Department of Physics
Prof. R. Bundschuh
The Ohio State University
Seventh Problem Set for Physics 847 (Statistical Physics II)
Winter quarter 2004
Important date:
Mar 16 9:30am11:18am final exam
Due date:
Tuesday, Feb 24
18. Spin
1
atom
12 points
An atom with spin 1 has a Hamiltonian
ˆ
H
=
A
ˆ
S
2
z
+
B
(
ˆ
S
2
x

ˆ
S
2
y
), where
ˆ
S
x
,
ˆ
S
y
, and
ˆ
S
z
are the
x
,
y
, and
z
component of the spin angular momentum operator. In the basis of eigenstates
of the operator
ˆ
S
z
these three operators have the matrix representations
ˆ
S
z
= ¯
h
1
0
0
0
0
0
0
0

1
,
ˆ
S
x
=
¯
h
√
2
0
1
0
1
0
1
0
1
0
,
and
ˆ
S
y
=
¯
h
i
√
2
0
1
0

1
0
1
0

1
0
.
At time
t
= 0 the atom is initially in an eigenstate of
ˆ
S
z
with eigenvalue +¯
h
.
a) Write the density matrix (in the basis of eigenstates of
ˆ
S
z
) at
t
= 0.
b) Compute the density matrix at time
t
in the basis of eigenstates of
ˆ
S
z
.
c) Compute the average
z
component of the spin at time
t
.
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 Winter '04
 BUNDSCHUH
 Physics, Angular Momentum, Fundamental physics concepts, Hilbert space, Statistical Physics II, density operators

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