Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas

Abstract

The well-known first two terms in the asymptotic density series for the ground-state energy of a Bose gas, ${E}_{0}=2\ensuremath{\pi}N\ensuremath{\rho}a[1+(\frac{128}{15\ensuremath{\surd}\ensuremath{\pi}}){(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}]$, where $a$ is the scattering length of the pair potential, is ordinarily obtained by summing an infinite set of graphs in perturbation theory. We show here how this same series may be obtained by elementary methods. Our method offers the advantages of simplicity and directness. Another advantage is that the hard-core case can be handled on the same basis as a finite potential, no pseudopotential being required. In fact, the analysis of the hard-core potential turns out to be simpler than for a finite potential, as is the case in elementary quantum mechanics. In an Appendix we discuss the high-density situation and show that for a certain class of potentials Bogoliubov's theory is correct in this limit. Thus, Bogoliubov's theory, which is never correct at low density unless a pseudopotential is introduced, is really a high-density theory.