () (i ay of its umerous alterative forms) is called the Redfield equatio, after the 965 work by A. Redfield. Note, however, that several other authors, otably icludig (i the alphabetical order) H. Hake, W. Lamb, M. La, W. Louisell, ad M. Scully, also made key cotributios ito the very fast developmet of the desity-matri approach to ope quatum systems i the mid-96s. 73 This correctio is frequetly called the Lamb shift, because it was first observed eperimetally i 947 by W. Lamb ad R. Retherford, as a mior, ~ GHz shift betwee eergy levels of s ad p states of hydroge, due to the electric-dipole couplig of hydroge atoms to the free-space electromagetic eviromet. (These levels are equal ot oly i the o-relativistic theory (Sec. 3.6), but also i the relativistic, Dirac theory (Sec. 9.7), if the Chapter 7 Page 4 of 58

338 Geeral result for dephasig rates T ' K F G cos cos m m' si si d, for ', m m m m m m' 'm 'm 'm 'm '' (7.) because it describes the effect abset without the eviromet: a epoetial decay of the off-diagoal matri elemets, i.e. dephasig. Comparig the first terms of Eq. () with Eq. (95), we see that the dephasig rates may be described by a very simple formula: T ' m m m m m' m' ' m ' m kbt S, for ', '' '' F (7.3) where the low-frequecy drag coefficiet is agai defied as lim ()/ - see Eq. (38). This result shows that two effects yield idepedet cotributios ito dephasig. The first of them may be iterpreted as a result of the virtual trasitios of the system to other eergy levels m; accordig to Eq. (87), it is proportioal to the stregth of couplig to eviromet at relatively high frequecies m ad m. (If the eergy quata of these frequecies are much larger tha the thermal fluctuatio scale k B T, oly the lower levels, with E m < ma[e, E ] are importat.) O the cotrary, the secod cotributio is due to low-frequecy, essetially classical fluctuatios of the eviromet, ad hece to the low-frequecy dissipative susceptibility. If the susceptibility (more eactly, the ratio = ()/) is frequecy-idepedet, both cotributios are of the same order, but their eact relatio depeds o the relatio betwee the matri elemets of a particular system. Returig agai to the two-level system discussed i Sec. 3, the high-frequecy cotributios vaish because of the absece of trasitios betwee its eergy levels, while the low-frequecy cotributio yields kbt kbt 4kBT σ σ, '' z z (7.4) T T thus eactly reproducig the result (4) of the Heiseberg-Lagevi approach. 74 Note also that Eq. (4) for T is very close i structure to Eq. (99) for T. For our simple iteractio model (7), the offdiagoal elemets of operator z i the statioary-state z-basis vaish, so that T. For the twowell implemetatio of the model (see Fig. 4 ad its discussio), this result correspods to a very high eergy barrier betwee the wells, that ihibits tuelig, ad hece ay chage of well occupacies W L electromagetic eviromet is igored.) The eplaatio of the shift, by H. Bethe i the same 947, has lauched the whole field of quatum electrodyamics to be briefly discussed i Chapter The first form of Eq. (3), as well as the aalysis of Sec. 3, imply that low-frequecy fluctuatios of ay other origi, ot take ito accout i ow curret calculatios (say, uitetioal oise from eperimetal equipmet), may also cause dephasig; such techical fluctuatios are ideed a serious challege at the eperimetal implemetatio of coheret qubit systems see Sec. 8.5 below. Chapter 7 Page 43 of 58

339 ad W R. However, T may become fiite, ad comparable with T, if tuelig betwee the wells is substatial. 75 Now let us briefly discuss dissipative systems with cotiuous spectrum. Ufortuately, for them the oly (relatively :-) simple results that may be obtaied from Eq. (8) are essetially classical i ature. As a illustratio, let us cosider the simplest eample of a D particle that iteracts with a thermally-equilibrium eviromet, but otherwise is free to move (ucofied). As we kow from Chapters ad 5, i this case the most coveiet basis is that of mometum eigestates p. I the mometum represetatio, the desity matri is ust the c-umber fuctio w(p, p ), defied by Eq. (54), that has already bee discussed i brief i Sec.. O the other had, the coordiate operator, that also participates i the right-had part of Eq. (8), has the form give by the first of Eqs. (5.64), i, (7.5) p dual to the coordiate represetatio formula (5.9). As we already kow, such operators are local see, e.g., Eq. (5.8b). Due to this locality, the whole right-had part of Eq. (8) is local as well, ad hece (withi the framework of our perturbative treatmet) the iteractio with eviromet affects essetially oly the diagoal values w(p, p) of the desity