bei48482_FM

5.7 Schrödinger’s Equation: Steady-State Form

11. Obtain Schrödinger’s steady-state equation from Eq. (3.5) with
    the help of de Broglie’s relationshiphmby letting y
    and finding ^2 x^2.

5.8 Particle in a Box

12. According to the correspondence principle, quantum theory
    should give the same results as classical physics in the limit of
    large quantum numbers. Show that as n→, the probability of
    finding the trapped particle of Sec. 5.8 between xand xx
    is xLand so is independent of x, which is the classical
    expectation.


13. One of the possible wave functions of a particle in the potential
    well of Fig. 5.17 is sketched there. Explain why the wavelength
    and amplitude of vary as they do.

198 Appendix to Chapter 5

of the wave functions for the n1 and n2 states of a parti-

cle in a box Lwide.

19. Find the probability that a particle in a box Lwide can be
    found between x0 and xLnwhen it is in the nth state.


20. In Sec. 3.7 the standard deviation of a set of Nmeasurements
    of some quantity xwas defined as



N

1



N

i 1

(xix^0 )^2

(a) Show that, in terms of expectation values, this formula can be

written as

(xx )^2 x^2 x 2

(b) If the uncertainty in position of a particle in a box is taken as

the standard deviation, find the uncertainty in the expectation

value x L2 forn 1. (c) What is the limit of xas n

increases?

21. A particle is in a cubic box with infinitely hard walls whose
    edges are Llong (Fig. 5.18). The wave functions of the particle
    are given by

Asin sin sin

Find the value of the normalization constant A.

nx1, 2, 3,...

ny1, 2, 3,...

nz1, 2, 3,...

nzz

L

nyy

L

nxx

L

y

z

L

L

L

Figure 5.18A cubic box.

x



x

∞ ∞

V

L

L

Figure 5.17

14. In Sec. 5.8 a box was considered that extends from x0 to
    xL. Suppose the box instead extends from xx 0 to x
    x 0 L, where x 0 ≠0. Would the expression for the wave func-
    tions of a particle in this box be any different from those in the
    box that extends from x0 to xL? Would the energy levels
    be different?


15. An important property of the eigenfunctions of a system is that
    they are orthogonalto one another, which means that
    
    
    nmdV 0 nm
    Verify this relationship for the eigenfunctions of a particle in a
    one-dimensional box given by Eq. (5.46).


16. A rigid-walled box that extends from Lto Lis divided into
    three sections by rigid interior walls at xand x, where x L.
    Each section contains one particle in its ground state. (a) What
    is the total energy of the system as a function of x?(b) Sketch
    E(x) versusx. (c) At what value of xis E(x) a minimum?


17. As shown in the text, the expectation value x of a particle
    trapped in a box Lwide is L2, which means that its average
    position is the middle of the box. Find the expectation value x^2.


18. As noted in Exercise 8, a linear combination of two wave func-
    tions for the same system is also a valid wave function. Find
    the normalization constant Bfor the combination

B sin sin

^2 x

L

x

L

22. The particle in the box of Exercise 21 is in its ground state of
    nxnynz1. (a) Find the probability that the particle will
    be found in the volume defined by 0 x L4, 0 y
    L4, 0 z L4. (b) Do the same for L2 instead of L4.


23. (a) Find the possible energies of the particle in the box of
    Exercise 21 by substituting its wave function in Schrödinger’s
    equation and solving for E. (Hint: Inside the box U0.)
    (b) Compare the ground-state energy of a particle in a one-
    dimensional box of length Lwith that of a particle in the three-
    dimensional box.

5.10 Tunnel Effect

24. Electrons with energies of 0.400 eV are incident on a barrier
    3.00 eV high and 0.100 nm wide. Find the approximate proba-
    bility for these electrons to penetrate the barrier.

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