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PHY 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu  Spring 2007
Problem Set 6
Due Thursday, April 26, in lecture
 Problem 1 [15 points]
A linear molecule of N identical atoms has a vibrational spectrum given by the angular frequencies,
ω_{m}=ω_{o}sin(mπ/2N),
for m = 1,2, ..., N1 (i.e. these are the frequencies of the normal modes of vibration).
a) Show, by explicit calculation, that the vibrational contribution to the specific heat of this molecule at very high temperature, T→∞, is
C = (N1)k_{B}
b) Show that for lower temperatures, such that 1/N << k_{B}T/(hbar ω_{o}) << 1,
C ~ T
c) How does C vary with temperature at very low temperatures, such that
k_{B}T/(hbar ω)_{o} << 1/N?
 Problem 2 [15 points]
Consider a degenerate Fermi gas of noninteracting, nonrelativisitic, particles in two dimensions (this might be a model for electrons in a thin metallic film).
a) Find the density of states g(ε).
b) Find the Fermi energy and the T=0 energy density.
c) Using the the fact that the particle density n is given by
find the chemical potential as a function of temperature, µ(T), for fixed density n, by doing this integral exactly.
You may have to look up an integral in an integral handbook! Using the exact expression for µ(T), find a simpler approximation that holds at low T<<T_{F}. Does µ(T) have a power series expansion in T at low T?
 Problem 3 [15 points]
N Fermions A of spin 1/2 are introduced into a large volume V at
temperature T. Two Fermions may combine to create a Boson with
spin 0 via the interaction,
A + A ↔ A_{2}
Creation of the molecule A_{2} costs energy ε_{o} > 0.
At equilibrium, the system will contain N_{F} Fermions and
N_{B} Bosons. Provide expressions from which the ratio
N_{B}/N_{F} can be calculated, and perform the
calculation explicitly for T=0. What would this (T=0) ratio be,
if the particles were classical (i.e. quantum statistics can be neglected).
Explain the difference.
 Problem 4 [15 points]
Consider a noninteracting gas of spin zero bosons, whose energymomentum relationship is given by ε(p) = Ap^{s}, for some fixed positive numbers A and s. The dimensionality of the gas is the number d, i.e. the volume of the gas is V = L^{d}, for a system of length L. In the following parts, we are considering the behavior in the thermodynamic limit of V → ∞.
a) For what values of s and d will there exist BoseEinstein condensation at sufficiently low temperatures?
b) For the case that there is BoseEinstein condensation, write an expression that gives how the condensate density n_{o} depends on the temperature T, the condensation temperature T_{c}, and the total density n.
c) Show that the pressure is related to the energy by, p = (s/d)(E/V)
d) Can a gas of photons in d = 3 undergo BoseEinstein condensation?

