The Density Distribution of a Weakly Interacting Bose Gas in an External Potential*

Spin-polarized atomic hydrogen (H$) at densities less than or of order 1019 cm -3 is a weakly interacting Bose gas. 1 To stabilize the ato~ns against recombination into molecules requires a strong magnetic field. In practice the magnetic field is nonuniform and serves to confine the gas. 2'3 It is of interest, therefore, to study theoretically the behavior of a weakly interacting Bose gas in an inhomogeneous potential. This problem has of course been examined previously, in the context of superfluid 4He, both at low temperatures, where the mean field equations of Gross and Pitaevskii apply, and near the critical point T = To, where scaling ideas a r e u s e d . 4 In 4He away from Tc healing lengths are very small and of largely theoretical interest. In H$, however, Walraven and Silvera pointed out that the change in the density profile that accompanies Bose condensation is dramatic and susceptible to direct experimental measurement ) '5 Our goal in this paper will therefore be to generalize the Gross-Pitaevskii theory to finite temperatures but avoid temperatures so close to T~ that critical fluctuations are important. This is naturally accomplished through the finite-temperature selfconsistent Har t ree -Foek equations that we derive variationally in Section 2.