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1 This cotet has bee dowloaded from IOPsciece. Please scroll dow to see the full text. Dowload details: IP Address: This cotet was dowloaded o /08/018 at 08:08 Please ote that terms ad coditios apply. You may also be iterested i: Quatitative Core Level Photoelectro Spectroscopy: Brief theory of photoemissio spectroscopy J A C Sataa Niels Bohr ad quatum physics A B Migdal Derivatio of the postulates of quatum mechaics from the first priciples of scale relativity Lauret Nottale ad Marie-Noëlle Célérier Quatum causality ad iformatio V P Belavki From physical assumptios to classical ad quatum Hamiltoia ad Lagragia particle mechaics Gabriele Carcassi, Christie A Aidala, David J Baker et al. From the origi of quatum cocepts to the establishmetof quatum mechaics M A El'yashevich Theory of electro, phoo ad spi trasport i aoscale quatum devices Hatef Sadeghi Aother look through heiseberg s microscope Stephe Bough ad Marcel Regiatto Coceptual problems i quatum mechaics V P Demutski ad R V Polovi

2 IOP Publishig Quatum Mechaics Mohammad Saleem Chapter 1 The failure of classical physics ad the advet of quatum mechaics Quatum mechaics has played a sigificat role i the developmet of various disciplies of physics. It was propouded i 195 ad has reiged supreme ever sice, extedig its domai over the years; of course, i its ow domai it has always bee i excellet agreemet with experimets. It has log sice become the laguage of physics ad ayoe who tries to uderstad the basic priciples of physics without havig a grasp of this subject is doomed to fade away i the darkess of igorace. I this book a attempt has bee made to provide a logical, lucid ad user-friedly treatmet of this elegat ad fasciatig subject. 1.1 A challege for classical physics Lookig through the lattice of history, we observe that the first quarter of the twetieth cetury was a challegig period for classical physics. The iterferece, diffractio ad polarisatio pheomea could oly be explaied by assumig that light had a wave ature. But some other pheomea, such as black body radiatio, the photoelectric effect ad Compto scatterig, defied the wave cocept of electromagetic radiatio. The black body radiatio spectrum was explaied by Plack who assumed that the atoms of the walls of the black body act as electromagetic harmoic oscillators. A oscillator ca radiate eergy oly i quata with E = h ν, where is a positive iteger or zero, ν is the frequecy of the oscillator ad h is a costat ow called Plack s costat. The remaiig two pheomea were explaied by Eistei ad Compto by assumig that radiatio itself, i particular light, cosists of particles, called photos, each photo possessig the eergy hν. Thus, light has a dual ature, sometimes exhibitig the behaviour of waves ad at other times showig the characteristics of particles. I 193, a Frech PhD scholar, de Broglie (proouced de Broy), exteded the idea of the duality of light to the duality of matter. This extesio of a cocept to cover ew realms is ot doi: / ch1 1-1 ª IOP Publishig Ltd 015

3 somethig ew i physics. We remember that Newto had show that the laws of mechaics had the same form i all iertial frames of referece. Eistei, whose geius, i the history of physics, is almost uparalleled i the twetieth cetury, exteded this idea to the etire field of physics by demadig that laws of physics should have the same form i all iertial frames of referece. Ad this became oe of the two basic postulates of special relativity. However, it must be emphasised that Newto proposed it oly as a characteristic of the secod law of motio but Eistei made it a criterio for the validity of ay law of physics. de Broglie wrote that after log reflectio i solitude ad meditatio, I suddely had the idea, durig the year 193, that the discovery made by Eistei i 1905 should be geeralised by extedig it to all material particles ad otably to electros. He assumed that if p is the magitude of the three-mometum of a particle of eergy E, ad λ is the wavelegth ad ν the frequecy of the associated wave, the, i additio to E = hν, we must have h p =. (1.1) ν Accordig to Eistei, this hypothesis of de Broglie s about the dual ature of matter was a first feeble ray of light o this worst of our physics eigmas. I 197 the experimet of Davisso ad Germer, i which electros were scattered by a crystal surface with typical diffractio effects, cofirmed this darig hypothesis which ultimately demolished the classical picture of physics. To get a taste of quatum theory, we aalyse the photoelectric effect ad Compto scatterig. 1. The photoelectric effect The photoelectric effect, the emissio of electros by a metal whe light falls o it, was discovered by Hertz i Experimets showed the followig characteristics of this effect. Whe light falls o a metal surface i a vacuum, the emissio of electros depeds upo the frequecy of the icidet light. There is a miimum frequecy of light which is required for the emissio of electros from a metal. The value of this threshold frequecy varies from metal to metal. The emissio of electros as well as the eergy of the emitted electros, photoelectros, does ot deped upo the itesity of the light source. However, if electros are emitted, the the magitude of their curret is proportioal to the itesity of the icidet light. Fially, the eergy of the photoelectros varies liearly with the frequecy of the light. The classical theory of electromagetic radiatio ca explai some of these characteristics but ot all of them. Credit for solvig this problem goes to Eistei who, i 1905, refied ad exteded the ideas Plack used to explai the black body radiatio spectrum ad assumed that light cosists of quata of eergy, called photos. I fact, Plack had itroduced the cocept of material resoators possessig quata of eergy hν, where is a iteger, while Eistei assumed that each quatum of light possesses the eergy hν. The absorptio of a sigle 1-

