26 Chapter2 TheParticle ina Box FIGURE 2.3 Graphs of for the three lowest-energ particle-in-a-box states. 以 FIGURE 2.4 Graphs of for the lowest particle- n-a-box states. 只 The wave function is zero at certain points;these points are called nodes.For each e of he value of the quantum number n,has more noc e existenc may s re I get fror one the ar the m pic par H in Chapter el and other n t be fully and ctly de- scribed in terms of co cepts of clas sics drawn from the macro c world. Figure 2.4 shows that the probability of finding the particle at various places in the box is quite different from the classical result.Classically,a particle of fixed energy in a box bounces back and forth elastically between the two walls.moving at constant speed. Thus it is equally likely to be found at any point in the box.Quantum mechanically,we find a maximum in probability at the center of the box for the lowest energy level.As we go to higher energy levels with more nodes,the maxima and minima of probability come closer together,and the variations in probability along the length of the box ultimately become undetectable.For very high quantum numbers,we approach the classical result of that in the limit of large quant numbers qu over in ntum m mechanic is known as inc hanics ho or mac s(moving at speed much ess thal he pic b opic bodies. f the Planck's constant ic bodies.Since Eq.(2.20),a macroscopic object in a macroscopic box having a macroscopic energy of motion would have a huge value for n.and hence.according to the correspondence prin- ciple,would show classical behavior. We have a whole set of wave functions,each corresponding to a different energy and characterized by the quantum number n.which is a positive integer.Let the subscript i denote a particular wave function with the value nfor its quantum number: t女，=0 elsewhere
26 Chapter 2 | The Particle in a Box The wave function is zero at certain points; these points are called nodes. For each increase of one in the value of the quantum number n, c has one more node. The existence of nodes in c and |c| 2 may seem surprising. Thus, for n = 2, Fig. 2.4 says that there is zero probability of finding the particle in the center of the box at x = l>2. How can the particle get from one side of the box to the other without at any time being found in the center? This apparent paradox arises from trying to understand the motion of microscopic particles using our everyday experience of the motions of macroscopic particles. However, as noted in Chapter 1, electrons and other microscopic “particles” cannot be fully and correctly described in terms of concepts of classical physics drawn from the macroscopic world. Figure 2.4 shows that the probability of finding the particle at various places in the box is quite different from the classical result. Classically, a particle of fixed energy in a box bounces back and forth elastically between the two walls, moving at constant speed. Thus it is equally likely to be found at any point in the box. Quantum mechanically, we find a maximum in probability at the center of the box for the lowest energy level. As we go to higher energy levels with more nodes, the maxima and minima of probability come closer together, and the variations in probability along the length of the box ultimately become undetectable. For very high quantum numbers, we approach the classical result of uniform probability density. This result, that in the limit of large quantum numbers quantum mechanics goes over into classical mechanics, is known as the Bohr correspondence principle. Since Newtonian mechanics holds for macroscopic bodies (moving at speeds much less than the speed of light), we expect nonrelativistic quantum mechanics to give the same answer as classical mechanics for macroscopic bodies. Because of the extremely small size of Planck’s constant, quantization of energy is unobservable for macroscopic bodies. Since the mass of the particle and the length of the box squared appear in the denominator of Eq. (2.20), a macroscopic object in a macroscopic box having a macroscopic energy of motion would have a huge value for n, and hence, according to the correspondence principle, would show classical behavior. We have a whole set of wave functions, each corresponding to a different energy and characterized by the quantum number n, which is a positive integer. Let the subscript i denote a particular wave function with the value ni for its quantum number: ci = a 2 l b 1>2 sina nipx l b, 0 6 x 6 l ci = 0 elsewhere n 5 1 n 5 2 n 5 3 x n 5 1 n 5 2 n 5 3 x 2 Figure 2.3 Graphs of c for the three lowest-energy particle-in-a-box states. Figure 2.4 Graphs of |c| 2 for the lowest particlein-a-box states
