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 T[rho] using the exact rho (r) and show that it locally reproduces the exact kinetic energy density tau (r). including the shell oscillations, surprisingly well everywhere except near the classical turning point. Fur the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of tau (TF)[rho (r)] yields the exact total kinetic energy.