Ground State of the Charged Bose Gas

Abstract

The ground-state energy of the charged Bose gas is calculated by the pair-correlation variational method of Girardeau and Arnowitt. The method is exact in the high-density limit ($r_s\ll 1$) and provides a variational extrapolation to intermediate densities. The leading terms of the high-density expansion, obtained by iteration of the variational integral equation, are $u_0=-0.804\mathrm{}{r_s}^{-\frac{3}{4}}-\left(\frac{1}{8}\right)\ln r_s+O\left(1\right)\mathrm{}$, where $u_0$ is the ground-state energy per particle in Rydbergs and $r_s$ is the ratio of the mean interparticle spacing to the Bohr radius. The first term was obtained previously by Foldy, but the logarithmic term is new; it is related to screening of the long-range correlations at a distance $r_0\sim {r_s}^{-\frac{1}{2}}{\rho }^{-\frac{1}{3}}$, in analogy with the logarithmic term in the correlation energy of the electron gas. Results of numerical solutions for the intermediate-density region are presented, ranging up to $r_s=10\mathrm{}$. On the basis of a comparison with the energy calculated from the known low-density expansion, it is estimated that the transition into Wigner's electron crystal (here a boson crystal) should occur at $r_s\sim 5$.