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Department of Physics
Prof. R. Bundschuh
The Ohio State University
Eighth Problem Set for Physics 847 (Statistical Physics II)
Winter quarter 2004
Important date:
Mar 16 9:30am11:18am Fnal exam
Due date:
Tuesday, Mar 2
21. Density operator of a free particle
10 points
A free particle is described by the Hamiltonian
ˆ
H
=
ˆ
~
p
2
/
2
m
. We assume that the particle is
in a cubic box of volume
V
=
L
3
.
a) Write the canonical density operator in the momentum base

~
k
i
, i.e., calculate
h
~
k
0

ˆ
ρ

~
k
i
.
When calculating the partition function you may replace the sum by an integral.
b) Write the canonical density operator in the coordinate base

~
r
i
, i.e., calculate
h
~
r
0

ˆ
ρ

~
r
i
.
You may again replace sums by integrals. (Hint: use
h
~
k

~
r
i
=
V

1
/
2
exp(

i
~
k~
r
).)
22. Single energy level
10 points
We consider one energy level of a large quantum system as the subsystem which we want
to describe while we summarize all the other levels of the system as the particle bath.
The
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 Winter '04
 BUNDSCHUH
 Physics

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