Ground state and excited states of a confined condensed 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 omega and oscillator length d(0)=root HBAR/m omega. By assumption, the self-consistent condensate has a macroscopic occupation number N-0 much greater than 1, with N-N-0 much less than N-0. A linearized hydrodynamic description yields operator forms of the particle-conservation law and Bernoulli's theorem, expressed in terms of the small density fluctuation operator <(rho)over cap>' and velocity potential operator <(Phi)over cap>', along with the condensate density n(0) and velocity v(0). For positive scattering length a and large stationary condensate (N-0 much greater than d(0)/a and v(0)=0), the spherical condensate has a well-defined radius R(0) much greater than d(0), and the low-lying excited states are irrotational compressional waves localized near the surface. Approximate variational energies E(0l) of the lowest radial modes (n=0) for successive values of orbital angular momentum l form a rotational band given by E(0l)approximate to E(0 . 0)+1/2HBAR(2)l(l+1)/mR(0)(2) with radial zero-point energy E(0 . 0)proportional to HBAR omega(R(0)/d(0))(2/3)=(HBAR(2)m omega(4)R(0)(2))(1/3).