PHY 4604 Fall 2009 – Homework 3
Due at the start of class on Wednesday, September 30. No credit will be available
for homework submitted after the start of class on Monday, October 5.
Answer all three questions. Please write neatly and include your name on the front page of
your answers. You must also clearly identify all your collaborators on this assignment. To
gain maximum credit you should explain your reasoning and show all working.
These questions are similar in length and difficulty to those that will appear on the first midterm exam. The exam will likely consist of one question requiring quantitative calculations
(like question 1 or question 2 below) and one question requiring a more qualitative answer
(like question 3).
Your may find useful the following integrals:
√
Z ∞
π(2n)!
2n −x2
x e dx = 2n+1
2
n!
0
Z
0
∞
2
x2n+1 e−x dx =
n!
2
1. Consider a particle of mass m oscillating in one dimension at the end of an ideal spring
of spring constant k, i.e., having a potential energy V (x) = 12 kx2 . Suppose that at
√
time t = 0, this system is in the quantum-mechanical state (1 − 2i)ψ2 (x)/ 5 (where
ψn with n = 0, 1, 2, . . . represents the nth stationary state).
(a) What is/are the most probable result/results of a position measurement performed
on this system at time t?
(b) What is the probability that a measurement of the position x performed on the
system at time t yields a value |x| greater than that possible for a classical particle
with the same total energy? You should express your answer in terms of variables
defined in the problem and a definite integral over a dimensionless variable. (You
do not need to evaluate this integral.)
2. A classic toy consists of a hard ball attached by elastic to the middle of a paddle.
We can model a variant of this setup quantum mechanically by the one-dimensional
potential
∞
for x < 0,
V (x) =
1
2 2
mω x for x > 0.
2
(a) Given that (i) V (x) coincides with the harmonic oscillator potential V0 (x) =
1
mω 2 x2 in the region x > 0, and (ii) stationary states of V (x) must vanish at
2
x = 0, express the normalized stationary states ψn (x) (n = 0, 1, 2, . . .) of V (x)
in terms of the familiar stationary states ψ0,n0 (x) (n0 = 0, 1, 2, . . .) of V0 (x).
(b) Calculate hxi in the state [3ψ0 (x) + 4iψ1 (x)] /5. The properties of the harmonic
oscillator ladder operators a± do not extend to V (x). You will need to evaluate
these expectation values using the explicit form of ψn (x). The resulting integrals
are simplified by working with the dimensionless variable ξ = x/a where a =
p
~/mω.
3. Consider a particle of mass m moving in one dimension under the influence of a potential

x < −a,
 0
−a < x < a,
−V0
V (x) =

x > a,
V0
with V0 = π 2 ~2 /ma2 .
(a) List the range(s) of energies E (if any) within which you expect stationary states
to be (i) forbidden; (ii) allowed at discrete energies only; (iii) allowed at all energies
within the range; (iv) doubly degenerate.
(b) List the range(s) of energies E (if any) within which you expect to be able to
construct stationary states having a nonzero probability current.
The remainder of this problem concerns a particular stationary state of this problem:
one of energy E = V0 /2, having a wave function ψ(x). Answer the questions below
using qualitative arguments wherever possible. It should not be necessary for you to
obtain a full quantitative solution for ψ(x).
(c) Write down the form of ψ(x) in the region x > a. Express the wavelength or
exponential decay length (whichever is appropriate) of ψ(x) in this region as a
multiple of a. (The exponential decay length is l in e−x/l .) Find the probability
current in this region.
(d) Write down the form of ψ(x) in the region −a < x < a. Express the wavelength
or exponential decay length of ψ(x) in this region as a multiple of a. Find the
probability current in this region.
(e) Write down the form of ψ(x) in the region x < −a. Express the wavelength
or exponential decay length of ψ(x) in this region as a multiple of a. Find the
probability current in this region.
(f) Based on the analogy with classical mechanics, explain whether you expect the
maximum amplitude of ψ(x) in the region x < −a to be greater than, less than,
or equal to the maximum amplitude of ψ(x) in the region −a < x < a.
(g) Sketch a graph of the probability density |ψ(x)|2 as a function of x. The horizontal
axis should range from x = −3a to x = 3a, and should have the points x = −a
and x = a explicitly labeled. You should be careful to represent correctly as many
features of the graph as possible (based on the information derived in the previous
parts of the problem).