Hyperspherical Description of the Degenerate Fermi Gas: S-wave Interactions

TL;DR

This paper introduces a novel theoretical framework for describing a spherically trapped degenerate Fermi gas at zero temperature using an effective one-dimensional Schrödinger equation based on the rms radius, providing insights into its ground state and excitation properties.

Contribution

It develops a hyperspherical approach that simplifies the many-body problem to a linear 1D Schrödinger equation, offering a new perspective on Fermi gas physics compared to Hartree-Fock methods.

Findings

Accurately predicts ground state energy, rms radius, and peak density.

The breathing mode frequency matches sum rule calculations.

Deviates from Hartree-Fock predictions for excitation frequencies.

Abstract

We present a unique theoretical description of the physics of the spherically trapped $N$-atom degenerate Fermi gas (DFG) at zero temperature based on an ordinary Schr\"{o}dinger equation with a microscopic, two body interaction potential. With a careful choice of coordinates and a variational wavefunction, the many body Schr\"{o}dinger equation can be accurately described by a \emph{linear}, one dimensional effective Schr\"{o}dinger equation in a single collective coordinate, the rms radius of the gas. Comparisons of the energy, rms radius and peak density of ground state energy are made to those predicted by Hartree-Fock (HF). Also the lowest radial excitation frequency (the breathing mode frequency) agrees with a sum rule calculation, but deviates from a HF prediction.