Lecture 03 Infinite Square Well
Introduction of Quantum Mechanics : Dr Prince A Ganai CHAPTER 6
The Schrödinger Equation
he success of the de Broglie relations in predicting the diffraction of electrons and 6-1 The Schrödinger Tother particles, and the realization that classical standing waves lead to a discrete Equation in One set of frequencies, prompted a search for a wave theory of electrons analogous to the Dimension 230 wave theory of light. In this electron wave theory, classical mechanics should appear 6-2 The Infinite as the short-wavelength limit, just as geometric optics is the short-wavelength limit of Square Well 237 the wave theory of light. The genesis of the correct theory went something like this, 6-3 The Finite accoWaverding tequationo Felix B forloc hmoving,1 who wparticlesas present at the time. Square Well 246 6-4 Expectation . . . in one of the next colloquia [early in 1926], Schrödinger gave a beauti- Values and fully clear account of how de Broglie associated a wave with a particle Operators 250 and how he [i.e., de Broglie] could obtain the quantization rules . . . by 6-5 The Simple demanding that an integer number of waves should be fitted along a Harmonic stationary orbit. When he had finished Debye2 casually remarked that he Oscillator 253 thought this way of talking was rather childish . . . [that to] deal properly 6-6 Reflection and with waves, one had to have a wave equation. Transmission of Waves 258 Toward the end of 1926, Erwin Schrödinger3 published his now-famous wave equation, which governs the propagation of matter waves, including those of elec- trons. A few months earlier, Werner Heisenberg had published a seemingly different theorIntroductiony to ex pofl Quantumain ato Mechanicsmic ph :e Drn oPrincemen Aa Ganai. In the Heisenberg theory, only measurable quantities appear. Dynamical quantities such as energy, position, and momentum are represented by matrices, the diagonal elements of which are the possible results of measurement. Though the Schrödinger and Heisenberg theories appear to be differ- ent, it was eventually shown by Schrödinger himself that they were equivalent, in that each could be derived from the other. The resulting theory, now called wave mechan- ics or quantum mechanics, has been amazingly successful. Though its principles may seem strange to us whose experiences are limited to the macroscopic world and though the mathematics required to solve even the simplest problem is quite involved, there seems to be no alternative to describe correctly the experimental results in atomic and nuclear physics. In this book we will confine our study to the Schrödinger theory because it is easier to learn and is a little less abstract than the Heisenberg theory. We will begin by restricting our discussion to problems with a single particle moving in one space dimension.
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232 Chapter 6 The Schrödinger EquationErwin Schrödinger. [Courtesy of the Niels Bohr Library, American Institute of Physics.] The Schrödinger Equation We are now ready to postulate the Schrödinger equation for a particle of mass m. In one dimension, it has the form then use this relation to work backward and see how the wave equation fo2r electrons 2 must differ from Equation 6-1. The total energy (nonrelativistic) of a particle of x, t x, t mass m is p2 V x, t x, t i 6-6 E V 6-4 2m 2 where V is the potential energy. Substituting the de Broglie relations 2in Emquation 6-4, x t we obtain 2k2 V 6-5 2m WThise diff ersw from iEqlualtio n n6-2 foor a wphoton bsecahuse iot conwtains t het pohtentaial etne rgyt Vh is equation is satisfied by a harmonic wave function in the and because the angular frequency does not vary linearly with k. Note that we get a factor of when we differentiate a harmonic wave function with respect to time and a spfacteor ocf k wihaen wle difcfereantiaste weith resopectf to posaitio n. Wfer exepecet, th erepfore,a thart ticle, one on which no net force acts, so that the potential the wave equation that applies to electrons will relate the first time derivative to the second space derivative and will also involve the potential energy of the electron. Finally, we require that the wave equation for electrons will be a differential enequaetionr thgat isy line ar in sthe wacve founctnion s (xt, t)a. Thnis etnsu,re s tVhat, i(f x(x,, t) atnd) V . First note that a function of the form cos(kx t) 1 0 2(x, t) are both solutions of the wave equation for the same potential energy, then any arbitrary linear combination of these solutions is also a solution—that is, (x, t) dao1 1e(x, t)s a2n 2(xo, t) its a ssoluationt, wiiths a1f anyd a2 betinhg aribitsrary ceonstqantsu. Sucah at ion because differentiation with respect to time changes the combination is called linear because both 1(x, t) and 2(x, t) appear only to the first power. Linearity guarantees that the wave functions will add together to produce con- structive and destructive interference, which we have seen to be a characteristic of comattser wiavnes aes we llt aso all o thaer w avse pihennomeena . Nbote uin patrti cutlarh thate (1) thse lien- cond derivative with respect to x gives back a cosine. Simi- earity requirement means that every term in the wave equation must be linear in (x, t) laanrd (2 ) trhate anIntroductionya dersivaotive nof i(xn, t) isg line aofr rin u Quantum(x,l t).e4 s ou Mechanicst the fo :r Drm Prince sin( Ak Ganaix t). However, the exponential form of the harmonic wave function does satisfy the equation. Let TIPLER_06_229-276hr.indd 231 8/22/11 11:57 AM x, t Aei kx t A c os k x t i sin kx t 6-7 where A is a constant. Then i A ei kx t i t and 2 2 i kx t 2 2 ik A e k x Substituting these derivatives into the Schrödinger equation with V(x, t) V0 gives 2 2 k V0 i i 2m or 2k2 V0 2m which is Equation 6-5. An important difference between the Schrödinger equation and the classical wave equation is the explicit appearance5 of the imaginary number i 1 1 2. The wave functions that satisfy the Schrödinger equation are not necessarily real, as we see from the case of the free-particle wave function of Equation 6-7. E vide ntly the wave function (x, t) that solves the Schrödinger equation is not a directly measur- able function like the classical wave function y(x, t) since measurements always yield real numbers. However, as we discussed in Section 5-4, the probability of finding the electron in some region dx is certainly measurable, just as is the probability that a flipped coin will turn up heads. The probability P(x) dx that the electron will be found in the volume dx was defined by Equation 5-23 to be equal to 2dx. This probabilistic interpretation of was developed by Max Born and was recognized, over the early and formidable objections of both Schrödinger and Einstein, as the appropriate way of relating solutions of the Schrödinger equation to the results of physical measure- ments. The probability that an electron is in the region dx, a real number, can be mea- sured by counting the fraction of time it is found there in a very large number of identical trials. In recognition of the complex nature of (x, t), we must modify
TIPLER_06_229-276hr.indd 232 8/22/11 11:57 AM 6-1 The Schrödinger Equation in One Dimension 235
