Thermodynamics, Statistical Physics, and Quantum Mechanics

138

The freeenergy associatedwith thecoordinate-dependentpart of the par-
tition function is

The averagepotentialenergy is given by

For we have aharmonicoscillator, and in agreementwith theequipar-
tition theorem(seeProblem4.42)

which alsoagreeswith the generalizedequipartition theorem.

Equipartition Theorem (Columbia, Boston)

a) For both ofthese averages the method is identical,since the Hamiltonian

depends on the samepower ofeither orq.Compose thefirst averageas

follows:

where theenergy is broken into the term and the rest of
the sum. Thesecondintegrals in thenumerator anddenominatorcancel,
so the remainingexpression may be written

where, asusual, Achange of variablesproduces apiecedependent
on and an integral that is not:

b) For and the averagepotentialenergy per particle

4.42

SOLUTIONS