A note on Fermi energy of Fermi gas trapped under generic power law potential in $d$-dimension

TL;DR

This paper calculates the average energy per fermion in a trapped Fermi gas under general power law potentials across different dimensions, showing how it approaches the Fermi energy as dimension increases, regardless of the potential's specifics.

Contribution

It generalizes previous results by deriving a formula for average energy in trapped Fermi gases with arbitrary power law potentials in any dimension.

Findings

Average energy depends on power law exponent at finite dimensions.

As dimension tends to infinity, average energy equals Fermi energy for all potentials.

The results unify free and trapped Fermi gases under a common framework.

Abstract

Average energy per fermion in case of Fermi gas with any kinematic characteristic, trapped under most general power law potential in $d$ dimension has been calculated at zero temperature. In a previous paper (M. Acharyya, Eur. J Phys. 31 L89 (2010)) it was shown, in case of free ideal Fermi gas as dimension increases average energy approaches to Fermi energy and in infinite dimension average energy becomes equal to Fermi energy at $T=0$. In this letter it is shown that, for trapped system at finite dimension the average energy depends on power law exponent, but as dimension tends to infinity average energy coincides with Fermi energy for any power law exponent. The result obtained in this manuscript is more general as we can describe free system as well as any trapped system with appropriate choice of power law exponent and true for any kinematic parameter.