2.2 Particle ina One-Dimensional Box 27 Since the wave function has been normalized,we have 娇dk=1证i=j (2.24 Use of Eq.(A.5)in the Appendix gives 7k=,-】.a+m1 =0 1L2(m-)m/1 2.25) 2(n;+n)T/1 since sin m=0for m an integer.We thus have 4=0， i≠j (2.26 When(2.26)holds,the %= (2.27 =8 (2.28) The property (2.27)of the wave functions is called orthonormality.We proved orthonor- alyony for the particle-in-a-box wave functons.we shall prove it more generally in might be puzzled by Eq.(2.26)and wonder why we would want p the one state by the wave func n of a different state.We will late all the 即eh of a system,ar A more rigo ous way to look at the particle in a box with infinite walls is to first treat the particle in a box with a finite jump in potential energy at the walls and then take the limit as the jump in Vbecomes infinite.The results,when the limit is taken.will be the same as(2.20)and (2.23)(see Prob.2.22). We have considered only the stationary states of the particle in a one-dimensional box.For an example of a nonstationary state of this system,see the example near the end of Section 7.8. Some online computer simulations of the particle in a box can be found at www .chem.uci.edu/undergraduate/applets/dwell/dwell.htm(shows the effects on the wave func when a barrier of variable height and width is introduc into the illia s.edu ingemann/Chem153/particl ng the s as the box length is varie stad.com/md/(shows both time- time-dependent states;see Prob
2.2 Particle in a One-Dimensional Box | 27 Since the wave function has been normalized, we have L - c*i cj dx = 1 if i = j (2.24) We now ask for the value of this integral when we use wave functions corresponding to different energy levels: L - c*i cj dx = L l 0 a 2 l b 1>2 sina ni px l b a 2 l b 1>2 sina nj px l b dx, ni nj Use of Eq. (A.5) in the Appendix gives L - c*i cj dx = 2 l c sin3(ni - nj )p4 2(ni - nj )p>l - sin3(ni + nj )p4 2(ni + nj )p>l d = 0 (2.25) since sin mp = 0 for m an integer. We thus have L - c*i cj dx = 0, i j (2.26) When (2.26) holds, the functions ci and cj are said to be orthogonal to each other for i j. We can combine (2.24) and (2.26) by writing L - c*i cj dx = dij (2.27) The symbol dij is called the Kronecker delta (after a mathematician). It equals 1 when the two indexes i and j are equal, and it equals 0 when i and j are unequal: dij K e 0 for i j 1 for i = j (2.28) The property (2.27) of the wave functions is called orthonormality. We proved orthonormality only for the particle-in-a-box wave functions. We shall prove it more generally in Section 7.2. You might be puzzled by Eq. (2.26) and wonder why we would want to multiply the wave function of one state by the wave function of a different state. We will later see (Section 7.3, for example) that it is often helpful to use equations that contain a sum involving all the wave functions of a system, and such equations can lead to integrals like that in (2.26). A more rigorous way to look at the particle in a box with infinite walls is to first treat the particle in a box with a finite jump in potential energy at the walls and then take the limit as the jump in V becomes infinite. The results, when the limit is taken, will be the same as (2.20) and (2.23) (see Prob. 2.22). We have considered only the stationary states of the particle in a one-dimensional box. For an example of a nonstationary state of this system, see the example near the end of Section 7.8. Some online computer simulations of the particle in a box can be found at www .chem.uci.edu/undergraduate/applets/dwell/dwell.htm (shows the effects on the wave functions and energy levels when a barrier of variable height and width is introduced into the middle of the box); web.williams.edu/wp-etc/chemistry/dbingemann/Chem153/particle .html (shows quantization by plotting the solution to the Schrödinger equation as the energy is varied and as the box length is varied); and falstad.com/qm1d/ (shows both timeindependent and time-dependent states; see Prob. 7.47)