which can also be written as Ct Ct Ct Ct t e iCt cos i sin cos 2 i sin 2 6-17b h h Th u s, we see that (t), whi ch descri bes t he time v ariatio n of (x, t), is an oscillatory 6-1 The Schrödinger Equation in One Dimension 235 function with frequency f C h. However, according to the de Broglie relation (Equation 5-1), the frequency of the wave represented by (x, t) is f E h; there- whifcohr eca, nw ael scoo bnec lwurdiett etnh aats the separa tion constant C E, the total energy of the particle, and we have Ct Ct Ct Ct t e iCt cos i sin cos 2 i sin 2 6-17b h h t e iEt 6-17c Th u s, we see that (t), whi ch descri bes t he time v ariatio n of (x, t), is an oscillatory funcftoiro na llw siothlu ftrieoqnuse tnoc yE qfuatioCn h6.- 6H ionwvoevlveirn, ga ctcimored-iinngd etop etnhde ednet pBortoegnltiiea lrse.l aEtiqouna tion 6-14 236 Chapter 6 The Schrödinger Equation (Eqtuhaetino nb e5c-o1)m, ethse, ofrne qmuuenltcipy loicfa tthioe nw bayv e r(exp)r,esented by (x, t) is f E h; there- fore, we conclude that the separa tion constant C E, the total energy of the particle, and we have 2 2 If eithe r (dx) orx d dx were not finite or not single valued, the same would be 2 V x x E x 6-18 true of (x,2 tm) andd xd dx i.E tA s we will see shortly, the predictions of wave mechanics t e 6-17c regarding the results of m ea surements involve both of those quantities and would thus SeparationEquation 6- 1of8 ithes re fTimeerred tando as tSpacehe time Dependencies-in dep en de nt Sc hofrö din ger equation. for all solutions ntoo Et qnuaetcioens s6a-r6i linyv opl rvein dgi ct itm fei-ninitdee poenr dednetf pinotieten tiψvala(sxl. u,Eetq)su aftioorn 6re-1a4l physical quantities. Such then becoTmhee st,i mone m-inudlteippleicnadtieonnt bSyc h r(öx)d,inger equation in one dimension is an ordinary dif- ferential equreastiuolnts iwn oounled vnaorita bbele a cx caenpdt aibs leth seirnecfoer me meauscuhr aebalsei eqru taon thiatinedsl,e s uthcahn atsh ea ngular momentum 2 general forman odf pEoq sui ta2itoidonn , a6xr-e6 . nTehvee rn oinrmfinaliitzea toior nm cuolntidpitlieo nv aolfu eEdq.u Aati ofinn a6l- 9r ecsatnri cbtei on on the form of V x x E x 6-18 expressed inth tee rwmasv2 omef f uthndecx t2imone - in(dxe)p iesn tdheantt i n (ox)r,d esirn tcoe othbee yti mthee dneopremndaelinzcaet ioofn tchoe ndition, (x) must absolute squaaprpe roofa tchhe wzearvo e fs uunfcftiicoine nctalnyc eflass. tW aes hxave so that normalization is preserved. For Equation 6-18 is referred to as the time-in dep en de nt Schrö di n ger{ equation. future reference, we may su imEt m arize tihEte conditions that the wave function (x) must The time-ind ep endxe, tnt Schxr,ö tdin ger e quaxtioen in o ne xdimeension is an oxrdi narxy d if- 6-19 ferential equatiomn eine to inne ovradriearb lteo x b aen da cics etphetarebfloer ea ms ufochll oeawsis e:r to handle than the genaernadl Efoqruma toiof nE q6u-9a t itohne n6 -b6 e.c Tohm ee nsorma liz ation cond iti on of Equation 6 -9 can be expressed in term1s .o f t(hxe) tmimues-tin edxeipsetn ad nendt s a(txi)s, fsyi ntchee tShec htirmöed idnegpeenr deeqnucea toiof nth.e x x dx 1 6-20 absolute square of2 t.h e w(xav) ea fnudn cdti on dcaxn cmeluss. tW bee h caoventinu ous. iEt iEt x3, .t (xx), tand d xdxe mus t b xe fe ini te. x x 6-19 andConditions Equation 6-9 t4h.e n b( exforc)o am ne dAcceptables d d x must b e s Waveingle val uFunctions ed . The form of t5h.e w(axv)e fun0c tfiaosnt en(xo)u tghha ta ssa xtis fies E q usaot itohna t6 t-h1e8 ndoerpmenadlisz oanti othne i nfotermgr al, Equation 6-20, x x dx 1 { 6-20 of the potential reenmeragiyn sf u b no cutinodn eVd(.x). In the next few sections we will study some Introductionsimple bu oft iQuantummporta nMechanicst problem : sDr i nPrince whic Ah Ganai V(x) is specified. Our example potentials will Conditionsbe approxima tforions tAcceptableo real physical po tWaveentials, sFunctionsimplified to make calculations easier. In some cases, the slope of the potential energy may be discontinuous, for example, The form of the wQuestionsave function (x) that satisfies Equation 6-18 depends on the form of tVhe( xp)o mtenatyia lh aevner goyn efu fnocrtimon inV (ox)n. eI nr etgheio nne xotf fsepwa csee ctainodns awnoe twheilrl fsoturdmy isno maen adjacent region. (This is a useful mathematical approximation to real situations in which V(x) simple but importa1n.t pLriokbele tmhse i nc lwashsicihc aVl( xw) aisv sep ecqiufiaedti.o Onu, rt ehxea mScphler öpodtienngtiealrs ewqiulla tion is linear. Why is this be avpaprrioexs imraaptiiodnlys toov reera l ap hsymsiaclal l rpeogtieonnti aolsf, sspimacpeli,f iseudc tho masa kaet ctahlec usluatrifoancse ebasoiuern. dary of a In smometea lc.a)s Tesh, et hper oscloeimpdeup rooef rittnha ens upt?cohte nctaisael se nise rtgoy s omlavye bthee d Siscchornötidniunoguesr, efqoru aetxiaomn psleep, arately in V(x)e amchay r ehgaivoen oon2fe .s pfToarhcme raienn idos ntnheo ern ef gariecoqtnou rior ife st hpaact et hae1n dso 1alun2 otiitnohn eEsr qjfoouiranmt i soimnn o a6on-t 1hal8dy.j aaDcte otnhete s pthoiisn tm oef an that (x) must regidoins.c (oTnhtisn uisi tay .usefbuel mreaathl?ematical approximation to real situations in which V(x) varies raSpiindclye otvheer par osmbaablli lrietyg ioonf ofifn sdpiancge , as up cahr taics l ea t ctahnen soutr fvaacery b oduisncdoanryti nouf oau sly from metpalo.)i nTth teo p proociend3t,u. rtehW ein h wsyua cvmhe uc fasusten tsch tieiso tenol es oc(ltxvr)ei c tm hfeuie sSltcd hb r eö( dxcio,n tng)te ibrn eeu qorueuaast.li9o? n SI sine ipctae pr atohteselsy iS bicnlh er töod ifningedr a nonreal wave eache qreugaitoionn o ifn svpoalcvee fasun tdnh cetht sieoencn ro etnqhdua idtr ees ratihtvaiast tftiihveees sdtoh2l uet icodlnxass2 sjo icina ls mw, otaohvtehe lf yier qsattu dtaheteir oipvnoa?itnitv eo f (which disciosn tthineu istylo.pe) 4m. uDste aslcsroi bbee h coownt itnhueo dues ;B throatg liise, hthyep ogrtahpehs iso fe n t(exr)s vinertsou tsh xe mScuhsrt öbdei nger wave sSminocoet ht.h (eI np rao sbpaebciliiatyl coafs ef iinnd winhg icah p tahreti cploet ecna tninaol te nvearrgyy d biseccoonmtineus oinusfilyn itfero, mth is restric- point to point, the waevqeu fautniocntio. n (x) must be continuous.9 Since the Schrödinger tion is relaxed. Since no particle can have infinite potential energy, (x) must be zero equation involves 5th. eW sechoantd w doeruivldat ibvee tdh2 e edfxfe2 c t o n ,t thhee fSircsht rdöedriivnagtievre e q u(wathiiocnh of adding a constant rest is thine rselogpioe)n sm wusht earles oV (bxe) cios nitninfiunoiutes.; Tthhaet ni sa, tt hthe eg braopuhn doaf r y( xo)f vseurcshu sa xr emguiostn ,b e may be discontinuous.) energy for a particle with mass to the total energy E in the de Broglie relation smooth. (In a special cfase inE whhi?ch the potential energy becomes infinite, this restric- tion is relaxed. Since no particle can have infinite potential energy, (x) must be zero in regions where V6(. x)D ies sincfriinbitee .i nT hweno radt sth we bhoautn idsa rmy eoaf nsut cbhy a nroegrimona,l iza mtiaoyn b oe f the wave function. discontinuous.)
TIPLER_06_229-276hr.indd 235 EXAMPLE 6-1 A Solution to the Schrödinger Equation Show that for a free 8/22/11 11:57 AM particle of mass m moving in one dimension the function (x) A sin kx B cos kx is a solution to the time-independent Schrödinger equation for any values of the TIPLER_06_229-276hr.indd 235 8/22/11 11:57 AM constants A and B.
SOLUTION A free particle has no net force acting on it, for example, V(x) 0, in which case the kinetic energy equals the total energy. Thus, p k 2mE 1 2. Differentiat- ing (x) gives d kA cos kx kB sin kx dx
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and differentiating again, d2 k2A sin kx k2B cos kx dx 2 k2 A sin kx B cos kx k2 x Substituting into Equation 6- 18, 2 k2 A sin kx B cos kx E A sin kx B cos kx 2m 2k2 x E x 2m and, since 2k2 2mE, we have E x E x and the given (x) is a solution of E qu ation 6- 18 .
6-2 The Infinite Square Well A problem that provides several illustrations of the properties of wave functions and is also one of the easiest problems to solve using the time-independent, one- dimensional Schrödinger equation is that of the infinite-square well, sometimes called the particle in a box. A macroscopic example is a bead free to move on a frictionless wire between two massive stops clamped to the wire. We could also build such a “box” for an electron using electrodes and grids in an evacuated tube as illustrated in Infinite potentialFi gwellure 6-1a. The walls of the box are provided by the increasing potential between the grids G and the electrode C as shown in Figures 6-1b and c. The walls can be
(a ) Electron
– C G G C – Classical Picture
FIGURE 6-1 (a) The electron V V placed between the two sets of electrodes C and grids G (b) Quantum Mechanical Realisation experiences no force in the region between the grids, Potential which are at ground potential. energy However, in the regions between each C and G is a Introduction of Quantum Mechanics : Dr Prince A Ganai C G G C x repelling electric field whose strength depends on the (c) magnitude of V. (b) If V is small, then the electron’s potential energy versus x has Potential energy low, sloping “walls.” (c) If V is large, the “walls” become very high and steep, becoming infinitely high for C G G C x V .
TIPLER_06_229-276hr.indd 237 8/22/11 11:57 AM 6-2 The Infinite Square Well 237 and differentiating again, d2 k2A sin kx k2B cos kx dx 2 k2 A sin kx B cos kx k2 x Substituting into Equation 6- 18, 2 k2 A sin kx B cos kx E A sin kx B cos kx 2m 2k2 x E x 2m and, since 2k2 2mE, we have E x E x and the given (x) is a solution of E qu ation 6- 18 .
