10.31.In exercise 10.30a, the wavefunction is not normal-
ized. Normalize the wavefunction and verify that it still satis-
fies the Schrödinger equation. The limits on xare 0 and 2.
How does the expression for the energy eigenvalue differ?
10.8 Particle-in-a-Box
10.32.Verify that equation 10.11 satisfies the Schrödinger
equation, and that equation 10.12 gives the values for energy.
10.33.The electronic spectrum of the molecule butadiene,
CH 2 CH–CHCH 2 , can be approximated using the one-
dimensional particle-in-a-box if one assumes that the conju-
gated double bonds span the entire four-carbon chain. If the
electron absorbing a photon having wavelength 2170 Å is
going from the level n2 to the level n3, what is the
approximate length of the C 4 H 6 molecule? (The experimen-
tal value is about 4.8 Å.)
10.34.How many nodes are there for the one-dimensional
particle-in-a-box in the state described by 5? by 10? by
100? Do not include the sides of the box as nodes.
10.35.Draw (at least roughly) the wavefunctions for the first
five wavefunctions for the particle-in-a-box. Now draw the
probabilities for the same wavefunctions. What similarities are
there between the wavefunctions and their respective proba-
bilities?
10.36.Show that the normalization constants for the general
form of the wavefunction sin (n x/a) are the same and
do not depend on the quantum number n.
10.37.Evaluate the probability that an electron will exist at the
center of the box, approximated as 0.495ato 0.505a, for the
first, second, third, and fourth levels of a particle-in-a-box. What
property of the wavefunction is apparent from your answers?
10.38.Is the uncertainty principle consistent with our de-
scription of the wavefunctions of the 1-D particle-in-a-box?
(Hint:remember that position is not an eigenvalue operator
for the particle-in-a-box wavefunctions.)
10.39.From drawings of the probabilities of particles existing
in high-energy wavefunctions of a 1-D particle-in-a-box (like
those shown in Figure 10.7), show how the correspondence
principle indicates that, for high energies, quantum mechan-
ics agrees with classical mechanics in that the particle is sim-
ply moving back and forth in the box.
10.40.Instead of x0 to a, assume that the limits on the
1-D box were x(a/2) to (a/2). Derive acceptable wave-
functions for this particle-in-a-box. (You may have to consult
an integral table to determine the normalization constant.)
What are the quantized energies for the particle?
10.9 Average Values
10.41.Explain how 2/asin(n x/a) isn’t an eigenfunc-
tion of the position operator.
10.42.Evaluate the average value of position, x , for 2 of
a particle-in-a-box and compare it with the answer obtained
in Example 10.12.
10.43.Evaluate px for 1 of a particle-in-a-box.
10.44.Evaluate E for 1 of a particle-in-a-box and show
that it is exactly the same as the eigenvalue for energy ob-
tained using the Schrödinger equation. Justify this conclusion.
10.45.Assume that for a particle on a ring the operator for
the angular momentum, pˆ, is i(/). What is the eigen-
value for momentum for a particle having (unnormalized)
equal to e^3 i? The integration limits are 0 to 2. What is the
average value of the momentum, p for a particle having this
wavefunction? How are these results justified?
10.46.Mathematically, the uncertainty Ain some observ-
able Ais given by A A^2 A 2. Use this formula to de-
termine xand pxfor 2/asin ( x/a) and show that
the uncertainty principle holds.10.11 & 10.12 3-D Particle-in-a-Box;
Degeneracy
10.47.Why do we define (1/X)(d^2 /dx^2 )Xas ( 2 mE/^2 ) and
not simply as E?
10.48.What are the units on (1/X)(d^2 /dx^2 )X? Does this help
explain your answer to the previous question?
10.49.Verify that the wavefunctions in equation 10.20 satisfy
the three-dimensional Schrödinger equation.
10.50.An electron is confined to a box of dimensions 2Å
3Å 5Å. Determine the wavefunctions for the five lowest-
energy states.
10.51.Assume a particle is confined to a cubical box. For
what set of three quantum numbers will there first appear de-
generate wavefunctions? For what sets of differentquantum
numbers will there first appear degenerate wavefunctions?
10.52.Determine the degeneracies of all levels for a cubical
box from the lowest-energy wavefunction, described by the
set of quantum numbers (1, 1, 1) to the wavefunction de-
scribed by the quantum number set (4, 4, 4). Hint:you may
have to use quantum numbers larger than 4 to determine
proper degeneracies. See Example 10.15.
10.53.From the expressions for the 1-D and 3-D particles-in-
boxes, suggest the forms of the Hamiltonian operator, ac-
ceptable wavefunctions, and the quantized energies of a par-
ticle in a two-dimensional box.
10.54.What are x , y , and z for 111 of a 3-D particle-in-
a-box? (The operators for yand zare similar to the operator
for x, except that yis substituted for xwherever it appears, and
the same for z.) What point in the box is described by these
average values?
10.55.What are x^2 , y^2 , and z^2 for 111 of a 3-D particle-
in-a-box? Assume that the operator xˆ^2 is simply multiplication
by x^2 and that the other operators are defined similarly. Check
the integral table in Appendix 1 for needed integrals.10.13 Orthogonality
10.56.Show that 111 and 112 for the 3-D particle-in-a-box
are orthogonal to each other.Exercises for Chapter 10 313