Quantum-Mechanical Equation of State of a Hard-Sphere Gas at High Temperature

The quantum-mechanical free energy $F$ of a hard-sphere gas at high temperature is a series in powers of the thermal wavelength $\lambda ={\left(\frac{2\pi {\hslash }^2}{\mathrm{mkT}}\right)}^{\frac{1}{2}}$; the coefficients of this series can be expressed in terms of the classical correlation functions. The result to first order is $\frac{F}{\mathrm{Nk}T}=\frac{F^{\mathrm{}\left(0\right)\mathrm{}}}{\mathrm{Nk}T}+\frac{\pi }{\sqrt{2}} g_2\mathrm{}\left(a\right)\mathrm{}{a}^2\rho \lambda ,$ where $F^{\left(0\right)}$ is the classical free energy, $N$ the total number of particles, $\rho$ the number density, $kT$ Boltzmann's factor times the temperature, $a$ the hard-sphere diameter, $g_2\mathrm{}\left(a\right)\mathrm{}$ the classical pair-correlation function at contact. The corresponding expression for the pressure is $p=p^{\mathrm{}\left(0\right)\mathrm{}}+\frac{3}{2\sqrt{2}}\frac{\lambda }{a}{\rho }^2\frac{\partial }{\partial \rho }\frac{p^{\mathrm{}\left(0\right)\mathrm{}}}{\rho }$ where $p^{\left( 0\right)}$ is the classical pressure. The principle of a systematic derivation of higher-order terms in $\lambda$ is given.