matri, i.e. the mometum probability desity w(p). Let us fid the equatio goverig the evolutio of this fuctio i time. Geerally, i the iteractio picture, matri elemets of operators ad ŵ acquire some time depedece, but i the limit p p, this dyamics lacks the high frequecies (86) that have bee so helpful for the derivatio of master equatios. As a result, the oly serious simplificatio of Eq. (8) is possible i the Markov approimatio, whe the time scale of the desity matri evolutio is much loger tha the correlatio time c of the eviromet, i.e. the time scale of fuctios K F () ad G(). I this approimatio, we may take the matri elemets out of the first itegral of Eq. (8), t K ( ), ( ), ( ),, F t t' dt' t t' w t' K F d w (7.6) kbt S,, w,, w, F ad calculate the double commutator i the Schrödiger picture. This may be doe either usig a eplicit epressio for the matri elemets of the coordiate operator, dual to Eq. (5.8b), or i a simpler way, usig the same trick as at the derivatio of the Ehrefest theorem i Sec. 5.. Namely, epadig a arbitrary fuctio f(p) ito the Taylor series i oe of its argumets (say, p), f f k k p ( ) p, (7.7) k k k! p ad applyig Eq. (5) to each term, we ca prove the followig simple commutatio relatio: 75 The tuelig may be described without alterig Eq. (7), ust by addig, to the uperturbed Hamiltoia (69), terms proportioal to other Pauli matrices. The reader is ecouraged to spell out the equatios for the time evolutio of the desity matri elemets of this system, ad aalyze their mai properties at least i the lowtemperature limit. Chapter 7 Page 44 of 58

340 Chapter 7 Page 45 of 58.!!,!, p f i p p f p k i kp i p f k p p f k f k k k k k k k k k k k k (7.8) Now applyig this result sequetially, first to w(p, p ) ad the to the resultig commutator, we get,,, p w p w i p i p w i w. (7.9) It may look like the secod itegral i Eq. (8) might be simplified similarly. However, it vaishes at p p, ad t t, so that i order to calculate the first ovaishig cotributio from that itegral for p = p, we have to take ito accout the small differece t t ~ c betwee the argumets of the coordiate operators uder that itegral. This may be doe usig Eq. (69) with the free-particle Hamiltoia cosistig of the kietic-eergy cotributio aloe: m p m p i H i t t' s,,, (7.) where the eact argumet of the operator i the right-had part is already uimportat, ad may be take for t. As a result, we may use the last of Eqs. (36) to reduce the secod term i the right-had part of Eq. (8) to w m p i w m p d G i dt' w t' t' t t' G t i t,,,, ) ( ), ( ), (. (7.) I the mometum represetatio, the mometum operator ad the desity matri w are ust c-umbers ad commute, so that, applyig Eq. (8) to product pw, we get w m p p i w m p w m p,,,, (7.) ad may fially reduce the itegro-differetial equatio Eq. (8) to a partial differetial equatio:. B p w T k w m p p t w (7.3) This is the D form of the famous Fokker-Plack equatio describig the classical statistics of motio of a free D particle i a medium with a liear drag characterized by the coefficiet. The first, drift term i the right-had part of Eq. (3) describes particle s deceleratio due to the average viscous force (37), F = -v = -p/m, provided by the eviromet, while the secod, diffusio term describes the effect of fluctuatios: particle s radom walk that obeys Eq. (85) with the diffusio coefficiet T k D B. (7.4) This fudametal Eistei relatio, 76 shows agai the itimate coectio betwee the dissipatio (frictio) ad fluctuatios, i this classical limit represeted by their thermal eergy scale k B T It was the mai result of A. Eistei s pioeerig aalysis of such Browia motio i 95. (The developmet of this aalysis i by M. Smoluchowski has led i 9 to the Fokker-Plack theory.) Fokker Plak equatio for free D particle Eistei relatio

341 Just for reader s referece, let me ote that the Fokker-Plack equatio (3) may be readily geeralized to the 3D motio of a particle uder the effect of a additioal eteral force F et (r, t): 78 w p p w p Fw D pw, with F Fet, (7.5) t m m where w = w(r, p, t) is the time-depedet probability desity i the 6D phase space, ad p is the abla/del operator of differetiatio over the mometum compoets, defied similarly to its coordiate couterpart. The Fokker-Plack equatio i this form is the basis for may importat applicatios; however, due to its classical character, its discussio is left for the SM part of my lecture otes. 79 To summarize our discussio of the two alterative approaches to the aalysis of quatum systems iteractig with a thermally-equilibrium eviromet, described i the last three sectios, let me emphasize that they give descriptios of the same pheomea, ad are characterized by the same two fuctios G(τ) ad K F (τ), but from two differet poits of view. Namely, i the Heiseberg-Lagevi approach we describe the system by operators that chage (fluctuate) i time, eve i thermal equilibrium, while i the desity-matri approach the system is described by o-fluctuatig probability fuctios, such as W (t) or w(p), that are statioary i equilibrium. I the (relatively rare) cases whe a problem may be solved by either method, they give idetical results for all observables. 