PHY 6645 Fall 2001 – Final Exam Instructions:

PHY 6645 Fall 2001 – Final Exam
Instructions: Please write your name in the space provided at the bottom of this page. At
the end of the exam, you should hand in your solutions with this page on top.
Attempt all five questions overleaf. The maximum score for each part of each question is
shown in square brackets. (The maximum overall score on the exam is 100.)
To gain full credit you should explain your reasoning and show all working.
You may consult Shankar’s “Principles of Quantum Mechanics,” your PHY 6645 lecture
notes, and/or solutions to exam or homework assignments from this semester’s course; you
may not use any other written or printed material. You must not 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.
PHY 6645 Fall 2001 – Final Exam
Follow the instructions overleaf.
1. A certain particle has spin s = 12 .
(a) A measurement is made of the instantaneous sum of the x and y components of
the particle’s spin angular momentum. List all the possible results (i.e., measured
values) that might be obtained from this measurement. [10]
(b) Immediately after the measurement described in (a), the x component of the
spin angular momentum is measured. For each of the possible results of the first
measurement, list all the possible results of the second measurement and their
respective probabilities. [20]
2. Suppose that two noninteracting spin- 12 particles are known to be in a spin-singlet
state. Let Sz(1) measure the z component of the spin angular momentum of particle
1. Let Sn(2) measure the component of particle 2’s spin angular momentum along
the direction n̂ described by polar angles (θ, φ), i.e., in Cartesian coordinates n̂ =
(sin θ cos φ, sin θ sin φ, cos θ). Find the expectation value hSz(1) Sn(2) i. [15]
3. Construct a (normalized) total angular momentum eigenstate |j = 12 , m = 12 i from
product states of the form |j1 = 1, m1 i ⊗ |j2 = 12 , m2 i. To gain full credit, you should
construct the eigenstate from first principles, i.e., without quoting any Clebsch-Gordan
coefficients. [15]
4. A point-like particle moves in a one-dimensional potential,
V (x) =
V >0
 +∞
for −a < x ≤ 0,
for 0 < x < a,
for |x| ≥ a.
Plot y = |ψ(x)|2 over the range −a ≤ x ≤ a for the first excited state of this problem (i.e., the energy eigenstate of second-lowest energy), assuming that the state has
an energy E > V0 . You should focus on getting the plot qualitatively (rather than
quantitatively) correct. Make sure that it is clear on your plot at which points (if any)
y is zero, and at which points (if any) dy/dx is zero. Also make clear the relative
magnitudes of y in the regions x < 0 and x > 0. [20]
5. Consider a one-dimensional harmonic oscillator that at time t = 0 is in a state
exp(−iP b/h̄)|0i, where |0i is the ground state of the harmonic oscillator, P is the
momentum operator, and b is a constant length. Calculate the position expectation
value at time t. [20]
Hints: (1) Recall (from Exam 2) that aH (t) = aS e−iωt . (2) You may use without proof
the formula (for operators A and B, and any scalar λ)
eλA Be−λA = B +
[A, B] + [A, [A, B]] + [A, [A, [A, B]]] + . . .