4 photo by a electro icreases the eergy of the electro by hν. Part of this eergy is used to remove the electro from the metal. This is called the work fuctio. The remaiig part of the eergy imparted to the electro icreases its velocity ad cosequetly its kietic eergy. Thus if hν, the eergy of a photo icidet o a metal is greater tha the eergy E required to separate the electro from the metal, ad v is the velocity of the emitted electro, the the followig relatio must hold: 1 hν = E + mv. (1.) All the characteristics of this effect are easily explaied by the cocept that light cosists of photos. The above formula shows that if the eergy of the icidet photo is less tha the work fuctio, the electros caot be separated from the surface of the metal ad therefore will ot be emitted. For a particular metal, the work fuctio E beig costat, the relatioship betwee the eergy of the icidet photo ad the kietic eergy of the emitted electro is liear. It is also clear that a more itese source of light will cause photos to be emitted at a greater speed ad this will produce a stroger electro curret. Thus Eistei was able to provide a completely satisfactory picture of the photoelectric effect by usig the cocept of the quatum ature of light. I fact, the dual ature of light is brilliatly reflected by the very assumptio Eistei made about the eergy of a photo. The frequecy is determied by the wave ature of light ad is used to defie the eergy of the particles costitutig the light. It is iterestig to ote that, i 191, Eistei was awarded the Nobel Prize i physics for his services to Theoretical Physics ad especially for his discovery of the law of the photoelectric effect ad ot for propoudig special relativity i 1905 ad geeral relativity i His extraordiarily remarkable work o relativity chaged the complexio of the etire field of physics ad esured him a seat amog the immortals of the subject, but surprisigly this magificet cotributio to the pool of kowledge was ever cosidered specifically for that eviable prize! 1.3 The Compto effect Compto was a America physicist who i 193 performed a crucial experimet which strogly cofirmed the corpuscular ature of light. The results of this experimet were explaied by Compto himself ad idepedetly by Debye. The experimet ot oly cofirmed the law of the coservatio of eergy, which was previously verified by the photoelectric effect, but also the law of coservatio of liear mometum. It was oticed that whe electromagetic radiatio of high frequecy is icidet o electros of a light elemet i which the electros are loosely boud to the ucleus ad ca be treated as free, the scattered radiatio is foud to have a smaller frequecy tha the radiatio of the origial frequecy. This is kow as the Compto effect. The experimet exhibits that the chage i the frequecy of icidet radiatio is idepedet of its iitial frequecy ad depeds oly upo the agle of scatterig. This ca be satisfactorily explaied by the quatum theory of light by makig use of relativistic expressios for various quatities. 1-3

5 Figure 1.1. The Compto effect. Cosider the scatterig of a photo of frequecy ν fallig o a electro of rest mass m 0, i a frame of referece i which the electro is at rest. Let ν be the frequecy of the scattered photo. Let p be the liear mometum of the electro after its collisio with the photo ad α the agle betwee the fial ad iitial directios of the photo. This is show i figure 1.1. Sice the icidet radiatio is of high frequecy, the icomig photo, by virtue of the relatio E = hν, is very eergetic. Therefore the electro which is lightly boud to its atom may be cosidered, to a good approximatio, as free. Thus the Compto effect may be treated as a problem of the collisio betwee a photo ad a statioary free electro. The iitial eergy of the photo is hν ad its iitial mometum is hν/c, while its fial eergy ad mometum are hν ad hν / c, respectively. Accordig to the laws of coservatio of eergy ad mometum, we have hν + m c = hν + E (coservatio of eergy), (1.3) 0 hν hν a = b + p (coservatio of mometum), (1.4) c c where a ad b are uit vectors i the directios of the icidet ad scattered photos, respectively, ad E is the eergy of the electro after the collisio. Rearragig the terms i (1.3), we have hν hν = E m c. 0 Squarig both sides of the above equatio, we obtai h ν + h ν h νν = E + m c Em c. (1.5) Rearragig the terms i (1.4) ad multiplyig through by c, we have hν a hν b = cp. Squarig the two sides of the above equatio, we obtai h ν + h ν h νν cos α = c p (1.6) 1-4

6 where α is the agle betwee a ad b. Substitutig the expressio for c p from the equatio E = m0 c 4 + c p ito (1.6), we obtai Subtractig (1.5) from (1.7), we have h ν + h ν h νν cos α = E m c. (1.7) ( ) 0 4 hνν (1 cos α) = mc E mc = mc( hν hν ) where, i obtaiig the expressio o the extreme right, we have made use of (1.3). The above equatio ca be writte as h ν ν λ λ α = 1 1 (1 cos ) = =, mc νν ν ν c c or 0 h λ λ = (1 cos α). (1.8) mc 0 The quatity h/m 0 c is called the Compto wavelegth ad has the value cm. Equatio (1.8) has bee foud to be cosistet with experimets. For his cotributio, Compto shared the Nobel Prize i physics i 197. Problem 1.1. Commet o the statemet that i the photoelectric effect the photo trasfers all of its eergy while i the Compto effect oly part of the eergy is trasferred to the electro. 1.4 Heiseberg s ucertaity priciple I classical physics, it is tacitly assumed that the operatio of observatio does ot appreciably disturb a system ad, at least i priciple, the disturbace caused by the measuremet process ca be rectified exactly. It required the igeuity of Heiseberg, oe of the most brilliat eve amog the Nobel laureates, to show that wave particle duality imposes restraits o simultaeous precise measuremets of positio ad mometum. The measuremet process i geeral disturbs a system by a amout which caot be predicted. For istace, cosider the hypothetical experimet show i figure 1., devised for the precise measuremet of the positio of a electro. The apparatus icludes a microscope with α as its aperture. The electro beam is movig i the positive directio of the x-axis with a well-defied mometum p x. I order to measure the positio of a electro, it has to be observed. For that purpose, we shie a beam of light alog the egative x-axis. Now, it is ecessary that at least oe photo after fallig o it should be scattered ito the microscope so that the observer sees it through the microscope. Due to the particle 1-5

7 Figure 1.. Measuremet of the positio of a electro. ature of light, as a photo strikes the electro, the latter is disturbed. The mometum of recoil of the electro could be calculated if the iitial ad fial mometa of the electro were kow. But because of the fiite aperture of the microscope, the photo ca eter it alog ay directio o the illumiatio coe of the observer. That is, the directio of the photo scattered ito the microscope is udetermied withi the agle subteded by the aperture. The ucertaity i the measuremet of the mometum of the electro is hν Δ p = si α. (1.9) x c We also otice that, usig stadard optical theory, the resolvig power of the microscope is give by Δ x = λ si α, (1.10) where λ is the wavelegth of light. This meas that Δx gives the ucertaity i the positio of the electro. The ucertaity Δx ca be made as small as we like by makig λ as small as we please ad/or makig the aperture α as large as we desire. But this will ehace the ucertaity i the measuremet of mometum. For istace, if we decrease λ, i.e. icrease the frequecy ν of the icidet light, it will certaily decrease the ucertaity Δx i the measuremet of the positio of the electro, but it will icrease the ucertaity Δp x because the photo strikig with greater eergy ( h ν) will disturb the electro to a greater extet. Multiplyig (1.9) ad (1.10), we obtai hν ΔxΔ p x = λ = h (1.11) c as λ = c/ ν. This meas that the product of the ucertaities i the simultaeous measuremet of the x-compoets of positio ad mometum is of the order of Plack s costat. This is just oe example. I fact, ay experimet desiged for a simultaeous precise measuremet of positio ad mometum will ecouter the 1-6

