Some exact results for a trapped quantum gas at finite temperature

TL;DR

This paper derives exact analytical formulas for particle and kinetic energy densities of noninteracting quantum gases in harmonic traps at finite temperature, and explores their properties and perturbations due to weak interactions.

Contribution

It provides new exact expressions for densities in 2D and 3D traps and analyzes the effects of weak interactions and temperature on energy level splittings.

Findings

Exact density expressions for 2D and 3D systems.

In 2D, the |l|-degeneracy remains unlifted at lowest order.

Zero-temperature densities reduce to Thomas-Fermi form at large N.

Abstract

We present closed analytical expressions for the particle and kinetic energy spatial densities at finite temperatures for a system of noninteracting fermions (bosons) trapped in a d-dimensional harmonic oscillator potential. For d=2 and 3, exact expressions for the N-particle densities are used to calculate perturbatively the temperature dependence of the splittings of the energy levels in a given shell due to a very weak interparticle interaction in a dilute Fermi gas. In two dimensions, we obtain analytically the surprising result that the |l|-degeneracy in a harmonic oscillator shell is not lifted in the lowest order even when the exact, rather than the Thomas-Fermi expression for the particle density is used. We also demonstrate rigorously (in two dimensions) the reduction of the exact zero-temperature fermionic expressions to the Thomas-Fermi form in the large-N limit.