3D Fokker- Plack equatio 7.7. Quatum measuremets Now we have got a sufficiet quatum mechaics backgroud for a brief discussio of quatum measuremets. 8 Let me start with remidig the reader the oly postulate of quatum mechaics that relates this theory with eperimet. I Chapter 4 it was formulated for a pure state described with ketvector a, (7.6) 77 This classical relatio may be derived usig several other ways icludig those much simpler tha used above. For eample, sice the Browia particle s motio may be described by a liear Lagevi equatio, Eq. (4) may be readily obtaied from the Nyquist formula (39) see, e.g., SM Sec Moreover, Eq. (3) may be geeralized to the motio i a additioal periodic potetial U(r). I this case, a aalog of Eq. (5) for the probability desity of quasi-mometum q (rather tha the geuie mometum p) icludes a additioal eergy bad ide (say, ), a additioal force F = -E (where E (q) is the eergy bad structure that was discussed i Secs..7 ad 3.4), ad a additioal term similar to the right-had part of Eq. (94), describig iterbad trasitios with quasi-mometum-depedet rates (q). These rates are still epressed by Eq. (96), but with the matri elemets replaced by those of the vector operator Ω r i of q iterbad trasitios, which was discussed i Chapter 5. For details ad a particular eample of a siusoidal potetial see, e.g., K. Likharev ad A. Zori, J. Low Temp. Phys. 59, 347 (985). 79 For a more detailed aalysis ad several eamples of quatum effects i dissipative systems with cotiuous spectra see, e. g., U. Weiss, Quatum Dissipative Systems, d ed., World Scietific, 999, or H.-P. Breuer ad F. Petruccioe, The Theory of Ope Quatum Systems, Oford U. Press, 7. 8 Quatum measuremets is a very ufortuate term; it would be more sesible to speak about measuremets of quatum mechaical observables. However, the former term is so commo ad compact that I will use it. Chapter 7 Page 46 of 58

342 where a ad A are, respectively, the eigestates of the operator of observable A, defied by Eq. (4.68). Accordig to the postulate, the outcome of each particular measuremet of observable A may be ucertai, 8 but is restricted to the set of eigestates A, with the probability of outcome A equal to W. (7.7) Sice we kow ow that the state of the system (or rather of the statistical esemble of similar systems we are usig for measuremets) is geerally ot pure, this postulate should be re-worded as follows: eve if the system is i the least ucertai state (6), the measuremet outcomes are still probabilistic, ad obey Eq. (33). 8 Quatum measuremet may be uderstood as a procedure of trasferrig the microscopic iformatio cotaied i coefficiets ito macroscopically available iformatio about the outcomes of particular eperimets, that may be recorded ad reliably stored say, o paper, or i a computer, or i our mids. If we believe that such trasfer may be always doe well eough, ad do ot worry too much how eactly, we are subscribig to the mathematical otio of measuremet, that was (rather reluctatly) used i these otes up to this poit. However, every physicist should uderstad that measuremets are performed by physical devices that also should obey the laws of quatum mechaics, ad it is importat to uderstad the basic laws of their operatio. The foudig fathers of quatum mechaics have ot paid much attetio to these issues, probably because of the followig two reasos. First, at that time it looked like the eperimetal istrumets (at least the best of them :-) were doig eactly what postulate (7) was tellig. For eample, had ot the z-orieted Ster-Gerlach eperimet tured two comple coefficiets ad, describig the icomig electro beam, ito particle couter clicks with rates proportioal to, respectively, ad? Also, the crude iteral ature of these istrumets made more detailed questios uatural. For eample, the electro rate coutig with a Geiger couter ivolves a effective disappearace of each icomig electro iside a zillio-particle electric discharge avalache. Thikig about such devices, it was hard to eve imagie measuremets that would ot disturb the quatum state of the particle beig measured. However, sice that time the eperimetal techiques, otably icludig high