Variational study of 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 N-0=N(0)a/d(0) characterizes the effect of the interparticle interactions, with N-0 much less than 1 for an ideal gas and N-0 much greater than 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 N-0, along with illustrative numerical values. The straightforward generalization to an anisotropic harmonic trap is considered briefly.