Introduction to Quantum Mechanics 171.303
Midterm Exam 11/3/05 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 (30 points)
Consider a spin-1/2 particle with charge q and mass m, in a magnetic field of strength B
oriented in the z
ˆ direction. At t=0, the particle is in the state
(
)
(
)
zx +−−= A0
ψ
(a) Compute the normalization constant A. (10 points)
(b) Compute z
S at t=0. (10 points)
(c) Compute y
S as a function of t. (10 points)
Problem 2 (20 points)
Consider a spin-3/2 particle which we shall describe in the basis of eigenstates for z
S
ˆ.
(a) Write the matrix representation for z
S
ˆ and +
S
ˆ (10 points)
(b) Write the matrix representations for x
S
ˆ (10 points)
Problem 3 (20 points)
Consider two particles with spin quantum number S1=3/2 and S2=1 respectively. Their
interaction can be described by a spin Hamiltonian 21
2ˆˆ
2
ˆSS ⋅= −J=H.
(a) Show that there exist simultaneous eigenstates of H
ˆ and
(
)
2
21
2ˆˆˆ SS +=
tot
S (10 points)
(b) Compute the eigenvalues of H
ˆ and the number of degenerate states for each energy
level. (10 points)
Problem 4 (30 points)
Consider a spin-1/2 particle that we describe in two different bases. The basis B1 consists
of eigenstates for z
S
ˆ and basis B2 consists of eigenstates for n
S
ˆ. The spherical
coordinates for the unit vector n
ˆ are
(
)
(
)
θφ
θφ
1
=
r so that a rotation of
φ
about z
ˆ
followed by a rotation of
θ
about y
ˆ brings z
ˆ into coincidence with n
ˆ.
(a) Write the representation of 2
ˆ
Sand n
S
ˆ in basis B2. (8 poins)
(b) Compute the transformation from the representation of a general state
ψ
in basis B1
to the representation of
ψ
in basis B2. (12 points)
(c) Compute the representation of z
S
ˆ in basis B2 and check your result by selecting
appropriate values of
θ
and
φ.
(10 points)
+Q