6-2 The Infinite Square Well A problem that provides several illustrations of the properties of wave functions and is also one of the easiest problems to solve using the time-independent, one- dimensional Schrödinger equation is that of the infinite-square well, sometimes called the particle in a box. A macroscopic example is a bead free to move on a frictionless wire between two massive stops clamped to the wire. We could also build such a “box” for an electron using electrodes and grids in an evacuated tube as illustrated in Figure 6-1a. The walls of the box are provided by the increasing potential between the grids G and the electrode C as shown in Figures 6-1b and c. The walls can be
(a ) Electron
– C G G C –
Infinite potential well FIGURE 6-1 (a) The electron V V 238 pChapterlaced betw e6e n thThee tw oSchr sets ödinger Equation of electrodes C and grids G (b) experiences no force in the region between the grids, V(x) made arbitrarily high and steep by increasing the potential V and reducing the separa- Potential which are at ground potential. energy However, in the regions tion between each grid-electrode pair. In the limit such a potential energy function between each C and G is a C G G C x looks like that in Figure 6-2, which is a graph of the potential energy of an infinite repelling electric field whsoqsue are well. For this problem the potential energy is of the form strength depends on the (c) magnitude of V. (b) If V is V x 0 0 x L small, then the electron’s 6-21 potential energy versus x has V x x 0 and x L Potential energy low, sloping “walls.” (c) If V is large, the “walls” Although such a potential is clearly artificial, the problem is worth careful study for 0 become veryL highx and steseepv, eral reasons: (1) exact solutions to the Schrödinger equation can be obtained with- becoming infinitely high ofourt the difficult mathematics that usually accompanies its solution for more realistic C G G C x V . potential functions; (2) the problem is closely related to the vibrating-string problem FIGURE 6-2 Infinite square familiar in classical physics; (3) it illustrates many of the important features of all Introduction of Quantum Mechanics : Dr Prince A Ganai well potential energy. For quantum-mechanical problems; and finally, (4) this potential is a relatively good 0 x L, the potential approximation to some real situations, for example, the motion of a free electron energy V(x) is zero. Outside inside a metal. this region, V(x) is infinite. TIPLER_06_229-276hr.indd 237 8/22/11Sin 11:57ce AMthe potential energy is infinite outside the well, the wave function is The particle is confined to the required to be zero there; that is, the particle must be inside the well. (As we proceed region in the well 0 x L. through this and other problems, keep in mind Born’s interpretation: the probability density of the particle’s position is proportional to 2.) We then need only to solve Equation 6-18 for the region inside the well 0 x L, subject to the condition that since the wave function must be continuous, (x) must be zero at x 0 and x L. Such a condition on the wave function at a boundary (here, the discontinuity of the potential energy function) is called a boundary condition. We will see that, mathemat- ically, it is the boundary conditions together with the requirement that (x) 0 as x { that leads to the quantization of energy. A classical example is that of a vibrating string fixed at both ends. In that case the wave function y(x, t) is the dis- placement of the string. If the string is fixed at x 0 and x L, we have the same boundary condition on the vibrating-string wave function: namely, that y(x, t) be zero at x 0 and x L. These boundary conditions lead to discrete allowed frequencies of vibration of the string. It was this quantization of frequencies (which always occurs for standing waves in classical physics), along with de Broglie’s hypothesis, that motivated Schrödinger to look for a wave equation for electrons. The standing-wave condition for waves on a string of length L fixed at both ends is that an integer number of half wavelengths fit into the length L: n L n 1, 2, 3, 6-22 2 c We will see below that the same condition follows from the solution of the Schrödinger equation for a particle in an infinite square well. Since the wavelength is related to the momentum of the particle by the de Broglie relation p h and the total energy of the particle in the well is just the kinetic energy p2 2m (see Figure 6-2), this quantum condition on the wavelength implies that the energy is quant ized and the allowed val- ues are given by p2 h2 h2 h2 E n2 6-23 2m 2m 2 2m 2L n 2 8mL2
Since the energy depends on the integer n, it is c us tomary to label it En. In terms of h 2 , the energy is given by 2 2 2 2 En n n E1 n 1, 2, 3, 6-24 2mL2 c
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which can also be written as Ct Ct Ct Ct t e iCt cos i sin cos 2 i sin 2 6-17b h h Th u s, we see that (t), whi ch descri bes t he time v ariatio n of (x, t), is an oscillatory function with frequency f C h. However, according to the de Broglie relation 238 Chapter 6 The Schr(Eqödingeruation Equation5-1), the frequency of the wave represented by (x, t) is f E h; there- fore, we conclude that the separa tion constant C E, the total energy of the particle, V(x) amnadd ew aer bhitaravreily high and steep by increasing the potential V and reducing the separa-6-2 The Infinite Square Well 239 tion between each grid-electrode pair. In the limit such a potential energy function 238 lo oks likChaptere that in 6F igTheure 6Schr-2, wödingerhich is Equationa g rapht of thee p oitEetn t ial energy of an infinite 6-17c Infinite potential well 10 wherseq uEar1e i ws etlhl.e F loorw theiss tp raolblolewme tdhe e pnoeterngtyial e annerdg yi si sg oifv tehne f boyrm V(x) for all solutions to EqVuamxtiaod ne a60r-b6it rianrvi 0lyo lh vixgin h g a2 ntLid2m stee-eipn bdye pinecnredaesnintg p tohete pnottieanltsia. lE Vq aunadt iroednu 6ci-n1g4 t he separa- then becomes, on multitpiolnic baetitwonee bn ye a c(hx g) ,rid -electrode pair. In the limit 6s-u2c1h a potential energy function V x Ex1 0 and x L 6-25 lo oks like that in F2igmurLe2 6-2, which is a graph of the potential energy of an infinite Although such a potential issq culae2raerl wy2 ealrlt.i fFicoira lt,h tihs ep rporbolbelmem t hise wpootretnht icaalr eefnuelr gstyu disy offo rth e form We nseovwer adl reeraisvoen st:h (1is) erxeascut lsto lf urto iomnsd ttho eth etxi Smcehr-öindidnegpere enqdueantiot nS ccahn rböed oibntgaienre de wquitah-tion (Equa- 0 L x V xV x x 0 E 0 x x L 6-18 tion o6u-t1 t8h)e, dwiffhicicuhlt mfoart hVem(xa)t i cs2 t0mh aits usudaxll2y accompanies its s olution for m ore r ealistic 6-21 potential functions; (2) the problem is c los ely relatedV to xthe vib ra tin g-xstring0 proanbdle m x L FIGURE 6-2 Infinite square Efaqmuialitairo nin 6c-la1s8si cisal rpehfyesrircesd; (t3o) aits i2 ltlhuset2 rtaitmese m-iann dye op fe tnh de e inmtp Sorcthanrtö f deiant ugreesr oefq aulal tion. Althoug h sudch a pxotential is clearly artificial, the problem is worth careful study for well potential energy. For quantum-mechanical problems; a nd f inally, (4) thisE p ot exntial is a relatively good The time-independseenvet rSalc2 rhmeraösodnidsn:x g(21e)r eexqaucta stoiolunti oinns o ton eth de iSmchernösdiiongne ri se qauna toiornd icnaanr bye doibft-ained with- 0 x L, the potential a0pproximation to Lsomx e real situations, for ex ample, the motion of a free electron energy V(x) is zero. Outside ferential equation in onoeut vthaer idaibffliecu xlt manadth eims atthicesr ethfaotr ues umalulyc hac ceoamsipearn iteos hitsa nsodlluet iothn afnor tmhoer e realistic or inside a metal. this region, V(x) is infinite. geneSrianlc ef otrhme poofte nEtiqaul aetnipeoorngtey n 6ti-isa6 li. n fufTinnhcitetei o nnoosu;tr s(mi2d)ae tlhithzee ap trwiooebnllle , cmtoh ines d cwiltaoivsoenl y f uornefcl atEitoeqndu tiaost itohen v6ib-r9a ticnagn-s tbrien g problem The particle is confined to theFIGURE 6-2 Infinite square erxeqpurieresds etod bien z eteror mthesr eo; ft hftaahtm eis i,lt itiahmre iepn-a ircntliadcsleesp imceaunl sdpt ehbnye stii nc ssi;(d xe(3) t,)h esit i wnileclluel.s t(trAhatsee wst eimm parenoy cd eoefdp tehne diemnpcoert aonft ftehaetu res of all region in the well 0 x L. well potential energy. For 2mE atbhrsooulguht eth sisq uanadr eo tohef rt hpreo bwqleuamavxnset, u kfmuee-nmpc eitnci o hmanni ncicda a nBl (copxerr)nlo’sbs. l eiWnmteesr;k p h2rae antvda(txe if)oi nn:a ltlhye, p(r4o)b athbiisli tpy otential 6i-s2 a6 relatively good Introduction0 x of L Quantum, the po tMechanicsential : Dr Prince A Ganai 2 density of the particle’s posaitpiopnro ixsi mpraotpioonrt i oton aslo tmo e re 2a.l) Wsiteu athtieonn sn,e efdo ro nexlya mtop sloel,v teh e motion of a free electron energy V(x) is zero. Outside iEt iEt Equation 6-18 for thxe ,r teg ioinn sixnid,s eitd ae mt heet awl .ell 0x e x L , suxbjeect to the con d itioxn th at x 6-19 wthheisr ree gwioen, hVa(xv) eis sinufbinsitteit. uted the Ssqinucaer eth oe f ptohtee nwtiaalv en nerugmy bise ri nkf,i nsiitnec oeutside the well, the wave function is Thes pinarcteic tlhe eis w coanvfei nfeudn tcot itohne must be continuous, (x) mu st be zero at x 0 and x L. and Equation 6-9 then breeqcuoirmede tso be zero there; that is, the particle must be inside the well. (As we proceed regiSonu cinh tah ec wonedlli t0i o n xo n tLh.e wav e fun ction at a bo u2ndary (he re, the discontinuity of the 2 p 2mE potential energy function) ist hcraolluegdh a tbhoisu nad nadr yo tchoenrd pitrioobnl.e Wmes, w kielle spe ein t hmati,n md aBthoermn’ast -interpretation: the probability k 2 6-27 density of the particle’s posi2tion is proportional to .) We then need only to solve ically, it is the boundary conditions togethe r withx th e r exqudirxem ent1 t hat (x) 0 as 6-20 x that leads to theE qquuaanttiiozna t6io-n1 8o f oern tehreg yr.e gAio cnl aisnssiicdael tehxea mwpellel 0is th axt of La, subject to the condition that { 2 2 and we have written x s ifnocre tthhee wseacvoe nfdun dcteiorinv matuivste b de c ontxinuoduxs, . (Ex)q umautsito bne 6z-e2ro6 aht axs 0 and x L. vibrating string fixed at both ends. In that cas e th e w av e function y(x, t) is the dis- solutpiloancesm oefn tt hoef tfhoer smtring. If tShuec sht rain cgo insd fiitxioend aotn x t h e 0w aavnde fxu nc tLio, nw ea th aa vbeo uthned asarym e(h ere, the discontinuity of the Conditionsboundary condition on fort he vpio bAcceptabletreantitniagl- setnrienrgg yw fauvnec ftiuonnc) t iWaveiosn c:a nllaemd eal yb ,oFunctions t uhnatd ay(rxy, ct)o bned izteioron . We will see that, mathemat- at x 0 and x L. These boicuanldlya,r yit c ios ntdhxiet iobn osu nlAedaa dsr iytno c kdoxins cdrietitoen asl ltoowgeedth ferre qwuietnhc tihees oref quirem6e-n2t8 t ahat (x) 0 as Tvhiber aftoiornm o fo tfh et hsetr iwnga. vIte w fxau sn tchtisi o {qnu a ntht(iaxzta) tliteohanad sto fst oafr tetihqseuf ieqenusca ienEstqi z(uwatahitoiicnoh n oaf l 6we-na1ye8rsg odyc.e cApure scn ldasss iocanl tehxea mfoplrem i s that of a andoffo rt hstea npdiontge nwtaiavel se inne crlgaysvs iifcbuarlan tcpinhtgiyo ssni tcr siV)n ,g (ax fl)oi.xn egIdn w atitht hbe o dtneh e Bexnrtdo gsf.el iIwen’s tshheayctp tociaothsnees it shw,e tehw aawt viel lf usntcutdioyn syo(xm, t)e is the dis- smimotpivlaet ebdu Stc ihmröpdoinrgtearn tto plorpoolkab cfleoemrm ae snw ti anov few tehhqeiu csathtri ioVnng (f.x oI)rf eitslhe esc tpsrtoerncinsi.gfi eisd f.i xOeud ra te xx a m 0p laen dp ox t en Lti,a wlse whailvle the same The standing-wave conbdoituionnd afroy r wcoaxnvdeist ioonn B ao nsct rotihnseg k voxif b lreantigntgh- Lst rfiinxge dw aatv beo fthu necntdios n: nam6e-l2y,8 tbhat y(x, t) be zero bies tahaptp arno ixnitmegaert inounmsb etor orfea hta axll f p wh 0ay vasenilcedna xgl t hps oL fti.e tT ninhtteioas etlh sbe,o lsueinnmdgatphrly Li fc:ioendd ittoio nms alekaed tcoa dlcisuclraetteio anllso weeads ifererq. uencies of where A and B are constants. The boundary condition (x) 0 at x 0 rules out the In some cases, the slopvei b oraft itohne o pf oth tee nsttriianlg .e Int ewrgasy thmisa qyu banet idzaitsicoon notfi nfrueoquuesn, cfieosr (ewxhaicmh pallew,a ys occurs cosin e solution (Equation 6n- 28b ) Lb ec auns e c1o, s2 ,0 3 , 1, so B must equal ze6ro-2. 2The bound- V(x) may have one fofrom2r sitnan doinnge wreagveios nin ocflc assspiacacle pahnysdic sa)n, oatlhonegr wfoitrhm d ei nB raong liaed’sj ahcyepnott hesis, that ary rceognidointi.o (nT hi(sx )i s a 0u saet fxum l omt iLav atghteievdm eSsachtircöadli nagperp troo xloiomk afotiro an w taov er eeaqlu astiitouna ftoior nelse citnro wnsh. ich V(x) We will see below that the same cTohned sittiaonnd fionlglo-wwas vfreo cmo nthdeit sioonlu tfioorn w oaf vthees Socnh arö sdtirningge ro f length L fixed at both ends veaqruiaetsio nr afopri da lpya rtoicvlee rin aa ns imnfianlilt e rseqguiaoren w oelfl . sSpinaccee t,h es wucavhe laensg taht ist hreel atseudr tfoa tchee boundary of a is tha t anL inte gerA n usminb kerL of ha0lf wavelengths fit into the length L6:-29 mmeotmale.n)t uTmh eof p throe cpeardtiucrlee biyn tshue cdhe Bcraosgelsie irse ltaoti osno lpve thhe S acnhdr töhde itnotgael re neeqrguya otifo n separately in Thiset ahcceo hpn adrretiitgciloieon inn i stoh fse aswtpiesallfc iiees djau nsitfd tkh teLh k eiinsn e artinecqy eu niienrrtege ygt hpea2rt t2itmmh e(e ssse oe lFu,i gttiuhoraent s6 i -sj2o,) ,ii ntfh ikss m qisuo aronettsuhtmlryi c atet dt hteo pthoein t of n L n 1, 2, 3, c 6-22 valudeciossn ckdoi tnigotiinvn oeunni t tybhe.y wavelength implies that the energy is qua2ntized and the allowed val- ues anre given by Since the probabiWlitey w oilfl sefien bdeilonwg thaa tp thaer tsiacmlee ccoanndnitiootn fvoallroyw sd firsocmo nthtein soulouutioslny o ff trhoem Sc hrödinger 2 2 2 2 p equatiohn f or a partihcle in an infinithe square we9ll. Since the wavelength is related to the p oint to point, thEe wave function (x) must ben 2 conti nuous. Sinc6-e2 3the Schrödinger mkn om enntu 2m of then pa rtic1l2,e 2 b,y 3th, ec de 2B roglie relation p h a6nd-3 t0he total energy of 2m 2m L 2m 2L n2 2 8mL equation involves the stehceo pnadrt idcleer iinv athteiv wee ldl i s judstx the kin etic, tehnee rfgiyr spt2 d2emri v(saetei vFeig ure (6w-2h)i, cthhi s quantum Since the energy depends on the integer n, it is customary to label it E . In terms of is the slope) must alsoc obnedi tcioonn otinn tuhoe uw sa;v ethle antg tihs ,im tphleie sg trhaapt hth neo efn er(gxy) i sv qeurasnut siz exd manud stht eb aell owed val- If we wrhite2 th, teh ew enaevreg yn ius mgivbeenr bky in terms of the wavelength 2 k, we see that s mooth. (In a special causees ainre wgihviecnh b ythe poten tial energy becomes infinite, this restric- Equation 6-30 is the same as E2q2uation 6-22 for standing waves on a string. The quan- tion is relaxed. Since no2 pa rticle c2 an have inpfi2nite pho2tential enhe2rgy, (x) muh2st be zero tized energy values, oErn e nenrg y ei2ge nnva Elu1 es , anre fo1,u 2n,d 3 ,fcrom Equation 6- 62-72,4 rep2lacing k 2mL E 2 2 n 2 6-23 by kin a rse ggiivoenns bwyh Eerqeu aVt(ixo)n i6s- 3in0f.i nWitee .t hTuhse hna avte th2em bou2nmd ary o2fm su2cLh na region,8 m L may be dniscontinuous.) Since the energy depends on the integer n, it is customary to label it En. In terms of 2 2 2 2 h 2 k, nthe ene2rgy i s given by2 En n n E1 2m 2mL2 2 2 2 2 En n n E1 n 1, 2, 3, 6-24 2mL2 c TIPLER_06_229-276hr.indd 238 which is the same as Equation 6-24. Figure 6-3 shows the energy-level diag8/22/11ram 11:57and AM the potential energy function for the infinite square well potential. The constant A in the wave function of Equation 6-28a is determined by the nor- TIPLER_06_229-276hr.indd 235 8/22/11 11:57 AM malization condition.
