Problem Set 7

Problem Set 7

Due Monday, April 29, in lecture

Problem 1 [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

n = ∫ dε g(ε)
e^β(ε-µ)+1

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, like we found in three dimensions with the Sommerfeld expansion?
Problem 2 [15 points]
a) Find the chemical potential µ for an ideal, non-relativistic, three dimensional Fermi gas at low temperature, to second order in T, at fixed pressure p. Note, this is different from what we did in lecture - there we computed µ at fixed density N/V. (Hint: what is E/V at fixed p? You may wish to review Lecture 23 to do this problem.)
b) The Gibbs free energy is related to the chemical potential by G(T,p,N) = Nµ(T,p). The entropy can be derived from the Gibbs free energy, and hence from the chemical potential, by using,
S = - (dG/dT)_p,N = - N(dµ/dT)_p
Using your result from part (a), find the entropy of the Fermi gas to lowest order in T. Does your result agree with that given in Pathria §8.1 Eq.(41), as derived there from the Helmholtz free energy?
Problem 3 [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 = 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 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 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 Bose-Einstein condensation?
Problem 4 [15 points]
N Fermions A of spin 1/2 (i.e. g_s=2) are introduced into a large volume V at temperature T. Two Fermions may combine to create a Boson with spin 0 (i.e. g_s=1) via the interaction,

A + A ↔ A₂
Creation of the molecule A₂ 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 could be neglected). Explain the difference.