vacuum, low temperatures, ad low-oise electroics, have much improved, ad evetually more iquisitive questios started to look ot so hopeless. I my scheme of thigs, these questios may be grouped as follows: (i) What are the mai laws of a quatum measuremet as a physical process? I particular, should it always ivolve time irreversibility? a huma/itelliget observer? (The last questio is ot as laughable as it may look see below.) (ii) What is the state of the measured system ust after a sigle-shot measuremet - meaig the measuremet process limited to a time iterval much shorter that the time scale of measured system s evolutio? This questio is aturally related to the issues of repeated measuremets ad cotiuous moitorig of system s state. 8 Besides the trivial case = (so that W = ), whe the system is i a certai eigestate (a ) of operator Â. 8 The reader i doubt is ivited to compare etropy S = - W lw, the measure of system s disorder (see, e.g., SM Sec..) of the pure state (S = ) with that i ay state with several ovaishig values of W (S > ). Chapter 7 Page 47 of 58

343 (iii) If a measuremet of observable A produced a certai outcome A, ca we believe that the system had bee i the correspodig state ust before the measuremet? The last questio is most closely related to various iterpretatios of quatum mechaics, ad will be discussed i the cocludig Chapter, ad ow let me provide some iput o the first two groups of issues. First of all, I am happy to report that these is a virtual cosesus of physicists o the two first questios of series (i). Accordig to this cosesus, ay quatum measuremet eeds to result i a certai, distiguishable state of a macroscopic output compoet of the measuremet istrumet - see Fig. 7. (Traditioally, its compoet is called a poiter, though its role may be played by a priter or a plotter, a electroic circuit sedig out the result as a umber, etc.). This requiremet implies that the measuremet process should have the followig features: - be time-irreversible, - provide large sigal gai, i.e. mappig the quatum process with its -scale of actio (i.e. of the eergy-by-time product) oto a macroscopic motio of the poiter with a much larger actio scale, ad - if we wat high measuremet fidelity, the process should itroduce as little additioal ucertaity as permitted by the law of physics. ecessary iteractio istrumet to huma observer quatum system back actio macroscopic poiter Fig.7.7. Geeral scheme of quatum measuremet. All these requiremets are fulfilled i a good Ster-Gerlach eperimet. However, sice the iteral physics of the particle detector at this measuremet is rather comple, let me give a eample of a differet, more simple sigle-shot scheme 83 capable of measurig the istat state of a typical twolevel system, for eample, a particle i a double quatum well potetial (Fig. 8). 84 Let the system be, at t =, i a pure quatum state described by ket-vector, (7.8) 83 This scheme may be implemeted, for eample, usig a simple Josephso-uctio circuit called the balaced comparator - see, e.g., T. Walls et al., IEEE Tras. o Appl. Supercod. 7, 36 (7), ad refereces therei. Eperimets by V. Semeov et al., IEEE Tras. Appl. Supercod. 7, 367 (997) have demostrated that this system may have measuremet accuracy domiated by quatum-mechaical ucertaity at relatively modest coolig (to ~ K). Oe of advatages of such implemetatio of this measuremet scheme is that it is based o eterally-shuted Josephso uctios devices whose quatum-mechaical model is i a quatitative agreemet with eperimet - see, e.g., D. Schwartz et al., Phys. Rev. Lett. 55, 547 (985). Colloquially, the balaced comparator is a istrumet with a well-documeted Hamiltoia icludig its part describig couplig to eviromet. 84 As a remider, dyamics of this system was discussed i Sec..6 ad the agai i Sec. 6.. Chapter 7 Page 48 of 58

344 where the compoet states ad may be described by wavefuctios localized ear the potetial well bottoms at s ~ see the blue lies i Fig. 8b. Let us rapidly chage the potetial profile of the system at t =, so that at t >, ad ear the origi, it may be well approimated by a iverted parabola (see the red lie i Fig. 8b): m U ( s ) s, at t, s f. (7.9) U ( s ) (a) (b) t f f s t t t Fig Potetial iversio o (a) macroscopic ad (b) microscopic scales of coordiate. It is straightforward to verify that the Heiseberg equatios of motio i such iverted potetial describe a epoetial growth of operator s i time (proportioal to ep{t} ad hece a similar growth of the epectatio value s ad its r.m.s. ucertaity s. 85 At this iflatio stage, the coherece betwee the two compoet states ad is still preserved, i.e. the time evolutio is reversible. Now