Statistical Mechanics - Physics at Oregon State University

A.1. SOLUTIONS FOR CHAPTER 1. 235
The energy eigenstates of a harmonic oscillator are ɛn = ω(n + 1
2 ) for n =
0, 1, 2, · · · Consider a system of N such oscillators. The total energy of this
system in the state {n1, n2, · · · , nN} is
where we have defined
U =
N
i=1
ɛni
M =
= (M + 1
2 N)ω
Calculate the multiplicity function g(M, N). Hint: relate g(M, N) to g(M, N +
1) and use the identity
N
i=1
ni
m
n + k n + 1 + m
=
n n + 1
k=0
Show that for large integers g(M, N) is a narrow Gaussian distribution in x =
M
N .
It is easy to show that
g(M, N) =
M
g(m, n)g(M − m, N − n)
m=0
because the states of parts of the system are independent. We also know that
g(M, 1) = 1
because for one particle there is only one way of obtaining M units of energy.
Therefore
g(M, N + 1) =
M
g(m, N)
m=0
Comparing with the hint we see that
n + M
g(M, N) =
n
for some value of n linearly related to N, or n = N + n0. Because g(M, 1) = 1
and 0+M
0 = 1 we find that n = N − 1 and hence
N − 1 + M
g(M, N) =
N − 1
Next we take the logarithms: