Leading corrections to local approximations II (with turning points)

TL;DR

This paper analyzes quantum corrections to Thomas-Fermi theory for 1D fermions, deriving universal formulas for particle distribution and energy, and tests their accuracy across various potentials, highlighting limitations near turning points.

Contribution

It introduces new analytical formulas for quantum corrections in semiclassical approximations, especially near turning points, and explores their accuracy and limitations in 1D fermionic systems.

Findings

Universal formulas for particle leakage beyond turning points.

Semiclassical densities are correctly normalized in the limit.

Approximations work well except at mid-phase points of evanescent regions.

Abstract

Quantum corrections to Thomas-Fermi (TF) theory are investigated for noninteracting one-dimensional fermions with known uniform semiclassical approximations to the density and kinetic energy. Their structure is analyzed, and contributions from distinct phase space regions (classically-allowed versus forbidden at the Fermi energy) are derived analytically. Universal formulas are derived for both particle numbers and energy components in each region. For example, in the semiclassical limit, exactly 1/(6\pi3^{1/2}) of a particle leaks into the evanescent region beyond a turning point. The correct normalization of semiclassical densities is proven analytically in the semiclassical limit. Energies and densities are tested numerically in a variety of one-dimensional potentials, especially in the limit where TF theory becomes exact. The subtle relation between the pointwise accuracy of the semiclassical approximation and integrated expectation values is explored. The limitations of the semiclassical formulas are also investigated when the potential varies too rapidly. The approximations are shown to work for multiple wells, except right at the mid-phase point of the evanescent regions. The implications for density functional approximations are discussed.