Lecture 04 finite Square Well Introduction of Quantum Mechanics : Dr Prince A Ganai 6-1 The Schrödinger Equation in One Dimension 235 which can also be written as Ct Ct Ct Ct t eiCt cos i sin cos 2 i sin 2 6-17b h h Thus, we see that (t), which describes the time variation 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 separation constant C E, the total energy of the particle, and we have Ct Ct Ct Ct t eiCt cos i sin cos 2 i sin 2 6-17b h h t eiEt 6-17c Thus, we see that (t), which describes the time variation 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 separation constant C E, the total energy of the particle, and we have 2 2 If either (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 dxi.E tA s we will see shortly, the predictions of wave mechanics t e 6-17c regarding the results of measurements involve both of those quantities and would thus SeparationEquation 6- 1of8 ithes re fTimeerred tando as tSpacehe time Dependencies-independent Sc hofrö dinger equation. for all solutions ntoo Et qnuaetcioens s6a-r6i linyv oplrveindgi ctitm fei-ninitdee poenr dednetf pinotieten tiψvala(sxl. u,Eetq)su aftioorn 6re-1a4l physical quantities. Such then becoTmhee st,i mone m-inudlteippleicnadtieonnt bSyc hr(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 pEoqsuita2itoidonn, 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 in (ox)r,d esirn tcoe othbee yti mthee dneopremndaelinzcaet ioofn tchoe ndition, (x) must absolute squaaprpe roofa tchhe wzearvoe fsuunfcftiicoine nctalnyc eflass. tW aes hxave so that normalization is preserved. For Equation 6-18 is referred to as the time-independent Schrödinger{ equation. future reference, we may su imEt m arize tihEte conditions that the wave function (x) must The time-independxe, tnt Schxr,ö tdinger equaxtioen in one xdimeension is an oxrdinarxy d if- 6-19 ferential equatiomn eine to inne ovradriearb lteo x b aen da cics etphetarebfloer ea ms ufochll oeawsise:r to handle than the genaernadl Efoqruma toiof nE q6u-9at itohne n6 -b6e.c Tohmee nsormalization condition of Equation 6-9 can be expressed in term1s .o f t(hxe) tmimues-tin edxeipsetn adnendt sa(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 cdtion dcaxn cmeluss. tW bee h caoventinuous. iEt iEt x3, .t (xx), tand d xdxe must bxe feinite. x x 6-19 andConditions Equation 6-9 t4h.e n b( exforc)o amne dAcceptables d dx must be s Waveingle val uFunctionsed. The form of t5h.e w(axv)e fun0c tfiaosnt en(xo)u tghha ta ssa xtisfies Eq usaot itohna t6 t-h1e8 ndoerpmenadlisz oanti othne i nfotermgr al, Equation 6-20, x x dx 1 { 6-20 of the potential reenmeragiyn sfu bnocutinodn 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 sthpaact 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 upcahr taicsl ea t ctahnen soutr fvaacery b oduisncdoanryti nouf oau sly from metpalo.)i nTth teo p proociend3t,u. rtehW ein h wsyua cvmhe u cfasusten tsch tieiso tenol esoc(ltxvr)ei c tm hfeuie sSltcd hb reö( 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 sdtoh2luet icodlnxass2 sjoicina 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 nt(exr)s vinertsou tsh xe mScuhsrt öbdei nger wave sSminocoet ht.h (eI np rao sbpaebciliiatyl coafs ef iinnd winhg icah p tahreti cploet ecnatninaol 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 tdh2e edfxfe2 ct on ,t thhee fSircsht rdöedriivnagtievre eq 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 ry( 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 TIPLER_06_229-276hr.indd 236 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 Equation 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.