On the harmonic oscillator wavefunction modified to account for variations in the width of a trapped Bose gas

We propose to augment the harmonic oscillator wavefunction so that it can describe changes in the width of a trapped Bose gas under variations of temperature and interactions. This is achieved by the inclusion of a variational parameter which describes the width of the Bose gas. The goal is to enable a simple evaluation of the thermodynamic properties by the application of the second quantized many-body Hamiltonian. It is demonstrated that the width of this wavefunction decreases with increasing temperature until the zeroth harmonic oscillator state vanishes completely close to the transition temperature. This is a manifestation of the depletion of the harmonic oscillator ground state. Beyond the critical temperature of the Bose gas, the width asymptotically approaches a constant value because the system cannot shrink below a certain limit due to the barrier of repulsive interactions between the bosons. Some previous work having made use of the standard harmonic oscillator wavefunction to describe strongly interacting systems at finite TT is mildly criticized. This is because the harmonic oscillator wavefunction in its common form does not allow changes in the width of a trapped Bose gas in response to changes in the interactions and/or temperature.