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Phys 404
Spring 2010
Homework 5,
CHAPTER 5
Due Thursday, March 25, 2010 @ 12:30 PM
General hint
:
Problems 1 – 3 involve classical statistical mechanics.
In this case, the sums over
quantum states are replaced by integrals over all position and momentum components, so that, for
example,
( ) ( )
-H(p,q)
NN
Nd
1
... X(p,q)e
dp
dq
h
X=
Z
τ
∫∫
( ) ( )
-H(p,q)
Nd
1
Z =
... e
dp
dq
h
τ
Here N is the number of particles, d is the spatial dimensionality of the problem, H(
p
,
q
) is the
Hamiltonian (total energy) in terms of the vector coordinates (
q
) and momenta (
p
) of all the
particles, and h is Planck’s constant.
Note that there is one integral for each component of
momentum and position for each particle, so that there are 2Nd integral signs indicated by the
...
in the formulas.
See the
Lecture 13 summary
for a review of classical statistical mechanics.
1
.
A pendulum hangs under gravity, has length
L
and mass
m
, and makes an angle
θ
with the
vertical direction.
Assuming that the amplitude of oscillation is small, find <
>, <
2

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