1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 443

11.3. Determine how the average energy of an ideal gas of identicalfermions in one

dimension at zero temperature depends on density. Repeat for a gas in two

dimensions.

11.4. Consider an ideal gas of massless spin-1/2 fermions in a cubic periodic box

of sideL. The Hamiltonian for the system is

Hˆ=

∑N

i=1

c|pˆi|,

wherecis the speed of light.

a. Calculate the equation of state in the high-temperature, low-density limit

up to second order in the density. What is the second virial coefficient?

What is the classical limit of the equation of state?

b. Calculate the Fermi energy,εF, of the gas.

c. Determine how the total energy depends on the density.

∗11.5. Problem 9.7 of Chapter 9 considers the case ofN charged fermions in a

uniform magnetic field. In that problem, the eigenfunctions and eigenvalues

of the Hamiltonian were determined. This problem uses your solution for

these eigenvalues and eigenfunctions.

a. Calculate the grand canonical partition functionZ(ζ,V,T) in the high-

temperature ( ̄hω/kT <<1) and thermodynamic limits. In this limit, it

is sufficient to work tofirst orderin the fugacity,ζ.

Hint:Beware of degeneracies in the energy levels besides the spin degen-

eracy.

b. The magnetic susceptibility per unit volume is defined by

χ=

∂M

∂B

,

whereMis the average induced magnetization per unit volume along the

direction of the magnetic field and is given by

M=

kT

V

(

∂lnZ

∂B

)

ζ,V,T

.

CalculateMandχfor this system. Curie’s Law for the magnetic suscep-

tibility states that|χ| ∝ 1 /T. Is your result in accordance with Curie’s

Law? If not, explain why it should not be.

c. 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.