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Set # 5
1. For a single quantum particle of mass m, spectra p2/2m in a volume V the partition
fundtion is Z1(m)=gV/3 with   h / 2mk BT . The particle has a spin degeneracy g
(g=2s+1 for spin s).
a) Calculate the partition function of two such particles if they are bosons and
also if they are fermions.
b) Calculate the corrections to the energy E, and the heat capacity C, due to Bose
or Fermi statistics.
c) Find the second virial coefficient a2, defined as PV=NkT[1+ a2n3] to leading
order in the small parameter n3.
2. Consider an ideal Bose gas in d dimensions whose single particle spectrum is
given by =|p|s, s>0.
a) Find the condition on s, d for the existence of Bose-Einstein condensation.
In particular show that for nonrelativistic particles in two dimensions
(s=d=2) the system does not exhibit Bose-Einstein condensation.
b) Show that
s E
d
P= d V
and CV(T∞) = s NkB
3. Consider an ideal Bose gas with spin 1 in a magnetic field B. The Hamiltonian
p2
for each particle is H 
 BS z where S z  1,0,1 and  is the gyromagnetic
2m
ratio. The average density is n and particle mass is m.
a) Particles of which S z can condense? Find an equation for the condensation
temperature Tc . Solve this equation explicitly for (i) B=0, and (ii) large B,
B  k BTc ; keep the lowest order correction.
b) If T is below but close to the value of Tc ( B  ) describe what happens as
B is increased from B=0. Find the critical B for condensation in the limit of
(a-ii).
c) Show that E  3PV / 2 and evaluate the specific heat at T  Tc in the limit
of (a-ii).
4. A cylindrical container of length L and base area A is divided in two by a freely
moving partition. To the left of the partition there are Na identical bosons of mass ma,
and to the right there are Nb bosons of mass mb.
One can assume that
i.
The state of the gas on the left may be described by a Boltzman
approximation
ii.
the gas on the right is in a condensate.
Under these assumptions
a. Find the location of the partition when the system is in equilibrium at
temperature T.
b. Find the condition that assumption i is valid in equilibrium.
c. Find the frequency of small oscillations of the partition around equilibrium.