Thermodynamics, Statistical Physics, and Quantum Mechanics

becomes a stepfunction. All thestatesabove acertainenergy,
are empty, andthe states below, are filled (seeFigure S.4.66). This
energy for an electron gas iscalled the Fermi energy. Physically, thisresults
from thesimplefactthat thetotalenergy of the gas should be aminimum.
However, we have toreconcile thiswith thePauli principle, whichprohibits
morethan oneelectron perquantumstate(i.e.,samemomentum and spin).
This means that thestates arefilled graduallyfromzero energy to the
limitingenergy, Thenumber ofstates accessible to afree particle with
absolute value ofmomentumbetween and is

In each ofthese states, we can put two electrons with opposite spin (up
and down), so if we consider the totalnumber of electrons,N,contained in
a box of volume V, then N is given by

Substituting we obtain

172

4.66 Nonrelativistic Electron Gas (Stony Brook,

Wisconsin-Madison, Michigan State)

SOLUTIONS

a) As the Fermi-Diracdistribution function