let the system be weakly coupled to a dissipative (e.g., Ohmic) eviromet. As we already kow, the eviromet performs two fuctios. First, it provides motio with the drag coefficiet (4), so that the system would evetually come to rest at oe of the relatively distat miima, f, of the iverted potetial (Fig. 8a). Secod, the dissipative eviromet esures state s dephasig o some time scale T. If we select the measuremet system parameters i such a way that ep{ T} f, (7.) the the process, after the potetial iversio, cosists of the followig stages, well separated i time: - the iflatio stage, preservig the compoet state coherece but providig a epoetial icrease of its eergy, 85 Somewhat couter-ituitively, the latter growth plays a positive role for measuremet fidelity. Ideed, it does ot affect the itrisic sigal-to-oise ratio s / s, while makig the itrisic (say, quatum-mechaical) ucertaity much larger that possible oise cotributio by the latter measuremet stage(s). Chapter 7 Page 49 of 58

345 - the dephasig stage, at which the coherece is suppressed, ad the desity matri of the system is reduced to a diagoal form describig the classical miture of the probability packets propagatig to the left ad to the right, ad - the stage of settlig to a ew statioary state a classical miture of two states located ear poits s = f, with probabilities (7) equal to, respectively, W = ad W = = -. If the fial states are macroscopically distiguishable (i.e. may play the role of a bistable poiter), as they are i the balaced-comparator implemetatio, there is absolutely o eed, at ay of these stages, to ivolve ay mysterious aother mechaism of wavefuctio chage (differet from the regular, Schrödiger evolutio) for the measuremet process descriptio. This may be the oly appropriate time to metio, very briefly, the famous - or rather ifamous Schrödiger cat parado so much overplayed i popular press. (The oly good aspect of this popularity is that the formulatio of this parado is certaily so well kow to the reader, that I do ot eed to repeat it.) I this thought eperimet, there is o eed to discuss the (rather comple :-) physics of the cat. As soo as the charged particle, produced at the radioactive decay, reaches the Geiger couter, the process rapidly becomes irreversible, so that the coheret state of the system is reduced to a classical miture of two possible states: decay o decay, leadig, correspodigly, to the cat alive cat dead states. So, despite attempts by umerous authors, typically without proper physics backgroud, to preset this situatio as a mystery whose discussio eeds the ivolvemet of professioal philosophers, hopefully by this poit the reader kows eough about dephasig to pay ay attetio. Let me, however, ote the two o-trivial features of this gedake eperimet, that are met i most real eperimets as well, icludig that with the potetial iversio (Fig. 8). First, the role of the measured coordiate of the system uder observatio (s) may be played ot by a coordiate of a sigle fudametal particle, but a certai combiatio of coordiates of may microscopic compoets of a macroscopic body. I particular, i Josephso uctio systems such as the balaced comparator we essetially measure the persistet electric curret ( supercurret ) - a certai liear combiatio of Cartesia compoets of the mometa of the electros that costitute the Bose- Eistei codesate of Cooper pairs. At that, the role of the local eviromet (that cotributes sigificatly to dissipative pheomea) is played by the same electros, with other liear combiatios of electro mometa playig the role of evirometal degrees of freedom - which were called {} i the last few sectios. This makes the couplig to eviromet somewhat less apparet (at least for the people who do ot kow what a liear combiatio is :-). Secod, oe may argue that eve after the balaced comparator (i our first eample) or the cat (i the secod eample) has reached its fial macroscopic state, huma observer s realizatio that i this particular eperimet the bistable poiter is i a certai state istatly decreases the probability (for the same observer!) of its beig i the opposite state to zero. However, as was already discussed i Sec..5, this is a very classical problem of the statistical esemble redefiitio that may be (or may be ot) performed at observer s will. Such redefiitio, if performed, is the oly possible role of a huma (or otherwise itelliget :-) observer i the measuremet process; if we are oly iterested i a obective recordig of results of a pre-fied sequece of eperimets, there is o eed to iclude such observer ito ay discussio. Chapter 7 Page 5 of 58