8 same costrait. This is ot a error i experimetal measuremet. It is iheret i ature i the sese that it is due to the uavoidable iteractio betwee the observer ad the observed durig the process of observatio. The above aalysis shows that a simultaeous precise measuremet of positio ad mometum is impossible. This is kow as Heiseberg s ucertaity priciple or priciple of idetermiacy. It ca be show that similar ucertaity relatios exist betwee eergy ad time ad agular mometum ad agle. I other words, a simultaeous precise measuremet of two caoically cojugate variables is impossible. 1.5 The correspodece priciple Although classical mechaics breaks dow whe applied to determiig the behaviour of tiy objects such as electros, protos, etc, it has bee providig correct aswers to mechaical pheomea at the macroscopic level. Therefore, at this level, quatum theory should be cosistet with classical mechaics. This is kow as Bohr s correspodece priciple ad is said to serve as a guide i discoverig the correct quatum laws. I fact, uder old coditios, a ew theory should always yield the same results as the old theory which it is replacig, because the origial theory has bee explaiig the experimetal data i its ow domai. I the case of quatum mechaics, this correspodece may be specified by claimig that, for large quatum umbers, quatum theory must be cosistet with classical physics. Moreover, if a quatum system has a classical aalogue, the for the limit h 0, it must yield the correspodig classical results. Thus, i the ucertaity priciple, as h 0 i the classical limit, the product ΔxΔp x 0 ad therefore a simultaeous precise measuremet of positio ad mometum at macroscopic level becomes permissible. The importace of the correspodece priciple lies ot i statig that quatum theory should yield the same results as classical mechaics at the macroscopic level, but i describig the coditios uder which it should happe. 1.6 The Schrödiger wave equatio There is o doubt that Plack s quatum theory of black body radiatio, Eistei s hypothesis of light quata for the explaatio of the photoelectric effect, Bohr s postulates regardig the iterpretatio of the spectrum of the hydroge atom ad de Broglie s hypothesis about the dual ature of matter were the milestoes i the progress of physics from 1900 to 193. But physicists desired a differetial equatio which could gover the behaviour of mechaical pheomea ad cosequetly explai various experimetal results. I 196, Schrödiger set up such a differetial equatio which is amed after him ad was supposed to replace Newto s secod law of motio as the basic law of ature i mechaics. The mechaics based o this differetial equatio was called wave mechaics ad is ow kow as quatum mechaics. Schrödiger s differetial equatio udoubtedly outshoe all the abovemetioed postulates ad started commadig immese attetio i the physics commuity immediately after its advet. We will ow set up this differetial equatio. It must be clearly stated that the Schrödiger wave equatio caot be logically derived. The historical developmet may make its presece somewhat 1-7

9 plausible. But we will establish it by adoptig a operatioal techique which is to-the-poit ad simple. We proceed as follows. I terms of the kietic eergy T ad the potetial eergy V of a particle, the orelativistic classical expressio for its total eergy E is give by E = T + V, where V is, i geeral, a fuctio of space ad time coordiates, V = V(r, t), but 1 p T = mv = m, where m is the mass, v is the velocity ad p is the liear mometum of the particle. This yields p E = + m V. (1.1) It is assumed that the trasitio from classical to quatum mechaics is made by iterpretig E, p ad V as operators such that E i ħ, (1.13) h where ħ = π, p ħ i, (1.14) ad V V. (1.15) The operator is give by = i + j + k, x y z where x, y, z are space coordiates ad i, j, k are uit vectors alog the x-, y- ad z-axes. The time coordiate, wherever it occurs, is deoted by t. Note that while E ad p are iterpreted as differetial operators, the potetial eergy is assumed to be oly a multiplicatio operator, its form remaiig uchaged whe movig from classical to quatum mechaics. Equatio (1.1) therefore yields where ħ = ħ i + V, t m + + x y z. 1-8

10 If a fuctio Ψ(x, y, z, t), which for coveiece we may also write as Ψ(r, t) or merely as Ψ, represets the particle uder cosideratio, the operatig this equatio o Ψ(, r t) Ψ ad iterchagig the two sides of the equatio thus formed, we obtai ħ + V Ψ = i ħ Ψ, (1.16 a) m or where HΨ= i ħ Ψ, (1.16 b) H + V m (1.16 c) is the Hamiltoia operator for the system. Equatio (1.16a), equivaletly (1.16b), is a partial differetial equatio i four idepedet variables ad is kow as the timedepedet Schrödiger wave equatio. It is called a wave equatio as it is similar to a differetial equatio for waves. It is assumed to be the fudametal differetial equatio goverig the behaviour of mechaical pheomea. It replaces Newto s secod law of motio i mechaics. However, ulike Newto s law of motio, it is ot garbed i words. This law of ature presets itself oly as a differetial equatio. The fuctio Ψ(r, t), a solutio of the time-depedet Schrödiger wave equatio for a system, is called a wave fuctio ad is ecessarily a complex fuctio because of the complex ature of the differetial equatio. It should ot be cosidered to be a physical wave. It is actually a mathematical fuctio cotaiig all the iformatio that ca be obtaied about the system it represets. The time-depedet Schrödiger wave equatio is a liear partial differetial equatio of the first order i the time derivative ad of the secod order i the spatial derivative. This implies that if its solutio at a particular time t 0 is kow, it ca be calculated at ay time t. But just as i the case of Newto s secod law of motio, there is o logical derivatio of the Schrödiger wave equatio. It was a brilliat guess of Schrödiger i the perspective of the dual ature of matter. The ultimate test of a theory comes from its cofrotatio with experimetal data. Ad of course, for o-relativistic mechaical pheomea, the theory based o Schrödiger s wave equatio has emerged successful. Wheever we have to solve a physical problem i quatum mechaics, we resort to this equatio, just as i classical mechaics we use the secod law of motio. To write the Schrödiger wave equatio for a particular system, we have to fid the classical expressio for the potetial eergy V of the system ad substitute it ito equatio (1.16a). This gives us the desired differetial equatio which may be solved to obtai a complex solutio Ψ(, r t) Ψ. Sice the differetial equatio cotais oly a first-order time derivative, the wave fuctio is uiquely prescribed, oce its value at a time t = t 0 is kow. But how is this fuctio Ψ(r, t) iterpreted so as to 1-9

