(1.1) 2. Press, This expression was used in a 1900 talk by Lord Kelvin (born W. Thomson) in reference to the blackbody

Transcription

1 Chapter 1. Itroductio This itroductory chapter briefly reviews the major 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 1900s, 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 just 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 1859, 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 >> 3 per small frequecy iterval is dk 4k dk dn d, (1.1) 3 3 c 3 where c 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 1D 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: 1 For remedial readig, I ca recommed, for eample, D. Griffith, Quatum Mechaics, d ed., Cambridge U. Press, 016. This epressio was used i a 1900 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. (1) 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 J/K. Note that i may theoretical papers (ad i the SM part of my otes), k B is take for 1, i.e. temperature is measured i eergy uits. K. Likharev

2 1 de k BT dn u kbt, (1.) 3 d 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 1%-scale accuracy, were compatible with the pheomeological law suggested i 1900 by Ma Plack: u. (1.3a) 3 c ep( / k T ) 1 The law may be recociled with the fudametal Eq. (1) if the followig replacemet is made for the average eergy of each field oscillator: kbt, (1.3b) ep( / k T ) 1 with a costat factor B B J s, (1.4) 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. 1. Plack radiatio law Plack s costat 10 u u /k B T Fig Blackbody radiatio desity u, epressed i uits of u 0 (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 1887 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 1905 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 1 Page of 6

3 Eergy vs frequecy threshold mi could be readily eplaied assumig that light cosisted of certai particles (ow called photos) with eergy E h, (1.5) 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 / > 0 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 194, Eq. (5) readily eplais 9 Plack s law (3). E -e mv E W Fig. 1.. Eistei s eplaatio of the photoelectric effect s frequecy threshold. (iii) The discrete frequecy spectra of radiatio by ecited atomic gases, kow sice the 1600s, 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 ~10-10 s due to electric dipole radiatio of electromagetic waves. 10 ) Especially challegig was the observatio by J. Balmer (i 1885) 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 just two positive itegers ad : 1 1, ' 0, (1.6) ' with 0 1, s -1. The Balmer series, icludig the value of 0, have foud its first eplaatio i the famous 1913 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 0. (1.7), ' ' 7 As a remider, A. Eistei received his oly Nobel Prize (i 19) 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 (e), so that the threshold correspods to ma = c/ mi = ch/w 300 m approimately at the border betwee the visible light ad ultraviolet radiatio. 9 See, e.g., SM Sec See, e.g., EM Sec. 8.. Chapter 1 Page 3 of 6

4 E ' E, ' E ' E Fig Electromagetic wave radiatio at system s trasitio betwee its two quatized eergy levels. Bohr showed that the correct 11 epressio for the levels (relative to the free electro eergy), E E H 0, (1.8) ad the correct value of the so-called Hartree eergy 1 Hydroge atom s eergy levels me e EH 0 7. e 4, (1.9) 0 (where e C is the fudametal electric charge, ad m e kg is electro s rest mass) could be obtaied, with a virtually oe-lie calculatio, from the classical mechaics plus just oe additioal postulate, equivalet to the assumptio that the agular mometum L = m e vr of the electro movig o a circular trajectory of radius r about hydroge s uclei (i.e. proto, assumed to stay at rest), is quatized as L, (1.10) 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. (10) together with the d Newto s law for the rotatig electro, v e m e, (1.11) r 4 0r 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. (1.1) 4 r (This o-relativistic approach to the problem is justified a posteriori by the fact the relevat eergy scale E H is much smaller tha electro s rest eergy, m e c ~ 0.5 Me.) By the way, the value of r, correspodig to = 1, i.e. to the smallest possible electro orbit, r B m e e / m, (1.13) Hartree eergy costat Agular mometum quatizatio Bohr radius 11 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. 1 Ufortuately, aother mae, Rydberg costat is also frequetly used for either this atomic eergy uit or its half, E H / 13.6 e. To add to the cofusio, the same term Rydberg costat is sometimes used for the reciprocal free-space wavelegth (1/ 0 = 0 /c) correspodig to frequecy 0 = E H /. Chapter 1 Page 4 of 6

5 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 (10) may be preseted as the coditio tha a iteger umber () of certai waves 13 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. (1.14) (iv) The Compto effect 14 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 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. (14): p photo k, (1.15) 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 15 E ( cp) ( mc ). (1.16) (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 ) 1/, mec ' e (1.17) ' cos p cos c c, (1.18) ' 0 si psi, (1.19) c 13 This fact was oticed ad discussed i detail i 193 by L. de Broglie, so that istead of discussig wavefuctios, especially of free particles, we are still frequetly speakig of de Broglie waves. 14 This effect was observed (i 19) ad eplaied a year later by A. Compto. 15 See, e.g., EM Sec Chapter 1 Page 5 of 6

