Ben Sauerwine
Practice for Qualifying Exams
Problem Source:
CMU Qualifying Exam August 2004
(1)
(a)
Write down the canonical partition function
for a quantum system
for the energy of state s, and
q
Z
s
E
T
k
B
1
=
β
(
is Boltzmann’s constant, T is
the absolute temperature).
Next write down the classical partition function
appropriate for a monatomic gas of N identical non-relativistic particles
of mass m in a volume V as an integral involving the Hamiltonian H.
B
k
c
Z
∑
−
=
s
E
q
s
e
Z
β
where the N and V dependencies appear in the energy states and analogously,
(
)
∫∫
−
=
q
pd
d
e
h
N
Z
N
N
q
p
H
N
c
3
3
,
3
!
1
v
v
β
(b)
Beginning with appropriate expressions for the averages
E
and
2
E
,
show that the variance of the energy E can be written in the form
(
)
N
V
E
E
E
,
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
−
=
−
β
for both the quantum and classical systems considered in part (a).
First, note that
(
)
2
2
2
2
2
2
E
E
E
E
E
E
E
E
−
=
+
−
=
−
.
Further,
∑
∑
∑
∑
−
−
−
−
=
=
s
E
s
E
s
q
s
E
s
E
s
q
s
s
s
s
e
e
E
E
e
e
E
E
β
β
β
β
2
2
and,
(
)
(
)
(
)
(
)
∫∫
∫∫
−
−
=
=
q
pd
d
e
q
p
H
h
N
E
q
pd
d
e
q
p
H
h
N
E
N
N
q
p
H
N
c
N
N
q
p
H
N
c
3
3
,
2
3
2
3
3
,
3
,
!
1
,
!
1
v
v
v
v
v
v
v
v
β
β

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