Physics 137B, Fall 2007
Quantum Mechanics II
Midterm II solutions
1. Consider a spinhalf system with
H
=
H
0
+
H
,
H
0
= (2
μB
0
/
¯
h
)
S
z
,
H
= 2
CS
x
, where
C
is a
nonzero constant and much less than
μB
0
/
¯
h
. Suppose that at
t
= 0 the system is in the spindown
state, i.e., the ground state of
H
0
. It may help to use the representations of spinhalf operators
S
z
=
¯
h
2
1
0
0

1
,
S
x
=
¯
h
2
0
1
1
0
.
(1)
(a) Write an expression for the transition probability from spinup to spindown as a function
of time. Hint: you should find some sort of oscillatory behavior.
We start from the formula
P
ba
(
t
) =
2

H
ba
(
t
)

2
¯
h
2
F
(
t, ω
ba
)
.
(2)
Here
ω
ba
is (
E
↑

E
↓
)
/
¯
h
= 2
μB
0
/
¯
h
.
All that remains is to compute
H
ba
. Using the supplied matrices,
H
↑↓
is found to be
C
¯
h
.
(b) Find the exact eigenvalues of the full twodimensional Hamiltonian for this twostate system
by solving a quadratic equation.
The determinant equation for the eigenvalues of
H
is
(
λ

μB
0
)(
λ
+
μB
0
) +
C
2
¯
h
2
= 0
⇒
λ
2
=
C
2
¯
h
2
+
μ
2
B
0
2
.
(3)
Hence the energy eigenvalues are
λ
=
±
C
2
¯
h
2
+
μ
2
B
0
2
.
(4)
(c) Is the initial state
 ↓
an eigenstate of the full Hamiltonian? A simple yes or no is fine; you
do not need to find the actual eigenstates.
No, since acting on it with the Hamiltonian gives a vector that has a nonzero up component.
(d) Show that the oscillation frequency expected from the exact solution is consistent with the
oscillation frequency from perturbation theory in the limit where perturbation theory is valid.
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 Fall '07
 MOORE
 mechanics, Kinetic Energy, ground state

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