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Semiclassical theory for spatial density oscillations in fermionic systems

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 $τ_{\text{TF}}[ρ]$ reproduces the oscillations in the quantum-mechanical densities to first order in the oscillating parts.