6 (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, (1 cos ), (1.0a) ' m c e which is traditioally represeted as the relatio betwee the iitial ad fial values of photo s wavelegth = /k = /(/c): ' (1 cos ) c (1 cos ), with c, (1.0b) m c m c e ad is i agreemet with eperimet. 16 (v) De Broglie wave diffractio. I 197, followig the suggestio by W. Elassger (who was ecited by de Broglie s cojecture 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, (1.1) where = /k = /p is the de Broglie wavelegth of electros, ad is a iteger. As Fig. 5 shows, this is just the well-kow coditio 17 that the optical path differece l = dsi betwee the de Broglie waves reflected from two adjacet crystal plaes coicides with a iteger umber of, i.e. of the costructive iterferece of the waves. 18 e Compto effect Bragg coditio d si d d si Fig Electro scatterig from a crystal lattice. 16 The costat c, which participates i this relatio, is close to m ad is called the Compto wavelegth of the electro. This term is somewhat misleadig: as the reader ca see from Eqs. (17)-(19), o wave i the Compto problem has such a wavelegth either before or after the scatterig. 17 Frequetly called the Bragg coditio, due to the pioeerig eperimets by W. Bragg with X-ray scatterig from crystals (that started i 191). 18 Later, spectacular eperimets with diffractio ad iterferece of heavier particles, e.g., eutros ad eve C 60 molecules, have also bee performed see, e.g., a review by A. Zeiliger et al., Rev. Mod. Phys. 60, 1067 (1988) ad a later publicatio by O. Nairz et al., Am. J. Phys. 71, 319 (003). 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, 30 (May 014), ad recet advaced eperimets by P. Hamilto et al., Phys. Rev. Lett. 114, (015). Moreover, quatum iterferece betwee differet parts ad differet quatum states of such macroscopic objects as supercoductig codesates of millios Cooper pairs has bee observed see Sec. 3.1 below for details. Chapter 1 Page 6 of 6

7 To summarize, all the listed effects may be eplaied startig from two very simple (ad similarly lookig) formulas: Eq. (5) for photos, ad Eq. (15) 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 190s) 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 ( ) 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 196 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. 1.. Wave mechaics postulates Let us cosider a spiless, 19 o-relativistic poit-like particle whose classical dyamics may be described by a certai Hamiltoia fuctio H(r, p, t), 0 where r is particle s radius-vector ad p is coordiate. 1 Wave mechaics of such Hamiltoia particles may based o the followig set of postulates that are comfortigly elegat - though their fial justificatio 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 19 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. 0 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 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 1 Page 7 of 6

8 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 d 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), (1.a) where sig * meas the comple cojugate. As a result, the total probability of fidig the particle somewhere iside a volume may be calculated as Probability via wavefuctio W wd r 3 * 3 d r. (1.b) I particular, if the volume cotais the particle defiitely (i.e. with the 100% probability, W = 1), Eq. (b) is reduced to the so-called ormalizatio coditio * 3 d r 1. (1.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 Aˆ 3 d r, (1.3) * 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, (1.4) 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 Hˆ, (1.5) 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 1 Page 8 of 6

9 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 just multiples the wavefuctio by this vector, while the operator of particle s mometum 6 is represeted by the spatial derivative: pˆ i, (1.6a) where is the del (or abla ) vector operator. 7 Thus i the Cartesia coordiates, r ˆ r, y, z, pˆ i,,. (1.6b) 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 ˆ pˆ H, (1.7a) m m so that, takig ito accout Eq. (6b), i the Cartesia coordiates, ˆ H. (1.7b) 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, (1.8) t m whose particular (but most importat) solutio is a plae, moochromatic wave, 9 i( krt ) ( r, t) ae, (1.9) 6 For a electrically charged particle i magetic field, this relatio is valid for its caoical mometum see Sec. 3.1 below. 7 See, e.g., Secs 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 Chapter 1 Page 9 of 6

