Exact coherent states of a noninteracting Fermi gas in a harmonic trap

TL;DR

This paper derives exact analytical expressions for various densities and distributions of a noninteracting Fermi gas in a harmonic trap, revealing how these properties evolve under trap modulation and opening.

Contribution

It provides closed-form solutions for densities and distributions of a Fermi gas in a harmonic trap, including their behavior during trap modulation and release.

Findings

Particle and momentum distributions share the same shape.

Distribution dispersion varies inversely with density dispersion.

Stationary momentum distribution can have arbitrarily large or small dispersion after trap opening.

Abstract

Exact and closed-form expressions of the particle density, the kinetic energy density, the probability current density, and the momentum distribution are derived for a coherent state of a noninteracting Fermi gas, while such a state can be obtained from the ground state in a $d$-dimensional isotropic harmonic trap by modulating the trap frequency and shifting the trap center. Conservation laws for the relations of the densities are also given. The profile of the momentum distribution turns out to be identical in shape with that of the particle density, however, %as an observable manifestation of the uncertainty principle, the dispersion of the distribution increases (decreases) when that of the particle density is decreased (increased). The expressions are also applicable for a sudden and total opening of the trap, and it is shown that, after the opening, the gas has a stationary momentum distribution whose dispersion could be arbitrarily large or small.