Semiclassical theory for spatial density oscillations in fermionic systems

TL;DR

This paper extends semiclassical methods to analyze spatial density oscillations in fermionic systems with spherical symmetry, demonstrating good agreement with quantum calculations and providing insights into oscillation origins.

Contribution

The authors generalize a semiclassical density theory to higher dimensions, regularize it near symmetry-breaking points and turning points, and validate it against quantum results for various potentials.

Findings

Semiclassical theory accurately reproduces quantum densities for moderate particle numbers.

Two types of oscillations are identified, linked to radial and non-radial orbits.

Thomas-Fermi functional captures first-order oscillations in densities.

Abstract

We investigate the particle and kinetic-energy densities for a system of $N$ fermions bound in a local (mean-field) potential $V(\bfr)$. We generalize a recently developed semiclassical theory [J. Roccia and M. Brack, Phys. Rev.\ Lett. {\bf 100}, 200408 (2008)], in which the densities are calculated in terms of the closed orbits of the corresponding classical system, to $D>1$ dimensions. We regularize the semiclassical results $(i)$ for the U(1) symmetry breaking occurring for spherical systems at $r=0$ and $(ii)$ near the classical turning points where the Friedel oscillations are predominant and well reproduced by the shortest orbit going from $r$ to the closest turning point and back. For systems with spherical symmetry, we show that there exist two types of oscillations which can be attributed to radial and non-radial orbits, respectively. The semiclassical theory is tested against exact quantum-mechanical calculations for a variety of model potentials. We find a very good overall numerical agreement between semiclassical and exact numerical densities even for moderate particle numbers $N$. Using a "local virial theorem", shown to be valid (except for a small region around the classical turning points) for arbitrary local potentials, we can prove that the Thomas-Fermi functional $\tau_{\text{TF}}[\rho]$ reproduces the oscillations in the quantum-mechanical densities to first order in the oscillating parts.