28 Chapter 2 The Particle in a Box 2.3 The Free Particle in One Dimension By a free particle.we mean a particle subject to no forces whatever.For a free parti- cle,integration of(1.12)shows that the potential energy remains constant no matter what the value ofx is.Since the choice of the zero level of energy is arbitrary,we may set V(x)=0.The Schrodinger equation(1.19)becomes +-0 (2.29 Equation (229)is the same as Eq.(2.10)(except for the boundary conditions).Therefore. the general solution of (2.29)is (2.13): 2.30 condition might we impose?It seems reasonable to stulate(since Fis less than zero.then this boundary condition will be violated since for E0 we hav dx represents a probability)that will remain finite i(2mE)2=i(-2mlEl)2=i-i-(2mlEl)=-(2mlEl)R and therefore the first term in (2.30)will bec me infinite a paches minus infinity infinity.Thus the boundary condition requires E≥0 2.31 for the free particle.The wave function is oscillatory and is a linear combination of a sine and a cosine term [Eq.(2.15)1.For the free particle,the energy is not quantized;all non- negative energies are allowed.Since we set=0.the energy E is in this case all kinetic energy.If we try to evaluate the arbitrary constants c and c by normalization,we will find that the integral(()dr is infinite.In other words,the free-particle wave function is not normalizable in the usual sense.This is to be expected on physical grounds because there is no reason for the probability of finding the free particle to approach zero as x goes to± The free-particle problem is an unreal situation because we could not actually have a particle that had no interaction with any other particle in the universe. 2.4 Particle in a Rectangular Well Consider a particle in a one-dimensional box with walls of finite height(Fig.2.5a).The potential-energy function is V=Vo forx<0.V=0 for 0sxsl.and V=Vo for >1.There are two cases to examine.depending on whether the particle's energy E is less than or greater than Vo. FIGURE 2.5 (a)Potential energy for a particle in a one-dimens onal rectan ar wel this potential.(c)The first excited-state wave function
28 Chapter 2 | The Particle in a Box 2.3 The Free Particle in One Dimension By a free particle, we mean a particle subject to no forces whatever. For a free particle, integration of (1.12) shows that the potential energy remains constant no matter what the value of x is. Since the choice of the zero level of energy is arbitrary, we may set V1x2 = 0. The Schrödinger equation (1.19) becomes d 2c dx 2 + 2m U2 Ec = 0 (2.29) Equation (2.29) is the same as Eq. (2.10) (except for the boundary conditions). Therefore, the general solution of (2.29) is (2.13): c = c1ei12mE21>2 x>U + c2e-i12mE21>2 x>U (2.30) What boundary condition might we impose? It seems reasonable to postulate (since c*c dx represents a probability) that c will remain finite as x goes to { . If the energy E is less than zero, then this boundary condition will be violated, since for E 6 0 we have i12mE21>2 = i1-2m E 21>2 = i # i # 12m E 21>2 = - 12m E 21>2 and therefore the first term in (2.30) will become infinite as x approaches minus infinity. Similarly, if E is negative, the second term in (2.30) becomes infinite as x approaches plus infinity. Thus the boundary condition requires E Ú 0 (2.31) for the free particle. The wave function is oscillatory and is a linear combination of a sine and a cosine term [Eq. (2.15)]. For the free particle, the energy is not quantized; all nonnegative energies are allowed. Since we set V = 0, the energy E is in this case all kinetic energy. If we try to evaluate the arbitrary constants c1 and c2 by normalization, we will find that the integral 1 - c*1x2c1x2 dx is infinite. In other words, the free-particle wave function is not normalizable in the usual sense. This is to be expected on physical grounds because there is no reason for the probability of finding the free particle to approach zero as x goes to { . The free-particle problem is an unreal situation because we could not actually have a particle that had no interaction with any other particle in the universe. 