Problem Set 8

Problem Set 8

Due Wednesday, April 29, 10am uploaded to Blackboard

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 T_c.
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=T_c?
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(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) 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. k_BT >> 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 T_c 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 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.
Problem 4 [15 points]
Consider a thermodynamic system consisting of two gases in volumes V₁ and V₂, separated by a thermally conducting, freely sliding wall, as shown in the diagram below. The gas in V₁ has N₁ particles, while the gas in V₂ has N₂ particles. Particles cannot pass through the separating wall. The system is in isolation from the rest of the universe, and the total volume V = V₁ + V₂ is fixed. In thermal equilibrium, the pressures of the gases on the two sides of the sliding wall must be equal.

If the system is initially in thermal equilibrium at temperature T, for each case below explain in which direction the wall between the two gases will move if the temperature is increased a small amount ΔT. You shouuld be able to get you answer with a simple graphical analysis, no algebra needed! But you must give a clear and sound argument.
a) The gas in V₁ is an ideal gas of fermions in the degenerate limit, i.e. k_BT << ε_F. The gas in V₂ is a classical ideal gas.
b) The gas in V₁ is an ideal gas of bosons in a Bose-Einstein condensed state, i.e. T < T_c. The gas in V₂ is a classical ideal gas.
c) The gas in V₁ is an ideal gas of fermions, and the gas in V₂ is an ideal gas of bosons. However both are now in the non-degenerate limit where their behavior is approaching that of a classical ideal gas (i.e. treat them in the small fugacity limit, keeping only the leading quantum correction to the classical equation of state).