Problem Set 10

Problem Set 10

Due Monday, April 29, uploaded to Blackboard by 11pm

Problem 1 [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) [5 pts] For what values of s and d will there exist Bose-Einstein condensation at sufficiently low temperatures? In particular, show that for s=2 (non-relativistic massive particles) there is no Bose-Eisntein condensation in d=2 dimensions.
b) [4 pts] For the case that there is Bose-Einstein condensation, write an expression that gives how the condensation temperature T_c depends on the total density of particles n, and how the condensate density n_o depends on temperature T.
c) [4 pts] Show that the pressure is related to the energy by, p = (s/d)(E/V)
d) [2 pts] Can a gas of photons in d = 3 undergo Bose-Einstein condensation?
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 U(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(n_x, n_y, n_z) = ℏω₀ (n_x + n_y + n_z + 3/2), where n_x, n_y, n_z = 0, 1, 2, . . . are integers.
a) [6 pts] 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 Problem Set 9 problem 3, but generalizing to 3D. You may assume that the thermal energy is much greater than the spacing between the energy levels, i.e. k_BT >> ℏω₀.
b) [3 pts] What is the largest value that the fugacity z can take?
c) [5 pts] Show that this system has Bose-Einstein condensation at sufficiently low temperature.
d) [3 pts] Find the Bose-Einstein condensation temperature T_c as a function of the number of paricles N.
e) [3 pts] Find how the number of particles N₀(T) in the condensed state varies with temperature for T=0 to T_c.
Problem 3 [20 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.