L TIPLER_06_229-276hr.indd 238 2 2 n x 8/22/11 11:57 AM n n dx An sin dx 1 6-31 0 L
TIPLER_06_229-276hr.indd 239 8/22/11 11:57 AM 6-2 The Infinite Square Well 239
6-2 The Infinite Square Well 239 10 where E1 is the lowest allowed energy and is given by where E is the lowest allowed energy102 an2d is given by 1 6-2 The Infinite Square Well 239 E1 2 2 6-25 2mL2 E1 2 10 6-25 where E1 is the lowest allowed2 menLergy and is given by We now derive this result from the time-independent Schr2öd2 inger equation (Equa- We now derive this result from the time-independent S c hrödinger equation (Equa- E1 2 6-25 tion 6-18), wthioicn h6 -f1o8r) ,V w(hxi)c h fo 0r Vis(x) 0 is 2mL We now d2erive2 this res2ult from the time-independent Schrödinger equation (Equa- d 2xd x tion 6-1 8), w hic h for V(x) E0 i s Ex x 2m2 d x 2 2m dx 2 2 d x E x or Infinite potential well 2 m dx 2 or or 2mE 2 x2m E 2 (x) k (x) 6-26 2 x (x) k 2 m(Ex) 2 6-26 2 x (x) k (x) 6-26 where we have substituted th e s q uare of the wave nu m2 ber k, since where we have substituted wthheer es wqeu haarvee osufb tshtiteu twedap tvhee2 s qnuuam2rem boEfe trh ek w, savine cneumber k, since k2 6-27 2 p 2 2mE 2 k2 6-27 2 p 2mE 2 2 2 and we have written kx for the seco nd der2iva tive d x dx . Equation 6-26 h6a-s2 7 solutions of the faonrdm we have writte n x for the seco nd derivative d2 x dx 2. Equation 6-26 has and we have written x so fluotrio tnh s eo f stehec of onrmd derivative d2 x dx 2. Equation 6-26 has x A sin kx 6-28 a solutions of the form x A sin kx 6-28 a and and x A sin kx 6-28 a Introduction of Quantum Mechanics : Dr xPrince A BGanai cos kx 6-28b x B cos kx 6-28b and where A and B awreh ceoren Ast aanndts .B T a hree cboo nustn adnatsr.y T choen bdo iutin odna r y (cxo)n di t0io na t x( x ) 0 0r ualte xs ou 0t trhuele s out the cosine solution (Ecoqsuinaeti soonl u6t-io2n8 b(E) qbueactiaouns e6 -c2o8sb )0 b ec a1u,s seo c Bos m0 u s t1 e, qsou aBl mzeursot .e Tquhael zbeoruon. Td-he bound- ary condition (xa)r y c o0n daitt i xo n x L(x )g i v e0sB at cx o s Lk xg ives 6-28b L A sin kL 0 6-29 where A and B are constants. The bo un daLry c onAd siinti koLn (0x ) 0 at x 0 rules o6u-t2 9the This condition is satisfied if kL is any integer times , that is, if k is restricted to the cosine solutioTnh i(sE cqonudaittiionn i6s -s2at8isbf)ie bde icf akuL sies acn oys i0nt e ge 1r , t ismo e sB m, tuhsatt eisq, uifa kl izse rreos.tr Tichteed btoo uthned - values kn given by ary conditionv alu(xe)s k n g 0iv aent xby L gives kn n n 1, 2, 3, 6-30 L A sin kL L0 c 6-29 kn n n 1, 2, 3, 6-30 L c This condition is satisfied Iiff wkeL w irs it ea nt hye winavtee gneurm btiemr ke sin te,r mths aotf tihse, wifa vke liesn grtehs tri ct2e d kt,o w teh ese e that values k giveIfn w bey write the wEqauvaet ionnu m6-b3e0r i sk t hine staemrme sa so Efq tuhaeti owna 6v-e2l2e nfogrt hst a nd ing2 w avke,s wone a s seteri ntgh.a Tt he quan- n Equation 6-30 is ttizheed s eanmereg ya sv Ealquuesa,t oior ne n6e-r2g2y feoigre sntvaanlduiens,g a wrea fvoeusn do nfr oam s tEriqnuga.t iTohn e6 - q2u7a, nre-placing k tized energy valubeys ,k no ar se gnievregny b eyi gEeqnuvaatiloune 6s,- 3a0r.e W foeu tnhdu sf rhoamve Equation 6- 27, replacing k by kn as given by Equation 6-30. We thus have 2k2 2 2 kn n n 1, 2, 3, nc 2 2 6-30 L En n 2 n E1 2 2 2 2m2 2mL kn 2 2 En n n E1 If we write the wave numwbheirc hk i s itnhe tsearmme sa2 sm oEfq utahtieo nw 26ma-2vL42e. lFeignugret h6 -3 sh ow2s the ekn,e rwgye-l esveeel dtiahgarat m and Equation 6-30 is the same athse Epoqteunatitailo enne r6g-y2 f2un fcotiro ns ftoarn tdhei ningfi nwitea vsqeusa roe nw eall sptortienngtia. l.The quan- which is the same asT Eheq ucoantisotann 6t A-2 i4n. tFheig wuarev e6 f-u3n csthioonw osf tEhqeu eantieorng 6y--2le8va eils ddeiategrrmamine adn bdy the nor- tized energy vthael upoetse,n otiar le enneermrggayly ifz ueatniigoctnei oncnovn afdoliruti otehnse., ianrfien iftoe usqnuda rfer owmel lE pqotueantiiaol.n 6- 27, replacing k by k as given byT Eheq cuoantsitoannt 6A- 3in0 t.h We wea vthe ufusn hctaivo ne of Equation 6L-28a is determined by the nor- n n x malization condition. 2 2 n n dx An sin dx 1 6-31 2 2 2 2 0 L k L n 2 2n x En n 2 2n E1 2 m n n dx 2mLA2n sin dx 1 6-31 0 L which is the same as Equation 6-24. Figure 6-3 shows the energy-level diagram and the potential energy function for the infinite square well potential. The constanTIPLER_06_229-276hr.inddt A in the wav e 239 function of Equation 6-28a is determined by the nor- 8/22/11 11:57 AM malization condition.
TIPLER_06_229-276hr.indd 239 8/22/11 11:57 AM L 2 2 n x n n dx An sin dx 1 6-31 0 L
TIPLER_06_229-276hr.indd 239 8/22/11 11:57 AM 6-2 The Infinite Square Well 239 6-2 The Infinite Square Well 239
10 where E1 is the lowest allowed energy an1d0 is given by where E1 is the lowest allowed energy and is given by 2 2 2 2 E 6-25 1 E1 2 6-25 2mL2mL2 We nowW dee nriovwe dtheriisv er etshuisl tr efrsoulmt frthome ttihme et-imined-einpdeenpdeenndte nSt cShcrhördöidnigngeer re eqqu6-2uaattii ooThenn ((EE qInfinitequuaa-- Square Well 239 6-2 The Infinite Square Well 239 tion 6-1t8io),n w6-h1i8c)h, wfohri cVh( xfo)r V( 0x) i s 0 is
10 2 2 where E1 is the lowest allowed energ2yd 2 a1 0nddx is gxiven by where E1 is the lowest allowed e nergy and is given b y 2 E x 2m 2 d2x 2 E x 2m dx 2 2 E 6-25 or 1E 1 2 6-25 or 2m2LmL2 2mE We nWowe n doewri vdee rtihvies threiss urlets ufrlto mfro tmh etx hteim t2iemm- ieEn -idnedpee pne(dxn)edne nt tS Sckch2hr röö(dxd)iin nggeerr eequuaattiioonn ((EEqquua6a---26 2 2 ti on 6ti-o1n8 )6,- 1w8h),i cwhh fiochr Vfo(rx )V ( x) 0 ix s0 is 2 (x) k (x) 6-26 where we have substituted t2he 2s2qu2are of the wave number k, since d d x x where we have substituted the s qu are of the w aEv Ee nxuxmber k, since 2m d2x 2 p 2 2mE 2m dkx2 6-27 p 2 2 or 2 2 m E or k 2 6-27 and we have written x for the se co nd deri vative d2 x dx 2. Equation 6-26 has 2mE 2mE 2 solutions of the form x 2 (x) k2 2 (x) 2 6-26 and we have written x xfo r the s eco2n d d(xe)riv at ivke d ( x) x dx . Equation 6-62-62 6has solution s of the form x A sin kx 6-28 a where we have substit ute d t he square of the wave number k , s ince where wea nhdave substituted the square of the wave number k, since x p 2A sin2 kmxE 6-28 a k2 2 6-27 p x 2Bm cEo2s kx 6-28b and k2 6-27 2 and wweh ehraev Ae wanrdit tBe na r e coxns tfaonrt sth. eT hseec bo o nudn ddearriyv actoinvdei tdio2 n x(x) d x 20. Eatq xu a ti o0n r 6u-le2s6 ohuats t he x B cos kx 6-28b and wsoel uhtaciovsneis nw eo rfsi othtleuent if oo nrm (Exqu aftoior nth 6e- 2s8ebc)o bnedc aduesrei vcaotsi v0 e d 12 , sox B mduxst2 .e qEuqaul azteiroon. T6h-2e 6b ohuansd - ary condition (x) 0 at x L gives sowluhteiroen sA o afn tdh eB f oarem constants. The bo un dary condition (x) 0 at x 0 rules out the x A sin kx 6-28 a cosine s olution (Equation 6-28b) bec auLse cosA 0 s in k1L, so B0 must equal zero. The bo6u-n2d9- x A sin kx 6-28 a ary acnodndition (x) 0 at x L gives This condition is satisfied if kL is an y integer times , that is, if k is restricted to the and x B cos kx 6-28b values kn given by L A sin kL 0 6-29 where A and B are constants. Th e bxoun daBry c coosn kdxit ion (x) 0 at x 0 rules ou6t- t2h8eb This condition is satisfied if kL is an y integer times , that is, if k is restricted to the cosin e solution (Equation 6-28bk)n b ecanu se co s n0 11,, s2o, B3, must equal zero. The bound6--30 wvhaelruee As kan gdi vBe nar be yconstants. The boundaLry condition (x)c 0 at x 0 rules out the ary cnondition (x) 0 at x L g ive s cosine solution (Equation 6-28b) because cos 0 1, so B must equal zero. The bound- Infinite potentialIf we write well the wave number k in terms of the wavelength 2 k, we see that ary co ndition (x) 0 at x L gi ve sL A sin kL 0 6-29 Equation 6-30 is the skamn e asn E q uat ionn 6 -221 f,o 2r ,s t3a,ncding waves on a string. The qu6a-n3-0 tized energy values, or energy Leigenvalues, are found from Equation 6- 27, replacing k This condition is satisfied if kL Lis an y Ain tseigne krL tim es0 , that is, if k is restricted to 6t-h2e9 by k as given by Equation 6-30. We thus have If wvea lwuersi tken n tghivee wn bayve number k in terms of the wavelength 2 k, we see that This condition is satisfied if kL is any integer times , that is, if k is restricted to the Equation 6-30 is the same as Eq ua tion 26-222 for st2an2ding waves on a string. The quan- kn 2 2 values kn given by E n n E kn nn n 1, 2, 32, 1 6-30 tized energy values, or energy eigenLva2lumes, are f2omunLd cfrom Equation 6-27, replacing k by kn as given by Equation 6-30. We thus have which is the same as Equation 6-24. Figure 6-3 shows the energy-level diagram and If we write the wave nukmn b er nk in te rmns of 1th, e2 ,w 3a,vceleng th 2 k, we see t6h-a3t0 the potential energy functionL f2or2 the infinit2e s2quare well potential. Equation 6-30 is the same as Equ atkionn 6-222 fo r standing 2waves on a string. The quan- The constant A inE tnhe wave fun ctinon of Equ ationn 6E-128a is determined by the nor- tized energy values, or energy eig2emnvalues, a2rem fLou2nd from Equation 6- 27, replacing k If we wrimtea ltihzaet iwona vcoen nduitmionb.er k in terms of the wavelength 2 k, we see that Equabtiyo nkn 6 a-s3 g0i vise nth bey s Eaqmuea taios nE 6q-u3a0t.i oWne 6 t-h2u2s fhoarv estanding waves on a string. The quan- which is the same as Equatio n 6-24. Figure 6L-3 shows the energy-level diagram and tized energy values, or energy eigenva2 lu2 es, are f2ou2nd fronm xEquation 6- 27, replacing k kn 2 2 the pote ntial energy function for t hn e ni dnxfin ite2 s qAuna srein we2ll potdexnt ial1. 