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Physics 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu ----- Spring 2004

Problem Set 5

Due Tuesday, April 20, in lecture

• Problem 1 [15 points]

  Consider our discussion of Pauli paramagnetism of the free, non-interacting, electron gas. We claimed that the chemical potential at finite magnetic field differed from its value at B=0 only to second order in the Zeeman energy, i.e.,

  µ(B) = µ(B=0) [ 1 + O(µ_BB/_F)² ]

  where µ_B is the Bohr magneton and _F is the Fermi energy.

  Explicitly demonstrate this result by computing µ(B) to second order in B, in the limit T=0.

• Problem 2 [15 points]

  a) Find the chemical potential µ for an ideal (non-relativistic) 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?)

  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]

  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 _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 non-interacting ideal gas of non-relativistic bosons (i.e. (p) = p²/2m) in d dimensions, where d is any number. Show that there is no Bose Einstein condensation for any d less than or equal to two.