Kirzhnits gradient expansion for a D-dimensional Fermi gas

TL;DR

This paper derives the gradient correction coefficient for the kinetic energy density of a D-dimensional ideal Fermi gas using the Kirzhnits semiclassical expansion, revealing that in two dimensions these corrections vanish entirely.

Contribution

It provides a new derivation of the gradient correction coefficient for the kinetic energy density in arbitrary dimensions using the Kirzhnits expansion, including the special case of two dimensions.

Findings

The correction coefficient is (D-2)/3D for dimensions D.

In two dimensions, the gradient corrections vanish to all orders.

The approach yields the differential equation for the density profile and the density functional.

Abstract

For an ideal D-dimensional Fermi gas under generic external confinement we derive the correcting coefficient $(D-2)/3D$ of the von Weizsacker term in the kinetic energy density. To obtain this coefficient we use the Kirzhnits semiclassical expansion of the number operator up to the second order in the Planck constant $\hbar$. Within this simple and direct approach we determine the differential equation of the density profile and the density functional of the Fermi gas. In the case D=2 we find that the Kirzhnits gradient corrections vanish to all order in $\hbar$.