PHYS5016 Statistical Mechanics

1 (a) In your own words explain the difference between a microcanonical ensemble and a canonical ensemble. [2]

(b) (i) Write down the partition function for a magnetic particle of spin 1 that has gyromagnetic ratio (ratio of the magnetic moment to the angular momentum) γ, in a magnetic field B. Assuming that they are distinguishable and non-interacting, show that the internal energy, U, for N such particles at temperature T is given by:

[5]

(ii) By considering the magnitude of the magnetic moment M = |U/B|, show that the system obeys Curie’s law at low values of B/T. [3]

(c) You are set the task of building a statistical mechanical model of a diatomic gas. Briefly list the steps that you would take and the principles that you would apply, and indicate how you would be able to obtain three relevant thermodynamic quantities. No detailed mathematical derivation is required. [6]

2 This question explores Bose-Einstein condensation in a harmonic potential. Consider a gas of N spin-zero, non-relativistic, identical, non-interacting bosons of mass m at temperature T. The corresponding distribution function fBE in energy is:

This gas is trapped in a 3-d harmonic potential (for example, a magnetic trap) given by:

for angular frequency ω.

(a) Explain why the single particle quantum states have energies:

for integers nx, ny, and nz. [2]

For large E , the number of states with energy less than E is given by:

(b) Briefly justify the form of this expression for N(E). (It is not necessary to derive it.) [2]

(c) Using the expression for N(E), show that the density of states, g(E), is given by: