  Quantum effects in statistical mechanics are important when the thermal
wavelength is of the order of, or greater than, the mean interatomic spacing.
This is examined at depth taking the example of a hard-sphere Boltzmann gas.
Using the virial expansion for the equation of state, it is shown that the
interaction energy of a classical hard-sphere gas is exactly zero. When the
(second) virial coefficient of such a gas is obtained quantum mechanically,
however, the quantum contribution to the interaction energy is shown to be
substantial. The importance of the semiclassical corrections to the interaction
energy shows up dramatically in such a system.
