Thermodynamics, Statistical Physics, and Quantum Mechanics

176 SOLUTIONS

where (S.4.68.9) defines the equation of state. To find quantum
corrections to the classical equation of state (which corresponds to the case
expand the integral in (S.4.68.9), using as a small
parameter:

The first term, which we may call corresponds to a Boltzmann gas with

(see Problem 4.39), and the second term gives the first correction

Using the fact that, for small corrections (see, for instance, Landau and

Lifshitz, Statistical Physics, Sect. 24),

we can write the first quantum correction to the free energy F. Using the

classical expression for in terms of and V gives the result to the same

Using

we obtain, from (S.4.68.13),

and

Using and substituting (S.4.68.10) into (S.4.68.9), we have

accuracy:

and