Exact and asymptotic local virial theorems for finite fermionic systems

TL;DR

This paper extends local virial theorems for fermionic systems to various potentials, demonstrating their exactness for linear potentials and their practical accuracy for moderate particle numbers through semiclassical theory and numerical tests.

Contribution

It generalizes local virial theorems to linear potentials and the 1D box, and introduces semiclassical-based generalized theorems supported by numerical validation.

Findings

Local virial theorems hold exactly for linear potentials in any dimension.

Generalized theorems are supported by semiclassical theory relating density oscillations to classical orbits.

Theorems are accurate for moderate particle numbers despite being asymptotic.

Abstract

We investigate the particle and kinetic-energy densities for a system of $N$ fermions confined in a potential $V(\bfr)$. In an earlier paper [J. Phys. A: Math. Gen. {\bf 36}, 1111 (2003)], some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point $\bfr$, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper we show that these {\it local virial theorems} (LVT) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are supported by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although they formally are only valid asymptotically for large particle numbers $N$, we show that they practically are surprisingly accurate also for moderate values of $N$.