Quantum effects in a system of Boltzmann hard spheres

The quantum contribution to the energy of a ‘Boltzmann’ gas consisting of hard spheres proves to be virtually constant up to very high temperatures where the thermal de Broglie wavelength constitutes only a small proportion of the hard sphere diameter. Consequently, the heat capacity of the system barely differs from the classical value of $(3/2)k_{\rm B}$ everywhere except in the lowest temperature region, where heat capacity as a function of temperature has the ‘Debye’ form but with a very low Debye temperature, of the order of several degrees. The line of equilibrium between a quantum crystal and liquid for a ‘Boltzmann’ system of hard spheres coincides with the classical one, with the exception of the very-low-temperature region. High-temperature quantum effects are revealed in the system under consideration in a kind of ‘bare’ form, while in the case of more realistic systems or models they can be masked by the complex behavior of other components of the total energy.