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Ground State and Excited States of a Confined Bose Gas

The Bogoliubov approximation is used to study the ground state and low-lying excited states of a dilute gas of $N$ atomic bosons held in an isotropic harmonic potential characterized by frequency $ω$ and oscillator length $d_0$. By assumption, the self-consistent condensate has a macroscopic occupation number $N_0 >> 1$, with $N-N_0 << N_0$. For negative scattering length $ -|a|$, a simple variational trial function yields an estimate for the critical condensate number $N_{0\,c} = \big({8π/25\sqrt{5}}\,\big)^{1/2}\,(d_0/|a|) \approx 0.671\,(d_0/|a|)$ at the onset of collapse. For positive scattering length and large $N_0 >>d_0/a$, the spherical condensate has a well-defined radius $R >> d_0$, and the low-lying excited states are compressional waves localized near the surface. The frequencies of the lowest radial modes ($n = 0$) for successive values of orbital angular momentum $l$ form a rotational band $ω_{0l} \approx ω_{00} + {1\over 2} l(l+1)\,ω\,(d_0/R)^2$, with $ω_{00} $ somewhat larger than $ω$.