Physics 480
Midterm Exam
Fall 2004
The total number of points for this exam is 24. You have 50 minutes to complete the
exam. Do both problems. Show all your work so that I can see how you arrived at
the answer. You may use Griffith ”Introduction to Quantum Mechanics” to look up the
relevant formulas.
1. Consider a set of identical harmonic oscillators with angular frequency ω.
(a) (4 points) You measure hEi to be 57
h̄ω. Give two possible (normalized) initial
50
states ΨA (x, 0) and ΨB (x, 0) in terms of eigenfunctions of the Hamiltonian
ψn (x) (use only the three lowest-energy eigenfuntions) so that
hxiA ≡ hΨA |xΨA i = 0
and
hxiB ≡ hΨB |xΨB i 6= 0.
(b) (2 points) Using your result from (a) give the formula for the time-dependent
state ΨA (x, t).
(c) (4 points) Using your result from (a) calculate hxiB .
(d) (2 points) For the state ΨB from (a), what is the probability to measure E =
3
h̄ω?
2
(e) (2 points) Assume you actually measure E = 32 h̄ω for a system in the state ΨB
from (a), what is hxiB after the measurement? Explain.
2. No calculator is allowed to solve this problem.
The matrix D describes rotations in two dimensions:
Ã
D=
cos θ − sin θ
sin θ
cos θ
!
(a) (2 points) Show that D is unitary.
(b) (4 points) Find the eigenvalues and normalized eigenvectors of D.
(c) (4 points) Find the matrix S that diagonalizes D via the similarity transformation SDS −1 . Perform the the similarity transformation SDS −1 explicitly
and show that it reduces D to diagonal form.
Good luck !
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