10 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). (1.30) m Costat a may be calculated, for eample, assumig that solutio (9) is eteded over a certai volume, while beyod it, = 0. The from the ormalizatio coditio (c) ad Eq. (9), we get 30 a 1. (1.31) 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 ; (1.3) m accordig to Eq. (30), 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, (1.33) ~ 1/ A A, (1.34) A of measuremet results from the average, i.e. the ucertaity of observable A. I applicatio to wavefuctio (9), these relatios yield E = 0, p = 0, while the particle coordiate r (at ) 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 (15), 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. (30), of as well), may be used to describe spatially localized pulses, called wave packets see Fig. 6. I Sec..1, I will prove (or rather reproduce H. Weyl s proof :-) that the wave Free particle s dispersio relatio Observable s variace Observable s ucertaity 30 For ifiite space ( ), Eq. (31) yields a 0, i.e. wavefuctio (9) vaishes. This formal problem may be readily resolved cosiderig sufficietly log wave packets see Sec.. below. Chapter 1 Page 10 of 6

11 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. (15), to the width p of the mometum compoet distributio as p. (1.35) Re Im (a) a k k (b) 0 the particle is (somewhere :-) here! 0 k 0 k p / Fig (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 0.1% spread of mometum of a 1 ke electro (p ~ kgm/s) allows a wave packet to be as small as ~ 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 1-3) 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 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, 10.1, ad 10.. 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 1905 work which essetially lauched the whole field! However, this fact has bee cofirmed by umerous eperimets, Chapter 1 Page 11 of 6

12 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. 31 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 just 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, (1.36) 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 1/5. 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, 1/39. 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 31 See, e.g., SM Secs. 5.8 ad 6., respectively. 1 Defiitio of probability Chapter 1 Page 1 of 6

13 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 1 lim M A M AW. (1.37) M 1 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 : dw d * 3 d r dt dt. (1.38) Assumig for simplicity that the boudaries of volume 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 cojugate, we get dw dt t * * ˆ * i ( H), (1.39) t * Hˆ Hˆ d r * 3 d r * t t d r i * (1.40) 3 Note that this requiremet is ot eteded to the relativistic quatum theory see Chapter 9 below. Chapter 1 Page 13 of 6

14 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, pˆ H U ( r,t) m m ˆ U ( r, t). (1.41) 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 Hˆ Hˆ ( a ib) Ha ˆ ihb ˆ, (1.4) Hamiltoia of a particle i a field where Ĥa ad Ĥb are also real, while This meas that Eq. (40) may be rewritte as dw dt 1 i * ˆ * ( ˆ) ( Ha ˆ ihb ˆ )* Ha ˆ ihb ˆ Hˆ ( a ib) H H. (1.43) Hˆ Hˆ d r m i * * 1 3 * * Now, let us use geeral rules of vector calculus 36 to write the followig idetity: 3 d r. (1.44) * * * * Ψ Ψ ΨΨ Ψ Ψ Ψ Ψ, (1.45) A compariso of Eqs. (44) ad (45) shows that we may write dw dt 3 ( j) d r, (1.46) where vector j is defied as i * * j ΨΨ c.c. Im Ψ Ψ, (1.47) m m where c.c. meas the comple cojugate 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 0, with I jd r, (1.48) 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 10. (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. 3.1 of this course.) 34 Historically, this was the mai step made (i 196) 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. (11.4a), combied with the del operator s defiitio. 37 See, e.g., MA Eq. (1.). Chapter 1 Page 14 of 6

15 Cotiuity equatio: differetial form where j is the projectio of vector j o the outwardly directed ormal to surface S that limits volume, i.e. the scalar product j, 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 j 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 j, (1.49) 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 j w k w wv, (1.50) 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, j 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 jd r 0. (1.51) t Now we may argue that this equality may is true for ay choice of volume oly if the epressio uder the itegral vaishes everywhere, i.e. if w j 0. (1.5) t This differetial form of the cotiuity equatio is sometimes more coveiet tha its itegral form (48) 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: Aˆ Aˆ ˆ ˆ 1 A1 A, ad associative: Aˆ Aˆ Aˆ Aˆ Aˆ Aˆ Chapter 1 Page 15 of 6

16 ˆ Aˆ Aˆ Aˆ, A (1.53) 1 1 c11 c Aˆ c11 Aˆ c c ˆ ˆ 1A1 c A A ˆ, (1.54) 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 rˆ 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 (1.55) is also a solutio of the same equatio. 40 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. (41) with time-idepedet U = U(r). First of all, let us prove that the followig product, T ( t) ( r), (1.56) 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 Hˆ T ( t) ( r) T ( t) Hˆ ( r), (1.57) ad dividig both parts of the equatio by = T, we get it Hˆ, (1.58) 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. 41 As a result, we are gettig two separate equatios for the temporal ad spatial parts of the wavefuctio: i T E T, (1.59) ariable separatio 40 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. 41 This argumetatio, leadig to variable separatio, is very commo i mathematical physics see, e.g., its discussio i EM Sec..5. Chapter 1 Page 16 of 6

