Quantum Hard-Sphere Gas. II

Abstract

Using a method due to the author which correctly reproduces the scattering of two hard spheres for all relative momenta and all partial waves, we study the properties of a low-density hard-sphere boson gas at $T=0\mathrm{}$°K. The system is described by a Hamiltonian $\mathcal{H}$ defined for all relative distances of the $N$ particles. The terms of $\mathcal{H}$ corresponding to the forward motion of a particle through the system lead to the notion of an effective mass $m^{*}$ for each particle in excess of its actual mass $m$, with $\frac{m^{*}}{m}$ given by a monotonically increasing function of $\rho a^3$, where $\rho$ and $a$ are the number of particles per unit volume and the hard-sphere diameter, respectively. The excitation spectrum is linear at low momenta, displays a roton-like behavior for momenta $p\gtrsim \frac{\hslash }{a}$, possesses further extrema at higher momenta, and asymptotically approaches $\frac{p^2}{2m^{*}}$. The pair distribution function vanishes in the low-density limit for particle separations $r\le a$.