Variational study of a dilute Bose condensate in a harmonic trap

A two-parameter trial condensate wave function is used to find an approximate variational solution to the Gross-Pitaevskii equation for $N_0$ condensed bosons in an isotropic harmonic trap with oscillator length $d_0$ and interacting through a repulsive two-body scattering length $a>0$. The dimensionless parameter ${\cal N}_0 \equiv N_0a/d_0$ characterizes the effect of the interparticle interactions, with ${\cal N}_0 \ll 1$ for an ideal gas and ${\cal N}_0 \gg 1$ for a strongly interacting system (the Thomas-Fermi limit). The trial function interpolates smoothly between these two limits, and the three separate contributions (kinetic energy, trap potential energy, and two-body interaction energy) to the variational condensate energy and the condensate chemical potential are determined parametrically for any value of ${\cal N}_0$, along with illustrative numerical values. The straightforward generalization to an anisotropic harmonic trap is considered briefly.