PHY 418: Statistical Mechanics I
Prof. S. Teitel firstname.lastname@example.org ---- 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,
for m = 1,2, ..., N-1 (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 = (N-1)kB
b) Show that for lower temperatures, such that 1/N << kBT/(hbar ωo) << 1,
C ~ T
c) How does C vary with temperature at very low temperatures, such that
kBT/(hbar ω)o << 1/N?
- Problem 2 [15 points]
Consider a degenerate Fermi gas of non-interacting, non-relativisitic, 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<<TF. 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 ↔ A2
Creation of the molecule A2 costs energy εo > 0.
At equilibrium, the system will contain NF Fermions and
NB Bosons. Provide expressions from which the ratio
NB/NF 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 non-interacting gas of spin zero bosons, whose energy-momentum relationship is given by ε(p) = A|p|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 = Ld, 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 Bose-Einstein condensation at sufficiently low temperatures?
b) For the case that there is Bose-Einstein condensation, write an expression that gives how the condensate density no depends on the temperature T, the condensation temperature Tc, 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 Bose-Einstein condensation?