Statistical Mechanics - Physics at Oregon State University

88 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES.
S(T, µ, V ) = NkB −
fM (ɛo; T, µ) log(fM (ɛo; T, µ))
where the sum is over orbitals.
Problem 5.
o
Consider a system of independent particles. The number of orbitals with
energy between E and E + dE is given by N(E)dE. The function N(E) is
called the density of states. One measures the expectation value of a certain
operator O. For a particle in an orbital o the value of the operator depends only
on the energy of the orbital, or Oo = O(ɛo). Show that in the thermodynamic
limit the ensemble average of the operator is given by
∞
< O >= O(E)N(E)f(E; T, µ)dE
−∞
where f(E; T, µ) is the distribution function for these particles.
Problem 6.
The orbital energies of a system of Fermions are given by ɛi = i∆ , with
∆ > 0 and i = 1, 2, 3, · · · , ∞. These energies are non-degenerate. If the system
has N particles, show that the low temperature limit of the chemical potential
gives ɛF = (N + 1
2 )∆.
Problem 7.
The entropy for a system of independent Fermions is given by
S = −kB (fF D log(fF D) + (1 − fF D) log(1 − fF D))
o
Calculate lim
T →0 fF D(ɛ, T, µ) for ɛ < µ ,ɛ = µ , and ɛ > µ.
The number of orbitals with energy ɛo equal to the Fermi energy ɛF is M.
Calculate the entropy at T = 0 in this case.
Explain your answer in terms of a multiplicity function. Pay close attention
to the issue of dependent and independent variables.