Thermodynamics, Statistical Physics, and Quantum Mechanics

139

The averageproceeds in precisely the sameway,yielding

b) The heat capacity, at constantvolume is equal to Frompart

(a), we have

where we now sumover the 3-space andmomentumdegrees offreedom per
atom. The heatcapacity,

is the law of Dulong and Petit.

c) Now take the average:

Integration by parts yields

where the prime on the productsign in thefirst term indicates that we

integrateover all except then the firstterm in the

THERMODYNAMICS AND STATISTICAL PHYSICS

If