11 relate it to physically measurable characteristics of the system? Certai prescriptios were proposed but after facig isurmoutable difficulties, o a suggestio by Bor, a cosesus was ultimately developed i the physics commuity. The complex fuctio Ψ(r, t) represetig the particle, beig itself ot directly observable, is iterpreted so that Ψ * (, r t ) Ψ(, r t )d x dd y z represets the probability of fidig the particle i a small volume dxdydz ( dτ) about the poit r ad at time t. It must be emphasised that this iterpretatio is oly a hypothesis. Its validity is established by the success of its predictios. No doubt, the time evolutio of the wave fuctio Ψ(r, t) is iextricably coected to probabilistic cocepts. It has bee rightly stated, i flowery laguage, that Ψ * Ψ is the widow through which we ca view the world of the atom. Schrödiger himself was shocked whe he was told about the statistical iterpretatio of quatum mechaics. He oce told Bohr If we are goig to stick to this damed quatum jumpig, the I regret that I ever had aythig to do with quatum theory. Bohr quipped But the rest of us are thakful that you did. To this iterpretatio, which demolishes the determiism of classical mechaics, Schrödiger ad Eistei could ot recocile themselves, eve to the last days of their lives. The total probability of fidig the particle i a volume i which it is cofied is ( τ) Ψ * ( r, t) Ψ( r, t) d τ Ψ * ( r, t) d ad must be 100%, i.e. 1, because the particle must be somewhere i that volume. Hece we may write Ψ * ( r, t ) Ψ ( r, t ) dτ = 1. (1.17) The wave fuctio Ψ(r, t) is the said to be ormalised to uity or simply ormalised. The above equatio is said to express the ormalisatio coditio. This equatio exists oly if the itegral Ψ * (, r t ) Ψ(, r t )dτ coverges, i.e. it is fiite; for istace, if the wave fuctio goes to zero sufficietly rapidly as r teds to ifiity. The fuctio Ψ(r, t)is the said to be square itegrable or to have a itegrable square. Symbolically, i this case, Ψ(r, t) 0as r.thisis the boudary coditio which is always satisfied whe the state is boud. For istace, a electro boud to the hydroge ucleus, costitutes the hydroge atom a boud state. For such a state, the particle ca ever go to ifiity. The itegral Ψ * t Ψ t τ Ψ t ( r, ) ( r, ) d ( r, ) dτ exists ad the fuctio Ψ(r, t) ca be ormalised. I certai cases, for istace for a free particle, the above itegral may diverge. The a somewhat differet formulatio 1-10

12 of the ormalisatio coditio is to be give which will be cosidered at a later stage. Ψ * (r, t) Ψ(r, t) is the probability desity, i.e. the probability per uit volume. Notice that the statemet i the above form about the probability is valid oly if Ψ(r, t) is ormalised. It may be emphasised that the above prescriptio icorporates the statistical iterpretatio i the basic differetial equatio, the Schrödiger wave equatio. Sice this is a homogeeous, liear differetial equatio, if Ψ(r, t) is a solutio of this differetial equatio, the CΨ(r, t) is also a solutio of the same where C is a costat. I most cases, the value of C ca be determied by usig the ormalisatio coditio. I oe-dimesioal space, Ψ * (x, t) Ψ(x, t)dx is iterpreted as the probability of fidig the particle i the legth dx betwee x ad x + dx at time t. It is importat to state at this stage that we have oly iterpreted Ψ so as to relate it to the probability distributio. Actually, we have to compute various physical quatities, represetig dyamical variables i classical mechaics, such as positio, liear mometum, eergy ad compoets of agular mometum, so that we may compare our theoretical results with experimetal data. Aother postulate is to be proposed for this purpose. Sice ew cocepts are ivolved, we will cosider it later i this chapter whe these cocepts have bee itroduced. 1.7 Costraits o solutios Every solutio of the Schrödiger wave equatio is ot physically acceptable. The above iterpretatio of Ψ * (r, t) Ψ(r, t) dτ as the probability of fidig the particle i the small volume dτ about r ad at time t imposes certai costraits o the solutio Ψ(r, t) of this secod-order partial differetial equatio. I fact, i order for the solutio to be physically acceptable, it must be well-behaved, i.e. the wave fuctio should be fiite, sigle valued ad cotiuous. Moreover, its first derivatives with respect to space co-ordiates must be cotiuous. This is aalysed below. 1. The fuctio Ψ(r, t) must be fiite for all values of x, y, z. I fact, Ψ(r, t) should be such that it vaishes sufficietly rapidly as ifiity is approached so as to give us o trouble; the fuctio remais square itegrable. This is so because otherwise the probability of fidig the particle i the small regio about a poit where the fuctio diverges will become ifiite which is physically uacceptable.. The fuctio Ψ(r, t) must be sigle-valued, i.e. for each set of the values of the variables it should have oly oe value. This is essetial because otherwise the probability of fidig the particle at a particular poit ad at a certai time will ot be uique; it will have more tha oe value, each value depedig upo the choice of the multivalued fuctio Ψ(r, t) such as ta 1 x. Strictly speakig, accordig to this argumet, it is Ψ * (r, t) Ψ(r, t) which should be sigle-valued. However, successful results for the characteristics of some physical quatities, such as the z-compoet of orbital agular mometum, require that the wave fuctio be sigle-valued. 3. The fuctio Ψ(r, t) ad its first derivatives with respect to space co-ordiates should be cotiuous i all parts of the regio. 1-11