6-31 by k as given by Equation 6-3E0n. W e thus hanve 2 n EL1 n The constant A in the w av e fun2cmtion of E20 mquLation 6-28a is determined by the nor- malization condition. 2 2 2 2 which is the same as Equation 6k-2n4. Figu2 re 6 -3 show2s the energy-level diagram and Introduction of Quantum Mechanics E : Drn Prince A Ganai n 2 n E1 the potential energy fu n ction fo2r mthe infiniLt2e msqLuare well potential. The constant A in the wave function of E2quat2ionn 6-x28a is determined by the nor- n n dx An sin dx 1 6-31 whichm aisli zthatei osna mcoen daist iEo nq. ua tion 6-24. Figu r0e 6-3 showsL the energy-level diagram and the potential energy function for the infinite Lsquare w ell po tential. TIPLER_06_229-276hr.indd 239 n x 8/22/11 11:57 AM The constant A in the wave function of Eq2uati2on 6-28a is determined by the nor- n n dx An sin dx 1 6-31 malization condition. 0 L L 2 2 n x n n dx An sin dx 1 6-31 0 L TIPLER_06_229-276hr.indd 239 8/22/11 11:57 AM
TIPLER_06_229-276hr.indd 239 8/22/11 11:57 AM
TIPLER_06_229-276hr.indd 239 8/22/11 11:57 AM 240 Chapter 6 The Schrödinger Equation 238 Chapter 6 The Schrödinger Equation Infinite potential well Energy
V(x) made arbitrarily high and steeV p by increasing the potential V and reducing the separa- tion between each grid-electrode pair. In the limit such a potential energy function looks like that in Figure 6-2n, which is a graph of the potential energy of an infinite square well. For this problem the potential energy is of the form 25E 1 5
V x 02 0 x L E n = n E 1 6-21 22 16E 4 E = –––––––––––––––––– 1 V x 1 2 x 0 and x L 2mL Although such a potential is clearly artificial, the problem is worth careful study for 9E 1 3 0 L x several reasons: (1) exact solutions to the Schrödinger equation can be obtained with-
out the d4iEf1ficult mathematics2 that usually accompanies its solution for more realistic
potential Ef1unctions; (2) the p1roblem is closely related to the vibrating-string problem FIGURE 6-2 Infinite square familiar in0 classVi =c 0al physicsL; (3x ) it illustrates many of the important features of all well potential energy. For FIGURE q6-3u aGnratpuh mof e-nmergye vcehrsuas nx fiocr aa pla rtpicrleo inb alne inmfinsite;l y adenepd w elfl.i Tnhae lployten,t ia(l 4) this potential is a relatively good Introduction0 x of L Quantum, the po tMechanicsential : eDrner gPrincey Va(xp) ip sA sr hGanaioowxni wmitha tthie oconlo retdo li ness.o Tmhe see t orfe alalolw esd ivtaulueast fioor tnhes p,a rtifcole’rs toetaxl ample, the motion of a free electron energy En as given by Equation 6-24 form the energy-level diagram for the infinite square well energy V(x) is zero. Outside potential. Classically, a particle can have any value of energy. Quantum mechanically, only inside a me2 ta2l.2 2 this region, V(x) is infinite. the values given by En n 2mL yield well-behaved solutions of the Schrödinger equation. As we Sbeicnomcee m orteh faem ilpiaro wtiethn enteiragyl- leveenl deiarggramys , tihse x aixnisf winilli btee o moitteudt. side the well, the wave function is The particle is confined to the Since three wqauvei fruendcti otno i sb zeero z ine rreogi otnhs eofr sep;ac teh wahetr ei sth,e tphoteen tpiaal ernteircgyl eis minfiu- st be inside the well. (As we proceed region in the well 0 x Ln.ite, the contributions to the integral from to 0 and from L to will both be zero. Thtuhs,r oonulyg thhe intthegirsa l afrnomd 0 otot Lh neere dps troo bbe elveamluaste,d . kIneteegprat iinng, wme oibntadin Born’s interpretation: the probability 1 2 2 An d2 eLnsi tiynd eopefn dtehnet o pf na. rTthiec lneor’msa lpizoeds witaivoe nfu nicsti opn rsoolputoiornts ifoorn tahils to .) We then need only to solve problem, also called eigenfunctions, are then E qu ation 6-18 for the region inside the well 0 x L, subject to the condition that 2 n x n x sin n 1, 2, 3, 6-32 since the wave fLuncLtion must bec continuous, (x) must be zero at x 0 and x L. These wSavuec fhun catio ncso a nre d exitaicotlyn t hoe snam teh aes thwe astavnedi nfgu-wnavcet ifuonnct ioants yan( xb) fooru ndary (here, the discontinuity of the the vibrpatointge-sntrtiniga lp reobnleemr.g Tyh ef wuanvce tfiuoncnti)on iss a ncda tlhlee dpr oaba biloityu ndidstraibruytio cn ondition. We will see that, mathemat- functions Pn(x) are sketched in Figure 6-4 for the lowest energy state n 1, called the ground state, and for the first two excited states, n 2 and n 3. (Since these wave ically, it is the bound2 ary conditions together with the requirement that (x) 0 as functions are real, Pn x n n n.) Notice in Figure 6-4 that the maximum 1 2 amplituxde s of ea ch{ of th et h na(xt) alree athde ss amtoe, th2 eL qu, aasn atriez thaotsieo onf Pon(xf) , e2nLe. rgy. A classical example is that of a Note, too, that both n(x) and Pn(x) extend to { . They just happen to be zero for x 0 anvdi xb ra Lt inn tghi s scatsrei.ng fixed at b ot h ends. In that ca se the wave function y(x, t) is the dis- Thep nluamcbeerm n einn tth eo efq utahtioen s satbroivne gis. c aIlfle dt ha equ asnttruimn ngu mibse rf. iItx sepedc ifaiets x 0 and x L, we have the same both the energy and the wave function. Given any value of n, we can immediately write dobwon uthne dwaavrey fu cncotinond aintdi othne e noenrg yt hofe th ev siybstreamt. iTnhge q-usatnrtiunmg n uwmbaerv ne function: namely, that y(x, t) be zero occurs baetc axus e of 0th ea bnoudnd xar y co nLdi.t ioTnsh e(xs)e b 0o atu xn d 0a arnyd xc o nL.d Witei woinll sse el ead to discrete allowed frequencies of in Section 7-1 that for problems in three dimensions, three quantum numbers arise, one assovciiabterda wtitoh nbo uondfa rtyh ceon dsittironins ogn. e aIcth cwooardsin atthe.is quantization of frequencies (which always occurs for standing waves in classical physics), along with de Broglie’s hypothesis, that motivated Schrödinger to look for a wave equation for electrons. The standing-wave condition for waves on a string of length L fixed at both ends TIPLER_06_229-276hr.indd 240 is that an integer number of half wavelengths fit in8/22/11to t11:57he AM length L: n L n 1, 2, 3, 6-22 2 c We will see below that the same condition follows from the solution of the Schrödinger equation for a particle in an infinite square well. Since the wavelength is related to the momentum of the particle by the de Broglie relation p h and the total energy of the particle in the well is just the kinetic energy p2 2m (see Figure 6-2), this quantum condition on the wavelength implies that the energy is quant ized and the allowed val- ues are given by p2 h2 h2 h2 E n2 6-23 2m 2m 2 2m 2L n 2 8mL2
Since the energy depends on the integer n, it is c us tomary to label it En. In terms of h 2 , the energy is given by 2 2 2 2 En n n E1 n 1, 2, 3, 6-24 2mL2 c
TIPLER_06_229-276hr.indd 238 8/22/11 11:57 AM 240 Chapter 6 The Schrödinger Equation
Energy
V
n
25E 1 5
2 E n = n E 1 22 16E 1 4 E 1 = –––––––––––––––––– 2mL2
9E 1 3
4E 1 2
E 1 1 0 V = 0 L x
FIGURE 6-3 Graph of energy versus x for a particle in an infinitely deep well. The potential energy V(x) is shown with the colored lines. The set of allowed values for the particle’s total
energy En as given by Equation 6-24 form the energy-level diagram for the infinite square well potential. Classically, a particle can have any value of energy. Quantum mechanically, only 2 2 2 2 the values given by En n 2mL yield well-behaved solutions of the Schrödinger equation. As we become more familiar with energy-level diagrams, the x axis will be omitted. Since the wave function is zero in regions of space where the potential energy is infi- nite, the contributions to the integral from to 0 and from L to will both be zero. Thus, only the integral from 0 to L needs to be evaluated. Integrating, we obtain 1 2 An 2 L independent of n. The normalized wave function solutions for this problem, also called eigenfunctions, are then Infinite potential well 2 n x 6-2 The Infinite Square Well 241 n x sin n 1, 2, 3, 6-32 L L c
