Statistical Mechanics - Physics at Oregon State University

22 CHAPTER 1. FOUNDATION OF STATISTICAL MECHANICS.
Problem 6.
The atomic spin on an atom can take 2S+1 different values, si = −S, −S +
1, · · · , +S − 1, +S. The total magnetic moment is given by M = N
si and
the total number of atoms is N. Calculate the multiplicity function g(N, M),
counting all states of the system for these values N and M. Define x = M
N and
calculate g(N, x) when N becomes very large.
Problem 7.
Twelve physics graduate students go to the bookstore to buy textbooks.
Eight students buy a copy of Jackson, six students buy a copy of Liboff, and
two students buy no books at all. What is the probability that a student who
bought a copy of Jackson also bought a copy of Liboff? What is the probability
that a student who bought a copy of Liboff also bought a copy of Jackson?
Problem 8.
For an ideal gas we have U = 3
2 NkBT , where N is the number of particles.
Use the relation between the entropy S(U, N) and the multiplicity function
g(U, N) to determine how g(U, N) depends on U.
Problem 9.
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 + 1
2 N)ω
M =
N
i=1
Calculate the multiplicity function g(M, N). Hint: relate g(M, N) to g(M, N+
1) and use the identity
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 .
i=1