Problem 1 [15 points]
Consider a gas of indistinguishable non-interacting, non-relativistic, bosons of mass m and spin zero in d dimensions.
a) Show that whether or not the gas will undergo Bose-Einstein condensation at sufficiently low temperature depends on the dimensionality d, and that in particular there is no Bose-Einstein condensation in d=2 dimensions.
b) For the cases where there is Bose-Einstein condensation, find the condensation temperature Tc.
c) For the cases where there is Bose-Einstein condensation, how does the density of particles in the condenstate vary with temperature as T varies from T=0 to T=Tc?
Problem 2 [20 points]
Consider a three-dimensional gas of N indistinguishable non-interacting spin zero bosons of mass m in an external isotropic harmonic potential V(r) = (1/2)mω02 |r|2, where r = (x, y, z). This might be taken as a model for bosons in a magnetic trap. The quantized single particle energy levels are given by ε(nx, ny, nz) = ℏω0 (nx + ny + nz + 3/2), where nx, ny, nz = 0, 1, 2, . . . are integers.
a) Compute the density of states g(ε). The density of states is such that g(ε)dε is the number of single particle states between ε and ε+dε.
Hint: You should try to do this the same way you found g(ε) in HW #7 problem 2, but generalizing to 3D. You may assume that the thermal energy is much greater than the spacing between the energy levels, i.e. kBT >> ℏω0.
b) What is the largest value that the fugacity z can take?
c) Show that this system has Bose-Einstein condensation at sufficiently low temperature.
d) Find the Bose-Einstein condensation temperature Tc as a function of the number of paricles N.
e) Find how the number of particles N0(T) in the condensed state varies with temperature for T=0 to Tc.