2.4 Particle in a Rectangular Well Consider a particle in a one-dimensional box with walls of finite height (Fig. 2.5a). The potential-energy function is V = V0 for x 6 0, V = 0 for 0 . x . l, and V = V0 for x 7 l. There are two cases to examine, depending on whether the particle’s energy E is less than or greater than V0. (a) (b) (c) x 5 0 x 5 l I II III V0 Figure 2.5 (a) Potential energy for a particle in a one-dimensional rectangular well. (b) The groundstate wave function for this potential. (c) The first excited-state wave function
2.4 Particle in a Rectangular Well29 （2 his is a =Cexp[(2m/2)P(Vo -E)Px]+Dexp[-(2m/)P(Vo -E)Px] m=Fexp[(2m/2)/P(%-E)P]+Gexp[-(2m/)P(%-E)/P] where C.D.F and G are constants As in Section 2.3,we must prevent from becoming infinite asx-.Since we are assuming E<Vo.the quantity (V-E)2 is a real,positive number,and to keep finite as x- -we must have D =0.Similarly,to keep finite asx+,we must have F=0.Therefore. 4=Cexp[(2m/)P(-E)x]. =Gexp[-(2m/h2)2(-E)x] In region II.V=0.the Schrodinger equation is (2.10)and its solution is(2.15): u A cos[(2m/)PEPx]Bsin[(2m/)EPx] (2.32 nditions.As with the par 10 The tants than th d A ell a e that its de ative ent we ote that if cha ed dis nt en its de Gits instantaneous rate of change)would hecome infinite at r for the particle in a rectangular well,the Schrodinger equation/dx?=(2m/h2)(V-E does not contain anything infinite on the right side.sod2d2 cannot become infinite. [For a more rigorous argument,see D.Branson,Am.J.Plrys.47,1000(1979).]Therefore. di/dx diu/dx at x =0 and diu/dx =dum/dx at x I. From v(0)=n(0).we get C=A.From()=(0).we get (Prob.2.21a) B =(Vo -E)A/E12.From n()=m(1).we get a complicated equation that al- lows Gt o be fo d in term I he constant is found by normalization 4.we get the Takin ()dividing it by ( 细( expressing B in terms of (2E-o)sim[(2mE)/2/h-2(VE-E2)/2cos[(2mE)1/2L/h] 2.33 (Prob.2.30).]Defining the dimensionless c ,since it gives ugh E =0 satisfies (2.33).it is not an allowed en g=E/%andb=(2m%)/21/h 2.34) we divide (2.33)by Vo to get (2e-1)sin(be2)-2(-82)2 cos(be)=0 2.35) Only the particular values of e that satisfy (2 33)gi e a wave function that is continu ous and has a continuous derivative.so the en els ar antized for E<Vo.To find the allowed energy ley can plot the le side of (2.35)versus e for 0<<1 and find the points where the curve crosses the horizontal axis (see also prob d ale)a detailed study (Merzbacher.Section 6.8)shows that the number of allowed energy levels with E<V is N.where N satisfies N-1<b/s N.where b =(2mV)I/h (2.36) For example,if Vo =h2/ml2,then b/=2(22)=2.83,and N 3
2.4 Particle in a Rectangular Well | 29 We first consider E 6 V0. The Schrödinger equation (1.19) in regions I and III is d 2c>dx 2 + 12m>U221E - V02c = 0. This is a linear homogeneous differential equation with constant coefficients, and the auxiliary equation (2.7) is s2 + 12m>U221E - V02 = 0 with roots s = { 12m>U221>21V0 - E21>2 . Therefore, cI = C exp312m>U221>21V0 - E21>2 x4 + D exp3- 12m>U221>21V0 - E21>2 x4 cIII = F exp312m>U221>21V0 - E21>2 x4 + G exp3- 12m>U221>21V0 - E21>2 x4 where C, D, F, and G are constants. As in Section 2.3, we must prevent cI from becoming infinite as x S - . Since we are assuming E 6 V0, the quantity 1V0 - E21>2 is a real, positive number, and to keep cI finite as x S - , we must have D = 0. Similarly, to keep cIII finite as x S + , we must have F = 0. Therefore, cI = C exp312m>U221>21V0 - E21>2 x4, cIII = G exp3- 12m>U221>21V0 - E21>2 x4 In region II, V = 0, the Schrödinger equation is (2.10) and its solution is (2.15): cII = A cos312m>U221>2 E1>2x4 + B sin312m>U221>2E1>2x4 (2.32) To complete the problem, we must apply the boundary conditions. As with the particle in a box with infinite walls, we require the wave function to be continuous at x = 0 and at x = l; so cI102 = cII102 and cII1l2 = cIII1l2. The wave function has four arbitrary constants, so more than these two boundary conditions are needed. As well as requiring c to be continuous, we shall require that its derivative dc>dx be continuous everywhere. To justify this requirement, we note that if dc>dx changed discontinuously at a point, then its derivative (its instantaneous rate of change) d2c>dx 2 would become infinite at that point. However, for the particle in a rectangular well, the Schrödinger equation d2c>dx 2 = 12m>U221V - E2c does not contain anything infinite on the right side, so d2c>dx 2 cannot become infinite. [For a more rigorous argument, see D. Branson, Am. J. Phys., 47, 1000 (1979).] Therefore, dcI>dx = dcII>dx at x = 0 and dcII>dx = dcIII>dx at x = l. From cI102 = cII102, we get C = A. From c I102 = c II102, we get (Prob. 2.21a) B = 1V0 - E21>2 A>E1>2 . From cII1l2 = cIII1l2, we get a complicated equation that allows G to be found in terms of A. The constant A is found by normalization. Taking c II1l2 = c III1l2, dividing it by cII1l2 = cIII1l2, and expressing B in terms of A, we get the following equation for the energy levels (Prob. 2.21b): 12E - V02 sin312mE) 1>2 l>U4 = 21V0E - E221>2 cos312mE21>2 l>U4 (2.33) [Although E = 0 satisfies (2.33), it is not an allowed energy value, since it gives c = 0 (Prob. 2.30).] Defining the dimensionless constants e and b as e K E>V0 and b K 12mV021>2l>U (2.34) we divide (2.33) by V0 to get 12e - 12 sin1be1>22 - 21e - e221>2 cos1be1>22 = 0 (2.35) Only the particular values of E that satisfy (2.33) give a wave function that is continuous and has a continuous derivative, so the energy levels are quantized for E 6 V0. To find the allowed energy levels, we can plot the left side of (2.35) versus e for 0 6 e 6 1 and find the points where the curve crosses the horizontal axis (see also Prob. 4.31c). A detailed study (Merzbacher, Section 6.8) shows that the number of allowed energy levels with E 6 V0 is N, where N satisfies N - 1 6 b>p . N, where b K 12mV021>2l>U (2.36) For example, if V0 = h2>ml 2 , then b>p = 2121>22 = 2.83, and N = 3
30 Chapter 2 The Particle in a Box Figure 2.5 shows for the lowest two energy levels.The wave function is oscillatory inside the box and dies off exponentially outside the box.It turns out that the number of nodes increases by one for each higher level. 化”eew生 oscillate (similar to the free-particle).We no longer have any reason to set D inand in equal to zero,and with onstants available to satisfy the ed to obtain properly behaved reall energiesab vo are am we state in wh alled a bound state.Fora bound s nd stot e exi te region E之aee5n油Eheo article in a nd and states with bound.For the free particle.all states are unbound. For an online simulation of the particle in a well,go to www.falstad.com/qmld and choose Finite Well in the Setup box.You can vary the well width and depth and see the effect on the energy levels and wave functions. 2.5 Tunneling ons I and III.wher e its total en Fis less than its ntial assi andT≥0.whei Tis the kinetic eneroy mean that ecannot he less than vin classical mechanics Consider a particle in a one-dimensional box with walls of finite height and finite thickness (Fig.2.6).Classically,the particle cannot escape from the box unless its energv is greater than the potential-energy barrier V However,a quantum-mechanical treatment (which is omitted)shows that there is a finite probability for a particle of total energy less than v to be found outside the box The term tunneling denotes the penetration of a particle into a classically forbidden region(as in Fig.2.5)or the passage of a particle through a potential-energy barrier whos height exceeds the particle's ene ergy.he ne 10T01 ore,tunneling is m e mass m,【 the funct Pm of away to zer ectrons tunnel quite e The eus involves tunneling of the alpha particles thro ugh the p harri uced hy the tive nuclear forces and the Coulombic rep ulsive force between the daughter nucleus and the alpha particle.The NH,molecule is pyramidal.There is a potential-energy barrier to inversion of the molecule,with the potential-energy maximum occurring at the pla- nar configuration.The hydrogen atoms can tunnel through this barrier,thereby inverting the molecule.In CHCH,there is a barrier to internal rotation,with a potential-energy dimensional box of finite height and thickness
