Physics 481
Midterm Exam
Spring 2005
The total number of points for this exam is 24. You have 50 minutes to complete the exam.
Do all problems. Show all your work so that I can see how you arrived at the answer.
You may use Griffith ”Introduction to Quantum Mechanics” and the lecture notes to look
up any relevant formulas.
1. Consider the quantum mechanical analog to the classical problem of a ball (mass
m) bouncing elastically on the floor:
(
V (x) =
mgx if x > 0,
∞
if x ≤ 0.
(a) WKB Approximation:
i. Sketch the potential. (2 points)
ii. Use the WKB approximation to find the allowed energies En in terms of
m, g and h̄. (4 points)
(b) Variational Method:
i. Why is the usual Gaussian trial wave function not an appropriate trial
function for this potential? (2 points)
ii. One possible group of (normalized) trial functions, defined for x > 0, are
ψ(x) = 2α3/2 xe−αx
Use this group of trial wave functions to find an estimate of the ground
state energy in terms of m, g and h̄. (6 points)
You may use the following information:
Z ∞
0
xn e−x dx = Γ(n + 1)
and
Γ(n + 1) = n!
2. Two noninteracting particles with the same mass m are in a one-dimensional harmonic oscillator potential.
(a) What are the values of the three lowest energies of the two-particle system for
the case that the particles are distinguishable? (2 points)
(b) What are the degeneracies of these energies if the two particles are
i. identical, with spin 12 ; (4 points)
ii. identical, with spin 1? (4 points)
Note that a degeneracy of 0 (i.e., no allowed states) is a possible answer. Do
not neglect the spin part of the complete wave function. Clearly justify your
answers.
Good luck !
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