13 Before cosiderig the last characteristic i detail, let us recall a mathematical theorem: a fuctio cotiuous at a poit is ot ecessarily differetiable there but a fuctio differetiable at a poit is ecessarily cotiuous there. Thus if a fuctio f is differetiable at a poit, i.e. if df/dx exists at a poit, the f must be cotiuous there. Similarly, if d f/dx exists, the df/dx must be cotiuous there. To make thigs simple, cosider the time-depedet Schrödiger wave equatio i oedimesio, ħ m x + V Ψ = i ħ Ψ. (1.18) Rearragig the terms, we have Ψ= ħ V Ψ + ħ Ψ m x i. We will cosider the case whe the potetial eergy of a physical system is fiite, whether cotiuous or with a umber of fiite discotiuities, because ifiite eergies do ot occur i ature. The the left-had side of the above equatio is fiite. Cosequetly, both the terms o the right-had side should be fiite. Thus, as Ψ/ x is fiite, i.e. the differetial coefficiet of Ψ/ x exists, the fuctio Ψ/ x must be cotiuous. Moreover, as Ψ/ x exists, i.e. the differetial coefficiet of Ψ exists, the fuctio Ψ must be cotiuous. Hece, the coditio that the wave fuctio ad its first space derivative should be cotiuous is a requiremet imposed by the fiiteess of Ψ ad cosistecy of the Schrödiger wave equatio. The aalysis ca easily be exteded to three dimesios. It ca be metioed that if Ψ is assumed to be cotiuous, the we eed ot assume that it is fiite because every cotiuous fuctio is fiite. Owig to the requiremet that the fuctio Ψ(r, t) must be well-behaved ad its first derivative with respect to the space coordiate should be cotiuous, all the mathematical solutios of the Schrödiger wave equatio are ot physically acceptable. This i tur meas that for a physical system, oly those eergies will be allowed which correspod to physically acceptable wave fuctios. We will fid that by the requiremet of admissible wave fuctios, eergy ad some other physical quatities are quatised. The time-depedet Schrödiger wave equatio is of first order i the timederivative. Therefore, if the wave fuctio Ψ(r, t 0 ) at ay iitial time t 0 is kow, the wave fuctio Ψ(r, t) at ay time t ca be calculated. Problem 1.. Commet o the remark that the assumptios that the wave fuctio be fiite ad sigle-valued at all poits i cofiguratio space may be more rigorous tha ecessary. 1-1

14 Remark If Ψ is fiite ad V is cotiuous or has fiite discotiuities, the the cosistecy of the Schrödiger wave equatio requires that Ψ ad its first space derivative should be cotiuous. It may be emphasised that it is ot a cosequece of the probabilistic iterpretatio of Ψ. This is a characteristic of certai types of secod-order differetial equatios. 1.8 Eigefuctios ad eigevalues Before we proceed further, we will defie a few terms ad illustrate them with the help of examples. We first defie the eigefuctios ad eigevalues of ay operator. Cosider a operator A operatig o a fuctio ϕ such that A ϕ = λϕ, (1.19) that is, it reproduces the fuctio ϕ multiplyig it with a costat λ. The the fuctio ϕ is called a eigefuctio (or characteristic fuctio) of the operator A with eigevalue (or characteristic value) λ or correspodig to the eigevalue λ. The equatio itself is called a eigevalue equatio (or characteristic equatio). Thus, i the eigevalue equatio d (si 3 x) = 9(si 3 x), dx 9 is the eigevalue of the operator d /dx correspodig to the eigefuctio si 3x. A operator has several eigefuctios with the correspodig eigevalues. These eigevalues may be discrete or cotiuous. If the eigevalue spectrum is discrete, the values are writte as, say, λ 1, λ, λ 3,. If the eigevalue spectrum is cotiuous, the values are deoted by λ, λ, λ,. The eigevalue spectrum may be partially discrete ad partially cotiuous. For simplicity, i geeral aalysis, as far as possible, we will be cosiderig oly discrete eigevalues. A set of fuctios ϕ 1, ϕ,, ϕ is said to be liearly idepedet if its liear combiatio a 1 ϕ 1 + a ϕ + + a ϕ caot be made equal to zero for all values of the variables except by takig all a equal to zero. For istace, si x ad cos x are two liearly idepedet fuctios as their liear combiatio, a 1 si x + a cos x, caot be made equal to zero for all values of the variable x except by takig a 1 = a = 0. Suppose that ϕ 1, ϕ,, ϕ are liearly idepedet eigefuctios of a operator A correspodig to the same eigevalue λ. The we may write Aϕ Aϕ Aϕ = λϕ, = λϕ, = λϕ. 1 1 (1.0) 1-13

15 The umber λ is called a -fold degeerate eigevalue of the operator A, correspodig to liearly idepedet eigefuctios ϕ 1, ϕ,, ϕ. These eigefuctios are called -fold degeerate, correspodig to the same eigevalue λ. The umber is called the degree of degeeracy of the eigefuctios. Problem 1.3. Show that 4 is a two-fold degeerate eigevalue of the operator d /dx correspodig to liearly idepedet eigefuctios si x ad cos x. 1.9 The priciple of superpositio We have see that the time-depedet Schrödiger wave equatio ca be writte as HΨ= i ħ Ψ. (1.16 b ) For coveiece, chagig the two sides of the above equatio, we obtai i ħ Ψ = HΨ. (1.1) Let Ψ 1 ad Ψ be two solutios (maybe belogig to differet values of eergy) of this differetial equatio so that ħ Ψ 1 i = HΨ1 (1.) ad i ħ Ψ = HΨ. (1.3) It ca easily be verified that a liear combiatio of these solutios, i.e. a 1 Ψ 1 + a Ψ, is also a solutio of the Schrödiger wave equatio. This is show below: ħ ( Ψ+ Ψ ) = ħ Ψ + ħ Ψ a a a 1 i 1 1 1i ai = ah 1 Ψ 1 + ahψ = Ha ( 1Ψ 1 + aψ). (1.4) I fact, we could directly state that as Schrödiger s secod order time-depedet partial differetial equatio is liear (because the fuctio Ψ ad its derivatives occur oly to the first degree ad ot as higher powers or products), every liear combiatio of its solutios is also a solutio of this differetial equatio. This is kow as the priciple of superpositio ad plays a very importat role i quatum mechaics. It ca be metioed that this is a characteristic of every homogeeous liear differetial equatio. 1-14

