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 uc fasusten 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. 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 FinitePotentialwellexample––11Dmodelofnucleus

FinitePotentialwellexample––QuantumQuantumwelllasers

Finite Potential Well

Introduction of Quantum Mechanics : Dr Prince A Ganai Finite Potential Well E > 0, V(x) = 0,|x| > L −L/2 L/2 I II III V(x) = − V0, |x| < L ΔxΔp = ℏ Bound State

ΔE = p2/2m = ℏ/2mL2

Strength of Potential V0/ΔE

Introduction of Quantum Mechanics : Dr Prince A Ganai Finite Potential Well Ψ(x, t) = ϕ(t)ψ(x) E > 0, ϕ(t) = e−iEnt/ℏ −L L −ℏ2 d2ψ(x) + V(x)ψ(x) = Eψ(x) I II III 2m dx2 V(x) = 0 For region I and III

−ℏ2 d2ψ(x) = Eψ(x) 2m dx2

d2ψ(x) 2mE 2mE + ψ(x) = 0 k2 = − dx2 ℏ2 ℏ2 [k] = L−1 Introduction of Quantum Mechanics : Dr Prince A Ganai Finite Potential Well kx −kx ψI(x) = Ae + Be

kx −kx ψIII(x) = Ee + Fe

−L L For wave functions to make sense B=0 and E=0

kx −kx I II III ψI(x) = Ae ψIII(x) = Fe

Again for region II

−ℏ2 d2ψ(x) − V ψ(x) = Eψ(x) 2m dx2 0 2 d ψ(x) 2m 2 2m Now Let = − [V + E]ψ(x) k′ = (V0 + E) dx2 ℏ2 0 ℏ2

2 d ψ(x) 2 = − k′ ψ(x) dx2 ψ (x) = Csink′x + Dcosk′x Introduction of Quantum Mechanics : Dr Prince A Ganai II 236 Chapter 6 The Schrödinger Equation

If either (x) or d dx were not finite or not single valued, the same would be true of (x, t) and d dx. As we will see shortly, the predictions of wave mechanics regarding the results of measurements involve both of those quantities and would thus not necessarily predict finite or definite values for real physical quantities. Such results would not be acceptable since measurable quantities, such as angular momentum Finite Potential Well and position, are never infinite or multiple valued. A final restriction on the form of the wave function (x) is that in order to obey the normalization condition, (x) must approach zero sufficiently fast as x { so that normalization is preserved. For future reference, we may summarize the conditions that the wave function (x) must

meet in order to be acceptable as follows: ψII(x) = Csink′x + Dcosk′x 1. (x) must exist and satisfy the Schrödinger equation. 2. (x) and d dx must be continuous. −kx kx ψIII(x) = Fe 3. (x) and d dx must be finite. ψI(x) = Ae 4. (x) and d dx must be single valued. 5. (x) 0 fast enough as x so that the normalization integral, Equation 6-20, { remains bounded.

1 ψI(−L) = ψII(−L) Questions dψ (−L) dψ (−L) 1. Like the cIlassical w=ave eqIIuation, the Schrödinger equat2ion is linear. Why is this important?dx dx 2. There is no factor i 1 1 2 in Equation 6-18. Does this mean that (x) must be real? 3 ψII(L) = ψIII(L ) 3. Why must the electric field (x, t) be real? Is it possible to find a nonreal wave function that satisfies the classical wave equation? dψII(L) dψIII(L) 4 4. Describe how the d=e Broglie hypothesis enters into the Schrödinger wave equation. dx dx Introduction of Quantum Mechanics : Dr Prince A Ganai5. What would be the effect on the Schrödinger equation of adding a constant rest energy for a particle with mass to the total energy E in the de Broglie relation f E h? 6. Describe in words what is meant by normalization of the wave function.

EXAMPLE 6-1 A Solution to the Schrödinger Equation Show that for a free 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 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 Finite Potential Well

−Lk 5 Ae = − Csink′L + Dcosk′L

−Lk 6 −Ake = − Ck′c osk′L − Dk′s ink′L

Lk 7 Fe = Csink′ + Dcosk′L

Lk Fke = Ck′c osk′L − Dk′s ink′L 8

Even parity ψ(−x) = ψ(x) −Lk Fe = Dcosk′L −Lk −Fke = − LDksink′L

k = k′t ank′L

Introduction of Quantum Mechanics : Dr Prince A Ganai Finite Potential Well

2 2mE 2 2m k = − k′ = (V0 + E) ℏ2 ℏ2

2 2 2mV0 k + k′ = ℏ2 L k′L = ζ ζ0 = 2mV0 ℏ 2 2 2 ζ + (kL) = ζ0

2 2 kL = ζ0 − ζ k = k′t ank′L

kL = k′L tank′L

2 2 ζtankζ = ζ0 − ζ

Introduction of Quantum Mechanics : Dr Prince A Ganai Finite Potential Well

Introduction of Quantum Mechanics : Dr Prince A Ganai 6-3 The Finite Square Well 249 Finite Potential Well Energy V (x) 248 Chapter 6 The Schrödinger Equation