These wave functions a re exac tly the s2ame as the standing-wave functions yn(FIGUREx) for 6-4 Wave functions 3 3 the vibrating-string problem. The wave functions and the probability distrib un(txio) na nd probability densities 2/L 2/L P x 2 x for n 1, 2, functions Pn(x) are sketched in Figure 6-4 for the lowest energy state n 1, calledn the n ground state, and for the first two excited states, n 2 and n 3. (Since thesea wnda v3 ef or the infinite square 0 /3 2 /3 x 0 2 /3 2 /3 x well potential. Though not functioLns are realL, Pn Lx n n n.) NoLtice inL FigureL 6-4 that the maximum amplitudes of each of the (x) are the same, 2 L 1 2, as are those of P (x),s h2owLn., n(x) 0 for x 0 n n and x L. Note, too, that both (x ) and P (x) ex2tend to . Th ey just happen to be zero for 2 n n 2 { x 0 and x L in this case. 2/L 2/L The number n in the equations above is called a quantum number. It specifies both the energy and the wave function. Given any value of n, we can immediately w0rite downL /2the wave fuLnctioxn and the 0energy of Lth/2e system.L The qxuantum number n occurs because of the boundary conditions (x) 0 at x 0 and x L. We will see in Section 7-1 that for problems in th re2 e dimensions, three quantum numbers arise, 1 1 one associated with boundary conditions on each coordinate. 2/L 2/L
0 L x 0 L x
Introduction of Quantum Mechanics : Dr Prince A Ganai
TIPLER_06_229-276hr.indd 240 8/22/11 11:57 AM Comparison with Classical Results Let us compare our quantum-mechanical solution of this problem with the classical solution. In classical mechanics, if we know the potential energy function V(x), we can find the force from Fx dV dx and thereby obtain the acceleration 2 2 a x d x dt from Newton’s second law. We can then find the position x as a func- tion of time t if we know the initial posi tion and velocity. In this problem there is no force whe n the particle is between the walls of the well because V 0 there. The par- ticle therefore moves with constant speed in the well. Near the edge of the well the potential energy rises discontinuously to infinity—we may describe this as a very large force that acts over a very short distance and turns the particle around at the wall so that it moves away with its initial speed. Any speed, and therefore any energy, is permitted classically. The classical description breaks down because, according to the uncertainty principle, we can never precisely specify both the position and momen- tum (and therefore velocity) at the same time. We can therefore never specify the ini- tial conditions precisely and cannot assign a definite position and momentum to the particle. Of course, for a macroscopic particle moving in a macroscopic box, the
energy is much larger than E1 of Equation 6-25, and the minimum uncertainty of momentum, which is of the order of L, is much less than the momentum and less than experimental uncertainties. Then the difference in energy between adjacent states will be a small fraction of the tot al energy, quantization will be unnoticed, and the classical description will be adequate.11 Let us also compare the classical prediction for the distribution of measure- ments of position with those from our quantum-mechanical solution. Classically, the probability of finding the particle in some region dx is proportional to the time spent in dx, which is dx v, where v is the speed. Since the speed is constant, the classical distribution function is just a constant inside the well. The normalized classical distri- bution function is 1 PC x L
TIPLER_06_229-276hr.indd 241 8/22/11 11:57 AM Infinite potential well 242 Chapter 6 The Schrödinger Equation
2
Quantum-mechanical distribution Classical distribution 1 P = –– 0 x L L
0 L x
FIGURE 6-5 Probability distribution for n 10 for the infinite square well potential. The dashed line is the classical probability density P 1 L, which is equal to the quantum- meIntroductionchanical d ofis Quantumtributio Mechanicsn averag e : dDr o Princever a Ar eGanaigion x containing several oscillations. A physical measurement with resolution x will yield the classic al result if n is so large that 2(x) has many oscillations in x.
In Figure 6-4 we see that for the lowest energy states the quantum distribution function is very different from this. According to Bohr’s correspondence principle, the quantum distributions should approach the classical distribution when n is large, that is, at large energies. For any state n, the quantum distribution has n peaks. The distribution for n 10 is shown in Figure 6-5. For very large n, the peaks are close together, and if there are many peaks in a small distance x, only the average value will be observed. 2 But the average value of sin kn x over one or more cycles is 1/2. Thus
2 2 2 2 1 1 n x av sin knx L av L 2 L which is the same as th e c las si cal dis tribution.
The Complete Wave Function The complete wave function, including its time dependence, is found by multiplying the space part by
e i t e i En t according to Equation 6-17. As mentioned previously, a wave function corresponding to a single energy oscillates with angular frequency n En , but the probability 2 distribution n x, t is independent of time. This is the wave-mechanical justifica- tion for calling such a state a stationary state or eigenstate, as w e have done earlier. It is instructive to l ook at the complete wave function for a particular state n: 2 i nt n x, t sin knx e L If we use the identity eiknx e iknx sin knx 2i we can write this wave function as
1 2 i knx nt i knx nt n x, t e e 2i L
TIPLER_06_229-276hr.indd 242 8/22/11 11:57 AM 242 Chapter 6 The Schrödinger Equation
2
Quantum-mechanical distribution 242 Chapter 6 The Schrödinger Equation Classical distribution 242 Chapter 6 The Schrödinger Equation 1 P = –– 0 x L L 2 2 242 Chapter 6 The Schrödinger Equation0 L x Quantum-mechanical Quantum-mechanical distribution distribution Classical distribution FIGURE 6-52 Probability distribution for n 10 for the infinite square well potential. The Classical distribution 1 dashed line is the classical probability density P 1 L, which is equal to the quantum- P = –– 0 x L Quantum-mechanical L mechanical distribution averaged over a region x condistributiontaining1 several oscillations. A physical measurement with resolution x will yield the classic alP re =su l––t if n0 i s sxo laLrge that 2(x) has 0 ClassicalL distribution L x many oscillations in x. 1 P = –– 0 x L 0 FIGUREL 6-5 PrxobaLbility distribution for n 10 for the infinite square well potential. The In Figure 6-4 we see that for the lowestd aesnheedrg liyn es itsa tthees c tlhases iqcaul apnrotbuambi ldityis dterinbsiutyt iPon fu1nLc,t wiohnic h is equal to the quantum- 0 mechLanical disxtribution averaged over a region x containing several oscillations. A physical FIGURE 6-5 Pisr ovbearyb idliitfyfe driesntrt ifbruotmio nth fios.r An c co 1rd0i fnogr ttmoh eeBa siounhrferimn’sei tnceto wsrqirteuhs arpersoeo nlwudteieolnnl c p eox p twerniilnlt icyaielp.l dlTe t,hh eteh c ela sqsuica anl rteusmult if n is so large that 2(x) has distributions should approach the classical distribution when n is large, that is, at large dashed line iFIGUREs the c l6-5ass iPcraolb pabriolibtya dbiisltirtibyu dtioenn fsoirt yn P 1 0 fo1mr tahLney, i nwofsihcniiltcleah tsi qoiunsas er ieqn w u eaxlll. ptoo tethnteia ql. uTahne tum- mechanical ddiaesstnhreeidbr guliinteieos i.ns tFahoev rec lraaasngsiyec adsl tpoarvoteeb arnb ai,l irtyeh gedie onqsnui tay n Pxtu cmo1 ndtLais,i ntwrihinbicguh t siiseo venqe urhaaal lst oo ntshc epi lqeluaaatknitsou.nm Ts-.h Ae pdhisytrsiibcualt ion for mechanical distribution averaged over a region x containing several oscillations. A physical n 10 is shown in Figure 6-5. For veIn r yF ilgaurrgee 6 -n4, wthee s epee tahkats f oarr eth ec lloosw2ee stto egneetrhgeyr s, taatneds thife quantum distribution function measuremenmt weaistuhre rmeesnot lwuittiho rnes oluxt iwoni l l xy wieillld y itehlde tchlea cslsasisciac al lr reessuulltt i fi fn nis isso slaorg lea trhgaet th2(ax)t h as (x) has there are many peaks in a small distanicse v eryx ,d iofnfelrye ntth fer oamv ethraisg. eA cvcaolrudein wg itoll Bboeh or’bs sceorrvreesdp.o ndence principle, the quantum many oscillamtiaonnys o isnci l latxio.ns in x. 2 But the average value of sin kn x over odniset roibru mtioonrse s chyoucllde sa pisp r1o/a2c.h Tthheu cslassical distribution when n is large, that is, at large energies. For any state n, the quantum distribution has n peaks. The distribution for In Figure 