16 This is the right momet to poit out that i classical mechaics, kowledge about the positio ad mometum of a particle at ay time describes what is called the state of the particle. If we kow the positio ad mometum of a particle at ay time, the we ca compute its positio ad mometum at ay other time by usig the secod law of motio. This implies that if the iitial state of the system is give, the values of all other variables ca be determied exactly for all times. I quatum mechaics, accordig to Heiseberg s ucertaity priciple, positio ad mometum caot be measured precisely at the same time. Therefore, the state of the particle i the classical sese caot be described. Hece, i quatum mechaics, there is o cocept of the trajectory of a particle as it is determied by a simultaeous precise kowledge of positio ad mometum at every momet; such a trajectory does ot exist whe quatum effects are importat as it is the impossible to keep track of the particle. However, all the accessible iformatio about the particle is cotaied i the wave fuctio Ψ. Naturally, the questio arises: how do we defie the state of a particle i quatum mechaics? I fact, we assume as a fudametal priciple of quatum mechaics that every physically acceptable solutio of the Schrödiger wave equatio represets a state of the system. The timedepedet Schrödiger wave equatio is a liear partial differetial equatio of first degree i the time variable. Therefore, if Ψ is kow at a particular time t 0, it ca be determied at ay time t. Suppose that Ψ 1 ad Ψ represet two states of a system which correspod to defiite eergy values, say E 1 ad E. These states are the called eigestates of the system. Sice a liear combiatio of these solutios, a 1 Ψ 1 + a Ψ, where a 1 ad a are (complex) umbers, is itself a solutio of this differetial equatio, it also represets a state of the system. However, this state does ot correspod to a defiite eergy. Such a state is called a quatum state of the system. It has o aalogue i classical mechaics where each state, like a eigestate i quatum mechaics, correspods to a defiite eergy. Now suppose that the fuctios Ψ 1 (t 0 ) ad Ψ (t 0 ) are two solutios of the Schrödiger wave equatio at time t 0. Let Ψ ( t ) = a Ψ ( t ) + a Ψ( t ) be a liear superpositio of Ψ 1 (t 0 ) ad Ψ (t 0 ) at time t 0. Suppose that the fuctios Ψ 1 (t 0 ) ad Ψ (t 0 ) develop with time ito fuctios Ψ 1 (t) ad Ψ (t). The, by virtue of the fact that the Schrödiger wave equatio is liear i Ψ, we must have Ψ () t = a Ψ () t + a Ψ(), t 1 1 i.e. at ay time t, Ψ(t) is the same liear combiatio of the fuctios Ψ 1 (t) ad Ψ (t). The fact that the first of these equatios etails the secod equatio is a cosequece of the liearity of the time developmet of a state. A overview of various disciplies of physics shows that the priciple of superpositio is oe of the most sigificat attributes of the wave cocept. It is commo to all types of waves. We may trasced a step higher tha this ad take it as a characteristic of every homogeeous liear differetial equatio: a liear 1-15

17 superpositio of the solutios of a homogeeous liear differetial equatio is also a solutio of this differetial equatio. Quatum mechaics which emphasises this formal aspect was developed by Dirac, a British Nobel laureate. It describes the possible states of a system by abstract quatities called state vectors which obey the priciple of superpositio. Ideed, the essece of this priciple as reflected by state vectors is much more attractive tha that predicted by wave fuctios. A quatity which ca be measured is called a observable. I classical physics, observables are represeted by ordiary variables. I quatum mechaics, however, observables are, by assumptio, represeted by operators. It is customary i the literature to use the same letter for the observable, variable ad operator. We will also adopt this covetio Complemetarity The statistical iterpretatio of the wave fuctio ad Heiseberg s priciple of ucertaity are the cocepts with which some emiet physicists of the good old days were fidig it difficult to come to terms. Bohr therefore devoted his full attetio to detailed aalyses of various ew cocepts which were leadig to ew treds i scietific attitudes ad its philosophical cosequeces. His aalysis that the wave ad particle aspects of matter are opposig but complemetary modes of its realisatio has bee amed the priciple of complemetarity. Thisistheessece of his views o the coceptual basis of quatum mechaics but it does ot help i makig calculatios i this field. I fact, classical physics, based upo the kowledge gaied from every day experiece, i.e. from the behaviour of macroscopic objects, tells us that a object i ature ca behave either as a particle or as a wave. It caot exhibit both characteristics. Accordig to the priciple of complemetarity, aalysis at a microscopic level reveals that a object ca behave both as aparticleadawavebutthetwomodescaotberealisedatthesametime. A measuremet which emphasises oe of the wave particle attributes does so at the expese of the other. A experimet desiged to exhibit the particle properties does ot give ay iformatio o the wave aspect ad vice versa. For istace, the observatio of cloud chamber tracks does ot ivolve wave aspect at all while iterferece ad diffractio experimets do ot cotai ay iformatio about the particles demeaour. It is iterestig to ote that all possibly available kowledge about the characteristics of a microscopic etity, say a electro, is cotaied i the wave fuctio. We will express it like this. A microscopic etity uder oe situatio shows those properties which at macroscopic level are attributed to particles ad we say that it is behavig as a particle. I some other situatio, the same etity exhibits characteristics which at macroscopic level are assiged to waves. The we say that the etity is behavig as a wave. Actually, the behaviour is determied by the same wave fuctio ad it is differet, as it must be, uder differet circumstaces. We are surprised simply because it is ot what we were expectig from our experiece i everyday life. But ature does behave that way ad sometimes it produces results eve agaist those expectatios which are based o very careful cosideratios ad aalysis! 1-16

18 1.11 Schrödiger s amplitude equatio We will ow show that the computatios i solvig the Schrödiger wave equatio are simplified if the potetial eergy V of the system does ot deped upo time explicitly: V V(x, y, z) V(r). Let us assume that i this case the wave fuctio Ψ(r, t) represetig the particle ca be expressed as a product of two fuctios, oe depedig oly o space coordiates ad the other depedig o time aloe. If we deote these fuctios, respectively, by ψ(x, y, z) (also writte as ψ(r)) ad ϕ(t), the we may write Differetiatig twice with respect to x, we obtai Ψ ( x, y, z, t) = ψ( x, y, z) ϕ( t). (1.5) Ψ ψ = ϕ. (1.6) x x where Ψ Ψ(x, y, z, t) Ψ(r, t), ψ ψ(x, y, z) ψ(r) ad ϕ ϕ(t). We ca obtai two similar expressios for the secod-order derivatives with respect to space coordiates y ad z. Next, differetiatig (1.5) with respect to t, we obtai Ψ d = ψ ϕ d t. (1.7) Notice that for the fuctio ϕ, we have used the ordiary derivative istead of the partial derivative as this fuctio depeds upo oe variable t oly. Substitutig the expressios from (1.6), etc, ad equatio (1.7) i equatio (1.16a), we obtai or ψ ψ ψ ϕ ħ ϕ + ϕ + d ϕ + Vx (, y, z) ψϕ = iħ ψ m x y z d t, Dividig throughout by ψϕ, we obtai ħ d ( ψ) ϕ + Vψϕ + iħψ ϕ m d t. ϕ ħ 1 1d ( ψ) + V = iħ m ψ ϕ d t. (1.8) The right-had side of this equatio is a fuctio of time oly while, as V depeds explicitly oly o space coordiates, the left-had side depeds upo space coordiates aloe. Therefore, a variatio i space coordiates will ot affect the right-had side while a variatio i time will ot affect the left-had side. This is possible oly if each side is equal to the same costat. We deote this costat by E. The we may write 1dϕ iħ = E ϕ dt (1.9) 1-17

