G25.2651: Statistical Mechanics

Problem set #7

Due April 28, 2003

1.

Consider a system of N negatively charged spin-1/2 fermions in an external magnetic field , i.e., pointing along the positive z direction. Each particle carries a charge -q (q;SPMgt;0), however, assume that Coulomb interactions between the particles can be neglected. Thus, the Hamiltonian for the system is

where c is the speed of light and is called the vector potential. is related to the magnetic field by

One possible choice for is

a.

The particles occupy a cubic box of side L with periodic boundary conditions. Find the energy levels of the system.

Hint: Show that the Schrödinger equation is separable and derive the single-particle wave equation. Try a solution to the single-particle equation of the form

and show that satisfies a harmonic oscillator equation with frequency and equilibrium position . Assume L is much larger than the range of .

b.

Calculate the grand canonical partition function, in the high temperature ( ) and thermodynamic limits. Note, in this limit, it is sufficient to work to first order in the fugacity, . WARNING!! Beware of degeneracies in the energy levels besides the spin degeneracy.

c.

The magnetic susceptibility per unit volume is defined by

where is the average induced magnetization per unit volume along the direction of the magnetic field and is given by

Calculate and for this system. Curie's Law for the magnetic susceptibility states that . Is your result in accordance with Curie's Law? If not, explain why it should not be.

d.

If the fermions are replaced by Boltzmann particles, does the resulting susceptibility still accord with Curie's Law?

Hint: Consider using the canonical ensemble in this case.

2a.

Show that the fluctuations in the average occupation number of a single-particle energy level for an ideal Fermi gas satisfies

2b.

Show that the fluctuations in the average occupation number of a single-particle energy level for an ideal Bose gas satisfies

What does this say about the size of the fluctuations possible in an ideal Bose gas?

3.

Show that Bose-Einstein condensation does not occur for an ideal Bose gas in one dimension. Is condensation possible in two dimensions?

4.

Consider a system of N identical bosons. Each particle can occupy one of two single particle energy levels with energies and . Calculate the temperature at which the average thermal occupation of the lower energy level is twice that of the higher energy level. Assume N;SPMgt;;SPMgt;1.

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