A Guide to Physics Problems. Part 2. Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 171

4.65 Two-Dimensional Fermi Gas (MIT,

Wisconson-Madison)

a) At the noninteractingfermions will be distributedamong the

available states so that the total energy is a minimum. The number of

quantumstatesavailable to afermion confined to a box of area Awith

momentumbetween and isgiven by

where the multiplicity and the spin The N fermions
at fill all the states of momentumfrom 0 to We can therefore
calculatethis maximum momentum from

The Fermi energy for thisnonrelativistic gas is simply

Using (S.4.65.2) and(S.4.65.3) weobtain

For and (S.4.65.5)becomes

b) The total energy of the gas

Substituting from(S.4.65.2) into (S.4.65.7)gives