Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 33

4.67 Ultrarelativistic Electron Gas (Stony Brook)

Derive the relation between pressure and volume of a free ultrarelativistic

electron gas at zero temperature.

4.68 Quantum Corrections to Equation of State

(MIT, Princeton, Stony Brook)

Consider anoninteracting, one-component quantum gas at temperature

with a chemical potential in a cubic volume V. Treat the separate cases

of bosons and fermions.

a)

b)

For a dilute system derive the equation of state in terms of tem-

perature pressureP,particle density and particlemass Do

this derivation approximately by keeping the leading and next-leading

powers of Interpret your results as an effective classical system.

At a given temperature, for which densities are your results valid?

4.69 Speed of Sound in Quantum Gases (MIT)

The sound velocity in a spin-1/2 Fermi gas is given at by

where is the mass of the gas particles, and is the number

density.

a) Show that

where is the chemical potential.

b) Calculate the sound velocity in the limit of zero temperature. Express

youranswer in terms of

c) Showthat

in a Bose gas below the Bose–Einstein temperature.