Bose-Einstein Gas with Repulsive Interactions: General Theory

Abstract

The properties of a Bose-Einstein gas with repulsive interparticle interactions are determined at low temperature. It is shown that the contribution to the energy arising from particle excitation with small momentum transfers computed in conventional perturbation theory is divergent, but that this difficulty can be avoided by an alternative procedure. An exact method is developed for dealing with that part of the Hamiltonian which gives rise to the perturbation divergence and an exact solution is obtained. The treatment of the region of large momentum transfers is then carried out by introducing the concept of the scattering length of the interaction. The energy is given by the well-known result of the optical theorem corrected by a series in powers of ${\left(\rho a^3\right)}^{\frac{1}{2}}$, where $\rho$ is the density and $a$ the scattering length. Examination of the processes contributing to the energy shows that 95.8% of the correction of order ${\left(\rho a^3\right)}^{\frac{1}{2}}$ results from a simple alteration of the perturbation energy denominators which takes into account the interaction energy of excited pairs with the unexcited pairs of the medium. The remaining 4.2% arises from multiple particle excitation. These methods also predict the existence of phonon excitation with phonon energies that are linear in the phonon momentum for small excitation and approach $\frac{q^2}{2M}$ for large momentum. The detailed consideration of this spectrum forms part of the following paper.