19 ad or ħ 1 ( ψ) + V = E, m ψ ħ + V ψ = Eψ. (1.30) m This is called the Schrödiger amplitude equatio. The operator i brackets i the above equatio is the Hamiltoia operator H of the particle. The amplitude equatio therefore ca be writte as Hψ = E ψ. (1.31) It is called the Hamiltoia form of the Schrödiger amplitude equatio. This is a eigevalue equatio. The costat E is the eigevalue of the Hamiltoia operator H correspodig to the eigefuctio ψ. The Schrödiger amplitude equatio thus takes the form of a eigevalue equatio for the Hamiltoia H ad this simplifies the aalysis of the problem. Let us first solve the differetial equatio (1.9) ivolvig oly time as a idepedet variable. Trasposig iħ to the right-had side of this differetial equatio ad itegratig with respect to time t, we obtai iet l ϕ ( t) = + K. ħ Choosig the iitial coditio that makes K = 0, we obtai or iet l ϕ ( t) =, ħ iet ϕ () t = exp. (1.3) ħ The expressio i brackets o the right-had side shows that E has the dimesios of eergy. We will assume here but will prove later o that E is the total eergy of the system. Let us ext cosider the differetial equatio (1.30) that ca be writte as m ψ + ( E V) ψ = 0. (1.33) ħ This is the time-idepedet Schrödiger wave equatio or the amplitude equatio or the steady-state Schrödiger equatio. This differetial equatio for a system ca be solved oly if the expressio for the potetial eergy of the system is kow. 1-18

20 The solutio of this differetial equatio is deoted by ψ(r). Equatio (1.5) ca ow be writte as iet Ψ (, r t) = ψ ()exp r, ħ where for coveiece we have writte r for x, y, z. If we are iterested i fidig the characteristics of a physical system whose potetial eergy does ot deped explicitly o time, the istead of usig the time-depedet Schrödiger equatio, which is relatively much more difficult to solve, we ca use the time-idepedet Schrödiger equatio which is easier to solve. We obtai the solutio ψ(r) of this differetial equatio ad multiply it by ϕ(t) ( exp( iet/ħ)) so as to obtai Ψ(r, t) which represets the system. The a wave fuctio of the system correspodig to a defiite eergy ca be writte as Thus iet Ψ ( r, t) = ψ( r) ϕ( t) = ψ( r) exp. (1.34) ħ Ψ * ( r, t) Ψ ( r, t) = Ψ * ( r) ψ( r). (1.35) The above aalysis shows that the probability desity for a particle whose potetial eergy does ot deped upo time explicitly is costat i time. For this reaso, a wave fuctio of the form (1.34) is said to represet a statioary state or a eergy eigestate of the particle. The eergy i a statioary state is said to be sharp or welldefied. It ca be said that although the wave fuctio Ψ(r, t) represetig the particle is time-depedet, the probability desity is idepedet of time. Therefore, the system would remai i that state idefiitely. Every measuremet will always give the same value of eergy. This iterpretatio is cosistet with the ucertaity relatio ΔEΔt ħ, as this meas that if the system is i a eigestate with defiite eergy so that ΔE = 0, the ulimited time should be available to make that measuremet. If the eergy spectrum is discrete, the lowest eergy state is called the groud state of the system. The higher eergy states are called excited states of the system. For a statioary state, the ormalisatio coditio, Ψ * (, r t ) Ψ (, r t )dτ = 1, is simplified to Ψ * i Et/ ħ ψ i Et/ ( r)e ( r)e ħ d τ = ψ * ( r) ψ( r) dτ = 1. (1.36) 1-19

21 I oe dimesio, the time-idepedet Schrödiger wave equatio reduces to the followig form: d ψ dx m + ( E V) ψ = 0, (1.37) ħ where ψ ψ(x). It is importat to ote that the amplitude fuctio ψ(x), a solutio of the Schrödiger amplitude equatio ad related to the wave fuctio Ψ(x, t) by iet Ψ ( x, t) = ψ( x) ϕ( t) = ψ( x) exp, (1.33') ħ should also be well-behaved. This is because if ψ(x) is ot fiite, the Ψ(x, t) will also be ot fiite ad if ψ(x) is ot sigle-valued, the Ψ(x, t) will also ot be siglevalued. This is due to the fact that the time fuctio ϕ(t) is always fiite ad does ot deped upo space coordiates. As far as the cotiuity of the fuctio ψ(x) ad its first derivative dψ/dx is cocered, we ote from (1.37) that as ψ(x) is always fiite ad ifiite eergies caot be achieved i ature, the first term o the left-had side of this equatio, i.e. the secod derivative of ψ(x), should also be fiite. Therefore, dψ/dx must be cotiuous. Moreover, as dψ/dx exists, the fuctio ψ(x) should be cotiuous. Thus, the very cosistecy of the Schrödiger amplitude equatio requires that both ψ(x) ad dψ(x)/dx should be cotiuous. Ifiite eergies do ot occur i ature. But let us see what will happe if at some poit the potetial jumps from a fiite to a ifiite value. Equatio (1.37) shows that i this case d ψ/dx will be ifiite. Therefore, dψ/dx may or may ot be cotiuous. That is, the coditio of cotiuity of the space derivative of the wave fuctio at the discotiuity of the potetial caot be imposed. However, as dψ/dx exists because otherwise the differetial equatio will ot hold, the fuctio ψ(x) will be cotiuous. I the solutio of the eigevalue equatio Hψ = Eψ for a physical system, we will fid that the eergy eigevalues very much deped upo the coditios imposed o the solutios to the eigevalue equatio. We take this opportuity to poit out that it is ot ecessary that the itegratio should always be over the etire space extedig from to The orthoormal set of fuctios Cosider a set of fuctios ϕ 1, ϕ, ϕ 3, which are idividually ormalised, i.e. ad mutually orthogoal, i.e. ϕ* ϕ dτ = 1, i = 1,, 3, (1.38) i i ϕ* ϕ dτ = 0, j i, i, j = 1,, 3, (1.39) i i 1-0

