Ground-state density profile of the weakly interacting Bose and Fermi gases confined in an arbitrary potential well

A theory is developed for the nonuniform ground-state density profile of weakly interacting quantum gases confined within an arbitrary potential well. Both Bose-Einstein and Fermi-Dirac cases are considered. This problem is of interest in connection with magnetically confined spin-aligned atomic hydrogen (${\mathrm{H}}_{\mathrm{\downarrow }}$) and deuterium (${\mathrm{D}}_{\mathrm{\downarrow }}$). The approach is based on a macroscopic formulation involving the principle of constancy of the chemical potential throughout the system. The main input is the internal chemical potential for which we use a local-density approximation (LDA). In turn we use for the LDA internal chemical potential standard results for the uniform interacting quantum gases. It is shown that generally, the systems mentioned develop macroscopically sharp surfaces (again, while in the gas phase). Density profiles are analytically calculated for the ideal Fermi gas and the leading-order weakly interacting Bose gas for the case of an arbitrary, symmetric power-law potential well. We also calculate the density profile including the next-highest-order correction in the interparticle interaction in the Bose case and the leading-order correction in the interparticle interaction in the Fermi case. The accuracy of the LDA density profiles is assessed. Numerical results are given appropriate for ${\mathrm{H}}_{\mathrm{\downarrow }}$ and ${\mathrm{D}}_{\mathrm{\downarrow }}$. For weakly interacting ${\mathrm{D}}_{\mathrm{\downarrow }}$ a notable contrast between the case of single hyperfine level occupation and the case of occupation of the lowest three hyperfine levels is pointed out.