FIGURE 6-12 Wave 3 2 V 3 functions n(x) and probability distributions 2 x for n 1, 2, and 3 n 0 L x 0 L x x for the finite square well. E Compare these with V = V0 Figure 6-4 for the infinite n = 4 2 2 square well, where the wave 2 FIGURE 6-14 Arbitrary well- functions are zero at x 0 L type potential with possible and x L. The wavelengths Infinite square well energy E. Inside the well are slightly longer than the 0 x 0 L x Finite square well [E V(x)], (x) and (x) corresponding ones for the have opposite signs, and the infinite well, so the allowed wave function will oscillate. 2 Outside the well, (x) and energies are somewhat 1 1 n = 3 smaller. (x) have the same sign, and, except for certain values of E, the wave function will not be 0 0 L x L x well behaved.

n = 2 determination of the allowed energy levels in a finite square well can be obtained from a detailed solution of the problem. Figure 6-12 shows the wave functions and the probability distributions for the ground state and for the first two excited states. n = 1 From this figure we see that the wavelengths inside the well are slightly longer than x 0 L the corIntroductionresponding wa ofve lQuantumengths for tMechanicshe infinite w e l:l Drof Princethe sam eA w Ganaiidth, so the corre- sponding energies are slightly less than those of the infinite well, aFIGUREs Figu 6-13re 6 -1 C3om parison of the lowest four energy levels of an infinite square well (broken illustrates. Another feature of the finite-well problem is that there areli noesn)l wy itah tfhionsiet eo f a finite square well (solid lines) of the same width. As the depth of the finite well decreases, it loses energy levels out of the top of the well; however, the n 1 level number of allowed energies, depending on the size of V0. For very small V0 there is remains even as V0 Tipler: 0. Modern Physics 6/e only one allowed energy level; that is, only one bound state can exist. This will be Perm fig.: 613 New fig.: 6-13 quite apparent in the detailed solution in the More section. First Draft: 2011-05-16

Note that, in contrast to the classical case, there is some probability of find2ing the 2 where k 2m E V x now depends on x. The solutions of this equation are particle outside the well, in the regions x L or x 0. In these regnio nlosn, gtehre s itmotpalle sine or cosine functions because the wave number k 2 varies energy is less than the potential energy, so it would seem that the kinetiwci tehn ex,r gbyu tm siunscte and have opposite signs, will always curve toward the axis be negative. Since negative kinetic energy has no meaning in classicaal npdh tyhsei csos,l uitti oinss will oscillate. Outside the well, will curve away from the axis so interesting to speculate about the meaning of this penetration of wave futnhcertieo wn iblle byeo nondl y certain values of E for which solutions exist that approach zero as the well boundary. Does quantum mechanics predict that we could mex aaspuprreo aac hneesg ian-finity. tive kinetic energy? If so, this would be a serious defect in the theory. Fortunately, we are saved by the uncertainty principle. We can understand this qualitatively as follows x (we will consider the region x L only). Since the wave function decreasMorees as e More, with given by Equation 6-34, the probability density 2 e2x becomes ver Iyn most cases the solution of finite-well problems involves transcen- dental equations and is very difficult. For some finite potentials, small in a distance of the order of x 1. If we consider (x) to be negligible however, graphical solutions are relatively simple and provide both beyond x L 1, we can say that finding the particle in the region x L is 1 insights and numerical results. As an example, we have included roughly equivalent to localizing it in a region x . Such a measurement introth-e Graphical Solution of the Finite Square Well on the home page: duces an uncertainty in momentum of the order of p h x h and a minimuwmw w.whfreeman.com/tiplermodernphysics6e. See also Equations 2 2 2 kinetic energy of the order of p 2m h 2m V0 E. This kinetic energ6y-3 6 through 6-43 and Figure 6-15 here. is just enough to prevent us from measuring a negative kinetic energy! The penetration of the wave function into a classically forbidden region does have important conse- quences in tunneling or barrier penetration, which we will discuss in Section 6-6. Much of our discussion of the finite-well problem applies to any problem in which E V(x) in some region and E V(x) outside that region. Consider, for exam- ple, the potential energy V(x) shown in Figure 6-14. Inside the well, the Schrödinger TIPLER_06_229-276hr.indd 249 8/22/11 11:57 AM equation is of the form x k2 x 6-35

TIPLER_06_229-276hr.indd 248 8/22/11 11:57 AM Curiosity Kills the Cat

Lecture 04 Concluded

Introduction of Quantum Mechanics : Dr Prince A Ganai