Ground State and Excited States of a Confined Bose Gas

Alexander L. Fetter

doi:10.1103/PhysRevA.53.4245

paper

Ground State and Excited States of a Confined Bose Gas

arXiv:cond-mat/9510037 · doi:10.1103/PhysRevA.53.4245

Abstract

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 $N_{0 c} = (8 π /25 5)^{1/2} (d_{0} /∣ a ∣) \approx 0.671 (d_{0} /∣ a ∣)$ 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 $ω_{0 l} \approx ω_{00} + \frac{1}{2} l (l + 1) ω (d_{0} / R)^{2}$ , with $ω_{00}$ somewhat larger than $ω$ .

11 pages, plainTEX, no figures

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Abstract

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