17 Statioary Schrödiger equatio Statioary state: time evolutio Statioary Schrödiger equatio for static potetial H ˆ. (1.60) E The first of these equatios is readily itegrable, givig E T cost ep i t, with ω, (1.61) ad thus substatiatig the fudametal relatio (5) betwee eergy ad frequecy. Pluggig Eqs. (56) ad (61) 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 * Aˆ d r = cost. (1.6) Due to this property, the states described by Eqs. (56), (60), ad (61), are called statioary. I cotrast to the simple ad uiversal time depedece (61), the spatial distributios (r) of the statioary states are ofte hard to fid, ad the solutio of the statioary (or time-idepedet ) Schrödiger equatio (60), 4 which describes the distributios, for various situatios is a major focus of wave mechaics. The statioary Schrödiger equatio (60), with time-idepedet Hamiltoia (41), U ( r) m E, (1.63) 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 1, if ', ' d r, ' (1.64) 0, 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 = 0, with a few eceptios), may be preseted as a uique epasio over the eigefuctio set: ( r,0) c ( r). (1.65) 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 1 Page 17 of 6

18 c * 3 ( r) ( r,0) d r. (1.66) Now let us cosider the followig wavefuctio E ( r, t) cak ( t) k ( r) c ( r)ep i t. (1.67) 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 (41) 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). (1.68) 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 : 0, for 0 a, 0 y a y, ad 0 z az, U ( r ) (1.69), otherwise. The oly way to keep the product U i Eq. (68) fiite outside the well, is to have = 0 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, (1.70a) 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 = 0, 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 1 Page 18 of 6

19 m, E y z 0, for 0 ad a ; for 0 a y 0 ad a y ;, 0 y a y z 0 ad a, ad z. 0 z a z, (1.70b) 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). (1.71) (It is coveiet to postpoe takig care of proper idices for a miute.) Pluggig this epressio ito the Eq. (70b) ad dividig by = XYZ, we get 1 d X 1 d Y 1 d Z E m X d Y dy Z dz. (1.7) 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 1D equatios 1 d X 1 d Y E, E m X d m Y dy with Eq. (7) turig ito the eergy-matchig coditio E y, 1 d Z m Z d E z, (1.73) E E E. (1.74) y All three ordiary differetial equatios (73), ad their solutios, are similar. For eample, for X(), we have a 1D Helmholtz equatio d X me k 0, with X k, (1.75) d ad simple boudary coditios: X(0) = X(a ) = 0. Let me hope that the reader kows how to solve this well-kow 1D 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 1/ si k a 1/ z si a, E k m ma E with 1 1,,..., (1.76). (1.77) 47 The frot coefficiet is selected i a way that esures the (ortho)ormality coditio (64). Chapter 1 Page 19 of 6

20 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,. (1.78) z m a a y az E E X ( ) / a Fig Eigefuctios (solid lies) ad eigevalues (dashed lies) of the 1D 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, (1.79) a a y az,, 1 y z Rectagular quatum well: geeral solutio with the coefficiets which may be readily calculated from the iitial wavefuctio (r, 0), 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 (1.80) k k that is the solutio to the statioary Schrödiger equatio (70a) if it is valid i the whole space. 48 The reader should ot be worried too much by the fact that the fudametal solutio (80) i free space is a travelig wave (havig, i particular, ovaishig value (50) of the probability curret j), 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 1 Page 0 of 6

21 Free particle: eigeeergies 3D umber of states Summatio over 3D states while those iside a quatum well are stadig waves, with j = 0, eve though the free space may be legitimately cosidered as the ultimate limit of a quatum well with volume = a a y a z. Ideed, due to the liearity of wave mechaics, two travelig-wave solutios (80) 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 j = 0. 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. (80) ca take ay real values. (This is why it is more coveiet to label the wavefuctios ad eigeeergies, k Ek 0, (1.81) 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 0), withi a volume d 3 k >> 1/ of the k space is just dn = d 3 k/(k k k ) = / 3. Sice i cotiuum it is more coveiet to work with travelig waves, we should take ito accout that, as was just 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: dn 3 d k 3. (1.8) For dn >> 1, this epressio is idepedet o the boudary coditios, 49 ad is frequetly preseted as the followig summatio rule 3 lim f f dn f d k k3 ( k) ( k) ( k), (1.83) 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. 50 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 1D mechaical waves), because it is valid for waves of ay ature. Chapter 1 Page 1 of 6