6-4 we see that for the lowest energy2 states the quantum2 di1stribut1ion function In Figure 6-4 we see that for the lo w2esxt energy snta stei ns12 0 tk hisex s qhouwann tiun mF i gduirset r6i-b5.u Ftioor nv efruyn lcartgioe nn , the peaks are close together, and if is very different from this. Accnording taov B ohr’tsh ceorerr easrepn omnadneyn cp ee pakrisn cinip a le s, mthael lq duiasntatunmce x, only the average value will be observed. L av L 2 L is very different from this. According to Bohr’s correspondence princi2ple, the quantum distributions should approach the classical distBriubtu tthioen a vwehreang en v iasl ulaer goef, stihnatk ni sx, oavt elar rognee or more cycles is 1/2. Thus distributionesn wsehrhgoiiceuhsl. diFs o atrph aepn rsyoa samtcaethe a tnsh, tethh ece lcqa lusaassn istcui acmal ldd diissitstr trirbiibubtuiuottniioo hnna. sw n h peena kns. iTsh lea rdgisetr,i btuhtaiotn i sfo, rat large n 10 is shown in Figure 6-5. For very large n, the peaks are close t2ogether, and 2if 2 2 1 1 energies. For any state n, the quantum distribution has n peaks. Tnhex daivst ribut isoinn kfnox r there are many peaks in a small distance x, only the average value will be observeLd. av L 2 L n 10 is shown in Figure 6-5. F2 or very large n, the peaks are close together, and if BTheut the a vCompleteerage value of sin k Waven x over one Functionor more cycles is 1/2. Thus there are many peaks in a small distance x, onwlyhi cthh eis athvee sraamgee a vs athleu cel awssiilcla lb dei sotrbibsuetirovne. d. The complete wa2ve 2function, in2clud2ing its tim2e1 depe1ndence, is found by multiplying But the average value of sin kn xn oxvearv o ne o sri nm konrxe cy cles is 1/2. Thus the space part by L Theav CompleteL 2 L Wave Function 2 i t i 2En 1 t 1 which is the same2 as the classical distrib2uteio n.Th e coem plet e wave function, including its time dependence, is found by multiplying n x av sin knx L the space Lpa 2rt by L according to Equation 6-17. As mentionaevd previously, a wave function corresponding The Complete Wave Function e i t e i En t which is the tsoa ma esi nags lteh een celragsys i ocsacli ldlaitsetsri bwuitthio ann.gular frequency n En , but the pro ba b ility Thdeis ctorimbpulteitoen w a ve fuxn,c ttio n2, iisn cinluddeinpge nitds etinmat ceoc dfo ertpdimeinnegd. et oTn chEeiq,s ui sia stfi oothunne 6d w- 1ba7yv. mAe-sum lmtiepeclnyhtianongni ecda pl rjeuvsiotiufsiclya,- a wave function corresponding the space part by n Infinite potential well to a single energy oscillates with angular frequency n En , but the probability tion for calling such a state a stationary state or eigenstate2, as we have done earlier. It i t idEistritbution x, t is independent of time. This is the wave-mechanical justifica- n n The Completeis instructive Wave to look at tFunctionhe coempl etee wave function for a particular state n: t io n for calling such a state a stationary state or eigenstate, as we have done earlier. It according to Equation 6-17. As mentioned previously, a wave func tion corresponding The complete wave functionThe, inc lcompleteuding its ti mWaveeis dinespt r2efunctionunctdiveen tco elo, oiks afto tuhen cdo mbypl emte uwlativpel fyuinncgtio n for a particular state n: to a single energy oscillates with angular frequency E i, bntut the probability n x, t sinn knx ne the space part by 2 2 distribution n x, t is independent of time. ThisL is the wave-mechanical justifica- i nt n x, t sin knx e tion for calling such a state a stationary state or eigenstate, as w e have done earlier. It L i t i En t isI ifn wstreu cutsivee tthoe l oidoke natt itthye comeplete wavee function for a particular state n: If w e use the identity eiknx e iknx according to Equation 6-17. As mentioned pre2viously, a wave function corresponidknixng iknx i nt e e n x, t si n knx sin knx e L 2i sin knx to a single energy oscillates with angular frequen cy n En , but the probabilit2yi 2 distributionI f w w ee ncuasexn , t whter i dtee nitsthi tiiysn wdeapveen fduennctti oonf taisme. This is the wave-mechanical justifica- we can write this wa ve function as tion for calling such a state a stationary statee ioknrx e igee inknxstate, as we have done earlier. It sin knx 1 2 1 2 i knx nt i knx nt is instructive to look at the complete wave functi2oini fknoxr an tpartic ui lkannxr xs, ttnat te n: e e n x, t e e 2i L 2i L we can write this wave function as 2 i nt n x, t sin k nx e 1 2 L i knx nt i knx nt n x, t e e 2i L If we use the identity Introduction of Quantum Mechanics : Dr Prince A Ganai ik x ik x TIPLER_06_229-276hr.indd 242 e n e n 8/22/11 11:57 AM sin knx TIPLER_06_229-276hr.indd 242 2i 8/22/11 11:57 AM
TIPLER_06_229-276hr.indd 242 we can write this wave function as 8/22/11 11:57 AM
1 2 i knx nt i knx nt n x, t e e 2i L
TIPLER_06_229-276hr.indd 242 8/22/11 11:57 AM 244 Chapter 6 The Schrödinger Equation
x L ––––– 2
2 L –– = sin 2
0 L /4 L /2 3L /4 L x
FIGURE 6-6 The probability density 2(x) versus x for a particle in the ground state of an infinite square well potential. The probability of finding the particle in the region 0 x L 4 is represented by the larger shaded area. The narrow shaded band illustrates the probability of finding the particle within x 0.01L around the point where x 5L 8. The probability that the electron would be found in the region specified is L 4 L 4 2 2 x P1 x dx sin dx 0 0 L L Letting u x L, hence dx L du , and notin g the appropriate change in the limits on the integral, we have that 4 4 2 2 2 u sin 2u 2 1 sin u du 0.091 0 2 4 0 8 4 Thus, if one looked for the pa rticle in a larg e number o f identica l searches, the elec- tron would be found in the region 0 x 0.25 cm about 9 percent of the time. This probability is illustrated by the shaded area on the left side in Figure 6-6. 6-2 The Infinite Square Well 243 (b) Since the region x 0.01L is very small compared with L, we do not need to integrate but can calculate the approximate probability as follows: Just as in the case of the standing-wave function for the vibrating string, we can con- sider this stationary-state wave function to be th2e sup2e rpxosition of a wave traveling to P P x x sin x the right (first term in brackets) and a wave of thLe sameL frequency and amplitude trav- eling to the left (second term in brackets). Substituting x 0.01L and x 5L 8, we obtain 5L 8 2 2 P sin 0.01L EXAMPLE 6-2 An Electron Lin a Wire L An electron moving in a thin metal wire is a reasonable approximation of a2 particle in a one- dimensi onal infinite well. The poten- tial inside the wire is const a nt on 0av.8e5ra4ge b0u.0t 1riLses sha0rp.0ly1 7at each end. Suppose the electron is in a wire 1.0 cm longL. (a) Compute the ground-state energy for the electron. T(bh)i Isf tmhee aelnesc trtohna’ts ethnee rgpyr oisb eaqbuil ailt yto othfe fiavnedriangge ktihnee tiecl eecnterrogny owf itthhei nm o0le.0c1uLle s ainro au nd xgas at5 TL 8 3i0s0 a Kbo, uabt o1u.7t 0p.0e3rc eeVn,t .w Thhaits i si sth iell eulsetcrtartoend’ si nq uFaingtumre n6u-m6,b werh ne?re the area of the shaded narrow band at x 5L 8 is 1.7 percent of the total area under the curve. SOLUTION EXAMPLE1. For que s6-4tion (Ana), tElectronhe ground -insta ant e e Atomic-Sizenergy is given bBoxy E qu (aa)t ioFnin 6d- 2th5e: energy in the ground state of an electron c2 o2nfined to a one-dimensional box of length L 0.1 nm. E1 (This box is roughly the 2smizLe2 of an atom.) (b) Make an energy-level diagram and find the wavelengths of the photons emitted for all transitions beginning at state 2 34 2 n 3 or less and ending at a l owe1r. 0e5n5er gy1 s0tate .J s 2 9.11 10 31 kg 10 2 m 2 34 15 6 .0 3 10 J 3.80 10 eV 2. For question (b), the electron’s quantum number is given by Equation 6-24: Introduction of Quantum Mechanics : Dr Prince A Ganai 2 En n E1 TIPLER_06_229-276hr.indd 244 8/22/11 11:57 AM 3. Solving Equation 6-24 for n and substituting En 0.03 eV and E1 from above yields
En n2 E1 or E 0.03 eV n n 15 E1 3.80 10 eV 6 2.81 10
Remarks: The value of E1 computed above is not only far below the limit of mea- surability, but also smaller than the uncertainty in the energy of an electron con- fined into 1 cm.
EXAMPLE 6-3 Calculating Probabilities Suppose that the electron in Exam- ple 6-2 could be “seen” while in its ground state. (a) What would be the prob- ability of finding it somewhere in the region 0 x L 4? (b) What would be the probability of finding it in a very narrow region x 0.01L wide centered at x 5L 8?
SOLUTION (a) The wave function for the n 1 level, the ground state, is given by Equation 6-32 as 2 x 1 x sin L L
TIPLER_06_229-276hr.indd 243 8/22/11 11:57 AM Curiosity Kills the Cat
Lecture 03 Concluded
Introduction of Quantum Mechanics : Dr Prince A Ganai