22 They are said to form a orthoormal set of fuctios. These two coditios for a orthoormal set ca be expressed as where the Kroecker delta is defied as ϕ* ϕ d τ = δ, (1.40) i j ij δ = 1, for j = I, ij δ = 0, for j i. ij 1.13 The equatio of cotiuity We kow that ay solutio Ψ(r, t) of the time-depedet Schrödiger wave equatio is such that ρ =Ψ * ( r, t ) Ψ( r, t ) (1.41) is iterpreted as the probability desity, i.e. the probability of fidig the particle i uit volume about the poit r ad at time t. Differetiatig with respect to time, we obtai But by virtue of equatio (1.16a), i.e. ρ = ΨΨ = Ψ * * * Ψ ( ) Ψ+Ψ. (1.4) t t iħ Ψ = ħ + V ψ m ad its complex cojugate iħ Ψ * = ħ * + V ψ, m (1.4) yields ρ ħ = ħ Ψ * * i i Ψ+ ( ħψ Ψ i ) t = ħ * * * ψ + ψ Ψ Ψ ħ V ψ + Vψ m m ( ) = ħ * * Ψ ψ Ψ ψ, m 1-1

23 or ( ) ρ ψ ψ = i ħ Ψ * * Ψ t m i = ħ Ψ ψ* * ( Ψ ψ). (1.43) m If we write the (1.43) ca be expressed as or i ħ * * Ψ Ψ Ψ Ψ = j, (1.44) m ( ) ρ + j t = 0, ρ + div j t = 0. (1.45) This is the well-kow equatio of cotiuity. This equatio also arises i electromagetic theory ad expresses the coservatio of charge. If ρ represets the charge desity, i.e. the charge per uit volume, ad j = ρv is the curret desity, i.e. the charge passig uit area ormal to its directio of motio i oe secod, the the equatio of cotiuity expresses the law of coservatio of charge. The charge which is decreasig with time i a bouded volume is accouted for by the charge which is crossig the surface of the bouded volume. I quatum mechaics, ρ is iterpreted as probability desity. Therefore, if j is iterpreted as the probability curret desity, the the equatio of cotiuity guaratees the coservatio of probability. Thus, if the probability of fidig a particle i a certai bouded regio decreases with time, it should correspod to the icrease i probability of fidig it outside that regio. The probability curret desity is also called the probability flux. It is give by ( ) ( ) i = ħ * * Ψ Ψ Ψ Ψ = ħ * j Im Ψ Ψ. (1.46) m m Problem 1.4. Prove that ( ) = ħ * j Im Ψ Ψ. (1.47) m 1-

24 We will ow prove that the total probability of fidig the particle i space is idepedet of time. For oe-dimesioal space, we have * ρ ( x, t) =Ψ( x, t) Ψ( x, t). (1.41') Itegratig with respect to x ad the differetiatig with respect to t, we obtai d dt But, i oe-dimesioal space, we have ħ m x d ρ ( x, t) dx = Ψ * ( x, t) Ψ( x, t) d x. (1.48) dt + Ψ = Ψ ( x, t) V( x) ( x, t) ih. (1.16 a ) Takig the complex cojugate of both sides of equatio (1.16a ), we obtai ħ m x + Ψ = Ψ * * ( x, t) V( x) ( x, t) ih. (1.49) Substitutig these expressios for Ψ * (x, t)/ ad Ψ(x, t)/ i (1.48) ad simplifyig, we obtai d dt ρ = ħ * i Ψ ( x, t) ( x, t) dx Ψ( x, t) m x = ħ * i Ψ ( x, t) Ψ( x, t) m * Ψ( x, t) +Ψ( x, t) dx * Ψ( x, t) +Ψ( x, t) = 0. (1.50) The last result has bee obtaied because the wave fuctio Ψ(x, t) 0asx so that the itegral Ψ * ( x, t) Ψ( x, t) dx may coverge. Equatio (1.50) shows that the total probability is coserved. This also esures that the ormalisatio is preserved: if the wave fuctio is ormalised, it will remai ormalised Complete sets of fuctios A set of fuctios ψ 1 (x), ψ (x), i a variable x is said to form a complete set if a arbitrary square itegrable fuctio ϕ(x) ca be expaded i terms of them: ϕ( x) = a ψ( x), (1.51) i i i 1-3

25 where a i are called expasio coefficiets. The values of a i ca be obtaied as follows. Multiplyig equatio (1.51) byψ j * (x) ad itegratig with respect to x, we obtai ψ* ( x) ϕ( x) d x = a ψ* ( x) ψ( x) d x. (1.5) j i i j i If the fuctios ψ 1 (x), ψ (x), form a orthoormal set, the Therefore, (1.5) reduces to ψ* j ( x) ψi( x) d x = δji. ψ* ( x) ϕ( x) d x = a δ ( x). j i ij i Summig over i, ad fially chagig j to i throughout the equatio, we obtai This equatio determies all the expasio coefficiets The quatum theory of measuremet a = ψ* i i ( x) ϕ( x) d x. (1.53) We will ow aalyse the process of measuremet i quatum mechaics i detail. We will cocetrate o how to compute the physical quatities which are to be compared with the experimet. I fact, wheever we wat to make a accurate measuremet of ay quatity, we measure that quatity a large umber of times ad take the arithmetic mea of the measured values. This is called the average or mea value of the variable. The average value of a variable x is deoted by x: x = x1 + x + + x. (1.54) I order to have a idea about the precisio of the various values we must kow the scatterig or dispersio of these idividual values about their average. The idividual deviatios from the mea are x x, x x,, x x. 1 But the average of these deviatios is equal to ( x x ) + ( x x ) + + ( x x ) = 1 1 x + x + + x x = x x = 0 This shows that whatever the deviatios of x from its mea value, the average of these deviatios is always zero. The average of these deviatios is therefore ot useful as a stadard for measurig dispersio. Its value, beig always zero, caot tell us whether the idividual values are close to or far away from the average. 1-4