346 The esemble redefiitio at measuremet leads to several other paradoes, of which the socalled quatum Zeo parado is perhaps most spectacular. 86 Let us retur to a two-level system with the uperturbed Hamiltoia give by Eq. (4.66), with / much larger tha the sigle-shot measuremet time, ad the system iitially (at t = ) is i a certai quatum well. The, as we kow from Secs..6 ad 4.6, before the first measuremet, the probability to fid state i the iitial state at time t is W ( t) cos Ωt. (7.) If the time is small eough (t = dt << /), we may use the Taylor epasio to write Ω dt W ( dt). (7.) 4 Now, let us retur the two-level system, after its measuremet, ito the same quatum well, ad let it evolve with the same Hamiltoia. Sice the occupatio of the opposite state is very small, the evolutio of W will closely follow the same law as i Eq. (), but with the iitial value give by Eq. () Thus, whe the system is measured agai at time dt, Ω dt Ω dt W (dt) W ( dt). 4 4 (7.3) After repeatig this cycle N times (with the total time t = Ndt still much less tha N / /), the probability that the system is still i the iitial state is Ω dt Ω t Ω t W ( Ndt) W ( t) 4 4N. (7.4) 4N Comparig this result with Eq. (), we see that the process of system trasfer to the opposite quatum well has bee slowed dow rather dramatically, ad i the limit N (at fied t), its evolutio is completely stopped by the measuremet process. There is of course othig mysterious here; the evolutio slowdow is due to statistical esemble s redefiitio. Now let me proceed to questio group (ii), i particular to the geeral issue of the back actio of the istrumet upo the system uder measuremet (symbolized with the back arrow i Fig. 7). I istrumets like the Geiger couter or the balaced comparator, such back actio is very large, because the istrumet essetially destroys ( demolishes ) the iitial state of the system uder measuremet. However, i the 97s it was realized that this is ot really ecessary. I Sec. 3, we have already N N 86 This ame, coied by E. Sudarsha ad B. Mishra i 997 (though the parado had bee discussed i detail by A. Turig i 954); is due to the apparet similarity of this parado to classical paradoes by aciet Greek philosopher Zeo of Elea. By the way, ust to have a miute of fu, let us have a look what happes whe Mother Nature is discussed by people to do ot uderstad math ad physics. The most famous of the classical Zeo paradoes is the Achilles ad Tortoise case: a fast ruer Achilles ca apparetly ever overtake a slower Tortoise, because (i the words by Aristotle) the pursuer must first reach the poit whece the pursued started, so that the slower must always hold a lead. For a physicist, the parado has a trivial resolutio, but let us liste what a philosopher (D. Burto) writes about it - ot i some year BC, but i AD: "Give the history of 'fial resolutios', from Aristotle owards, it's probably foolhardy to thik we've reached the ed. For me, this is a sad symbol of moder philosophy. Chapter 7 Page 5 of 58

347 discussed a eample of a two-level system coupled with eviromet (i our curret cotet, with measuremet istrumet) ad described by Hamiltoia so that, with H a, H f, H H H H (7.5) s it e, H it s z it H s. (7.6) Comparig this equality with Eq. (67) we see that i the Heiseberg picture, the Hamiltoia operator (ad hece the eergy) of the system of our iterest does ot chage with time. O the other had, the iteractio ca chage the state of the istrumet, so it may be used to measure its eergy or aother observable whose operator commutes with the iteractio Hamiltoia. Such trick is called either the quatum o-demolitio (QND) or back-actio-evadig (BAE) measuremets. 87 Let me preset a fie eample of a real measuremet of this kid - see Fig z (a) (b) Fig QND measuremet of sigle electro s eergy by Peil ad Gabrielse: (a) the core of eperimetal setup, ad (b) a record of the thermal ecitatio ad spotaeous relaatio of Fock states. 999 APS. I this eperimet, a sigle electro is captured i a Peig trap a combiatio of a (virtually) uiform magetic field B ad a quadrupole electric field. 89 Such electric field stabilizes cyclotro orbits but does ot have ay oticeable effect o electro motio i the plae perpedicular to the magetic field, ad hece o its Ladau level eergies (see Sec. 3.): 87 For a detailed survey of this field see, e.g., either V. Bragisky ad F. Khalili, Quatum Measuremets, Cambridge U. Press, 99, or H. Wisema ad G. Milbur, Quatum Measuremet ad Cotrol, Cambridge u. Press,. 88 S. Peil ad G. Gabrielse, Phys. Rev. Lett. 83, 87 (999). 89 Similar to the oe discussed i EM Sec..4 (see i particular Eq. (.77) ad Fig..7), but with additioal rotatio about oe of the aes either or y. Chapter 7 Page 5 of 58