PHY 4604 Fall 2008 — Exam 1
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Instructions: Attempt all three questions. The maximum possible credit for each part
of each question is shown in square brackets. Please try to write your solution neatly and
legibly.
You will receive credit only for knowledge and understanding that you demonstrate in
your written solutions. It is in your best interest to write down something relevant for every
question, even if you can’t provide a complete answer. To maximize your score, you should
briefly explain your reasoning and show all working. Give all final algebraic answers in terms
of variables defined in the problem and h̄ (the reduced Planck constant).
During this exam, you may use one formula sheet. You are not permitted (a) to consult
any other books, notes, or papers, (b) to use any electronic device, or (c) to communicate
with anyone other than the proctor. In accordance with the UF Honor Code, by turning in
this exam to be graded, you affirm the following pledge: On my honor, I have neither given
nor received unauthorized aid in doing this assignment.
Print your name where indicated below, and sign to confirm that you have read and
understood these instructions. Please do not write anything else below the line.
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Question
1
2
3
Total
Score
1. A particle is in a one-dimensional state described by a wave function



2Ax



for 0 < x < a,
Ψ(x, t = 0) =  A(3a − x)



 0
for a < x < 3a,
otherwise.
(a) [10 points] Find a value of A that ensures the wave function is normalized.
(b) [10 points] What is the probability that a measurement of the particle’s position
at t = 0 yields a result x < a?
(c) [10 points] What is the expectation value of x at t = 0?
(d) [10 points] What is the expectation value of the momentum p at t = 0?
2. A one-dimensional harmonic oscillator of mass m and angular frequency ω is in an
initial state
√
Ψ(x, t = 0) = [iψ2 (x) − 2ψ7 (x)] / 5,
where ψn (x) (n = 0, 1, 2, . . .) is the nth stationary state as conventionally defined.
(a) [15 points] What is the expectation value of the oscillator’s energy at a later time
t = T?
(b) [15 points] What is the smallest positive, odd integer power of the momentum
p that can have a nonzero expectation value in the state Ψ(x, T )?
3. A particle of mass m moves under the influence of the one-dimensional potential
V (x) =



V0 > 0



−V0




 0
for |x| < a/2,
for a/2 < |x| < a,
for |x| > a.
(a) [15 points] Sketch a graph showing the qualitative features of the particle’s probability density when it is in a state of energy E = −V0 /2. You may assume that
this is the particle’s first excited state.
(b) [15 points] Sketch a graph of the particle’s probability density when it is in a state
of energy E = V0 /3.
Do not attempt a quantitative solution for the algebraic form of the probability density
in part (a) or (b). In each case, your graph should make clear (and you should also
provide a brief written explanation of) qualitative features such as (i) the relative
magnitude of the probability density in different spatial regions and (ii) the relative
wavelengths in those regions where the probability density is oscillatory. Make sure
that you label the points x = ±a/2 and x = ±a on the horizontal axis.
2