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Simple Analytical Particle and Kinetic Energy Densities for a Dilute Fermionic Gas in a d-Dimensional Harmonic Trap

We derive simple analytical expressions for the particle density $ρ(r)$ and the kinetic energy density $τ(r)$ for a system of noninteracting fermions in a $d-$dimensional isotropic harmonic oscillator potential. We test the Thomas-Fermi (TF, or local-density) approximation for the functional relation $τ[ρ]$ using the exact $ρ(r)$ and show that it locally reproduces the exact kinetic energy density $τ(r)$, {\it including the shell oscillations,} surprisingly well everywhere except near the classical turning point. For the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of $τ_{TF}[ρ(r)]$ yields the {\it exact} total kinetic energy.