30 Chapter 2 | The Particle in a Box Figure 2.5 shows c for the lowest two energy levels. The wave function is oscillatory inside the box and dies off exponentially outside the box. It turns out that the number of nodes increases by one for each higher level. So far we have considered only states with E 6 V0. For E 7 V0, the quantity 1V0 - E21>2 is imaginary, and instead of dying off to zero as x goes to { , cI and cIII oscillate (similar to the free-particle c). We no longer have any reason to set D in cI and F in cIII equal to zero, and with these additional constants available to satisfy the boundary conditions on c and c , one finds that E need not be restricted to obtain properly behaved wave functions. Therefore, all energies above V0 are allowed. A state in which c S 0 as x S and as x S - is called a bound state. For a bound state, significant probability for finding the particle exists in only a finite region of space. For an unbound state, c does not go to zero as x S { and is not normalizable. For the particle in a rectangular well, states with E 6 V0 are bound and states with E 7 V0 are unbound. For the particle in a box with infinitely high walls, all states are bound. For the free particle, all states are unbound. For an online simulation of the particle in a well, go to www.falstad.com/qm1d and choose Finite Well in the Setup box. You can vary the well width and depth and see the effect on the energy levels and wave functions. 2.5 Tunneling For the particle in a rectangular well (Section 2.4), Fig. 2.5 and the equations for cI and cIII show that for the bound states there is a nonzero probability of finding the particle in regions I and III, where its total energy E is less than its potential energy V = V0. Classically, this behavior is not allowed. The classical equations E = T + V and T Ú 0, where T is the kinetic energy, mean that E cannot be less than V in classical mechanics. Consider a particle in a one-dimensional box with walls of finite height and finite thickness (Fig. 2.6). Classically, the particle cannot escape from the box unless its energy is greater than the potential-energy barrier V0. However, a quantum-mechanical treatment (which is omitted) shows that there is a finite probability for a particle of total energy less than V0 to be found outside the box. The term tunneling denotes the penetration of a particle into a classically forbidden region (as in Fig. 2.5) or the passage of a particle through a potential-energy barrier whose height exceeds the particle’s energy. Since tunneling is a quantum effect, its probability of occurrence is greater the less classical is the behavior of the particle. Therefore, tunneling is most prevalent with particles of small mass. (Note that the greater the mass m, the more rapidly the functions cI and cIII of Section 2.4 die away to zero.) Electrons tunnel quite readily. Hydrogen atoms and ions tunnel more readily than heavier atoms. The emission of alpha particles from a radioactive nucleus involves tunneling of the alpha particles through the potential-energy barrier produced by the short-range attractive nuclear forces and the Coulombic repulsive force between the daughter nucleus and the alpha particle. The NH3 molecule is pyramidal. There is a potential-energy barrier to inversion of the molecule, with the potential-energy maximum occurring at the planar configuration. The hydrogen atoms can tunnel through this barrier, thereby inverting the molecule. In CH3CH3 there is a barrier to internal rotation, with a potential-energy V0 x Figure 2.6 Potential energy for a particle in a onedimensional box of finite height and thickness