Statistical Mechanics · Problem 5.7

The Theory of Simple Gases — R. K. Pathria

Physics — Statistical Mechanics · classical-limit-partition-function-interacting

Problem

Show that the quantum-mechanical partition function of a system of

N

interacting particles approaches the classical form

Q_N(V,T) = \frac{1}{N!h^{3N}} \int e^{-\beta E(q,p)}\,(d^{3N}q\,d^{3N}p)

as the mean thermal wavelength

\lambda

becomes much smaller than (i) the mean interparticle distance

(V/N)^{1/3}

and (ii) a characteristic length

r_0

of the interparticle potential.