Prof. Jennifer Curtis
School of Physics, Georgia Tech
Spring 2012
Statistical Mechanics 4142:
Problem Set 3
Due date:
Friday, Feb. 3,
2:05pm
.
Comments:
You may use the following
definite integral of a Gaussian function
:
[1] Quantum Harmonic Oscillator, Multiplicity and Total Energy. (K&K #2.3)
(a) (2 pts.)
Find the entropy of a set of N oscillators of frequency
ω
as a function of the total quantum
number n. Use the multiplicity function for a quantum harmonic oscillator (given in class or in K.K.
1.55, where n=q).
You will need to use typical approximations, assuming N is large.
(b)
(2 pts.)
Let
U
denote the total energy
q
ω
of the oscillators.
Express the entropy as
σ
U
,
N
(
)
. Show
that the total energy at temperature t is:
U
=
N
ω
exp
ω τ
(
)
−
1
.
This is the Planck result; it is derived again in Ch. 4 (K&K) by a powerful method that does not
require us to find the multiplicity function.
[2] Multiplicity of a Large Twostate Paramagnet. (Schroeder 2.24)
For a single,
large
twostate paramagnet, the multiplicity function is very sharply peaked about
N
↑
=
N
2
.
(a) (2 pts.)
Use Stirling’s approximation to estimate the height of the peak in the multiplicity function.
(b) (2 pts.)
Use the methods introduced in class to derive a formula for the multiplicity function in the
vicinity of the peak, in terms of
x
≡
N
↑
−
N
2
. Check that your formula agrees with your answer to
part (a) when x=0.
(c) (2 pts.)
How wide is the peak in the multiplicity function?
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 Spring '10
 Davidovic
 mechanics, Normal Distribution, Statistical Mechanics, Entropy, pts, multiplicity function

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