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 $\rho (r)$ and the kinetic energy density $\tau (r)$ 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 $\tau [\rho ]$ using the exact $\rho (r)$ and show that it locally reproduces the exact kinetic energy density $\tau (r)$, {\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 ${\tau }_{TF}[\rho (r)]$ yields the {\it exact} total kinetic energy.