Thermodynamics, Statistical Physics, and Quantum Mechanics

170 SOLUTIONS

where the last equationrepresents theideal gas lawat constanttempera-

ture. This yields the equation

Solvingthis cubic equation in thesmall limitgives

Substituting(S.4.63.9) into(S.4.63.4) yields (in the same approximation)

Quantum Statistics

4.64 Fermi Energy of a 1D Electron Gas

(Wisconsin-Madison)

For a one-dimensional gas thenumber of quantumstates in the interval

where andLis the “length” of themetal. The total number

of electronsN(which in this case is equal to thenumber of atoms) is

Therefore,

where is theatomic spacing. The Fermienergy

where is the electronmass.

is