Introduction to Quantum Mechanics 171.303
Midterm Exam 11/2/06 1-2
Check the attached formula pages. Start each problem on a fresh page and please give
detailed reasoning. Please ask your proctor for clarification if the text is unclear.
Problem 1 (25 points)
After passing through one channel of a Stern Gerlach apparatus with a field gradient in
the x-z plane, a spin-1/2 particle is found in the following quantum state:
(
)
zz −−+= 3A
ψ
(a) Compute the normalization constant A. (10 points)
(b) What can be said about the direction of the field gradient? (15 points)
Problem 2 (25 points)
An electron is placed in a uniform magnetic field with strength B.
(a) Which component of angular momentum has a time independent expectation value?
A full explanation is needed for full credit. (10 points)
A measurement of angular momentum is carried out in a direction perpendicular to the
field direction at time t0.
(b) For which future times is it possible to predict the exact outcome of a subsequent
measurement of angular momentum along this same direction? (15 points)
Problem 3 (30 points)
Consider two particles with spin quantum number s1=3/2 and s2=1 respectively. Their
interaction can be described by the Hamilton operator 21
2ˆˆ
2
ˆss ⋅= =
A
Hwhere 0>A.
(a) By rewriting
H
ˆ in terms of
(
)
2
21
2ˆˆ
ˆss +=Scompute the eigenvalues of
H
ˆand the
number of degenerate states for each eigenvalue. (15 points)
The system is in its lowest energy state when a measurement of the projection of the total
angular momentum along the z-direction is performed. The measurement yields 2/==
z
S.
(b) Determine the possible results for a subsequent measurement of z
s2 and the
respective probabilities. (15 points) (hint: use the Clebsch Gordan table)
Problem 4 (20 points)
Consider a spin-1 object subject to the following hamiltonian:
(
)
22 ˆ
/
ˆz
SDH == where
D>0.
(a) Determine the energy eigen-states and eigen-values. (10 points)
(b) Assume the spin is in the lowest energy state. Suggest a technique to create a
transition into an excited state. (10 points) (Hint: Emulate NH3 in an AC E-field.)
+Q
