PHY 4604 Fall 2008 — Final Exam
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Instructions: Attempt all four questions. The maximum possible credit for each part
of each question is shown in square brackets. Please try to write neatly!
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. (This is true even in the case
of the multiple-choice questions.) Give all final algebraic answers in terms of variables
defined in the problem and h̄ (the reduced Planck constant). For numerical problems, take
h̄ = 1.05 × 10−34 J s.
During this exam, you may use the formula sheet provided, your own three formula sheets,
and an electronic calculator. You are not permitted (a) to consult any other books, notes,
or papers, (b) to use any electronic device other than a calculator, 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.
Name (printed):
Signature:
Question
1
2
3
4
Total
Score
1. [10 points total] For each part of this question, your final answer should be a letter
from A through E. You must show your reasoning to get full credit.
(a) [4 points] A free electron has a momentum of 5.0 × 10−24 kg m/s. The wavelength
of its wave function is:
(A) 1.3 × 10−8 m,
(D) 2.1 × 10−13 m,
(B) 1.3 × 10−10 m,
(E) none of these.
(C) 2.1 × 10−11 m,
(b) [6 points] An electron is in a quantum state for which there are seven allowed
values of Lz , the z component of the angular momentum. The magnitude squared
of the angular momentum, L2 , is:
(A) 3h̄2 ,
(B) 7h̄2 ,
(C) 9h̄2 ,
(D) 12h̄2 ,
(E) 14h̄2 .
2. [25 points total] A spinless particle has a wave function
√
ψ(r, θ, φ) ∝ r e−r/a 1 + 6 sin θ cos φ ,
where a is real and positive.
(a) [10 points] What is the most probable value of the radial coordinate r?
(b) [15 points] What is the expectation value of L2 , the magnitude squared of the
orbital angular momentum?
3. [30 points total] An electron is a spatial state characterized
by orbital quantum numbers
√
l = 1, ml = 1. Its spin state is |χi = (2 |↑i − |↓i)/ 5, where |↑i and |↓i are eigenkets
of the spin-z operator Ŝz .
(a) [10 points] Find the expectation value of the x component of the electron’s spin
angular momentum.
(b) [20 points] Find the possible results of a measurement of the magnitude of the
electron’s total angular momentum, and the probability of obtaining each result.
4. [35 points total] Two particles of mass m moving in one dimension are confined by
hard walls to the region 0 < x < a, inside which their Hamiltonian is
2
h̄2
∂2
∂
Ĥ = −
.
+
2m ∂x21 ∂x22
Each particle has the maximum possible z component of its spin angular momentum.
(a) [10 points] Suppose that the two particles are distinguishable. List the normalized
spatial wave functions—written in terms of single-particle stationary states ψn (x),
where n is a non-negative integer—and the energies for the three two-particle
stationary states of lowest energy.
(b) [10 points] Redo (a) for identical bosons instead of distinguishable particles.
(c) [10 points] Redo (a) for identical fermions instead of distinguishable particles.
(d) [5 points] State in words how would your answers to (a)–(c) would be affected if
the two particles were instead in a state of zero total spin. You may find it helpful
to look at the table of Clebsch-Gordan coefficients.
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