Corrections to Thomas-Fermi densities at turning points and beyond

TL;DR

This paper derives simple, accurate semiclassical formulas for fermion densities near turning points in one-dimensional potentials, improving upon Thomas-Fermi theory without complex sums or derivatives.

Contribution

It introduces closed-form, uniform semiclassical approximations that include leading corrections to Thomas-Fermi densities for non-interacting fermions.

Findings

High accuracy of the derived formulas near turning points

Formulas are simple and involve no sums or derivatives

Applicable to various one-dimensional potentials

Abstract

Uniform semiclassical approximations for the number and kinetic-energy densities are derived for many non-interacting fermions in one-dimensional potentials with two turning points. The resulting simple, closed-form expressions contain the leading corrections to Thomas-Fermi theory, involve neither sums nor derivatives, are spatially uniform approximations, and are exceedingly accurate.