Statistical Mechanics - Physics at Oregon State University

5.4. PROBLEMS FOR CHAPTER 5 117
(C) For T = ∞. (Note: this one you can get without any detailed calculation).
(D) For kBT ≫ ɛF , one more term beyond (C).
Problem 5.
The virial expansion is given by p
kT = ∞ j=1 Bj(T )
N j
V with B1(T ) = 1.
Find B2(T ) for non-interacting Fermions in a box.
Problem 6.
The energy of relativistic electrons is given by ɛp,s = p 2 c 2 + m 2 c 4 , which
is independent of the spin s. These particles are contained in a cubical box,
sides of length L, volume V .
(1) Calculate the Fermi energy ɛF = µ(T = 0) as a function of N and V .
(2) Calculate the internal energy U.
(3) Expand the integral in (2) in a power series, assuming that the density N
V
is very low.
(4) Expand the integral in (2) in a power series, assuming that the density N
V
is very high.
Problem 7.
Landau diamagnetism. The orbits of an electron in a magnetic field are also
quantized. The energy levels of the electron are now given by
ɛ(pz, j, α, s) = p2 z
2m
eB 1
+ (j +
mc 2 )
with pz = 2π
L l, l = 0, ±1, ±2, · · ·, and j = 0, 1, · · · . The quantum number
α counts the degeneracy and α runs from 1 to g = eBL2
2πc . The magnetic field
is along the z direction. We have ignored the energy of interaction between the
spin and the magnetic field.
(1) Give an expression for the grand partition function ζ(T, µ, V ).
(2) Calculate the magnetic susceptibility for T → ∞ ( no, zero is not an
acceptable answer, one more term, please) in a weak magnetic field.