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) an d the kinetic energy density tau (r) for a system of noninteracting fermion s in a d-dimensional isotropic harmonic oscillator potential. We test the T homas-Fermi (TF, or local-density) approximation for the functional relatio n T[rho] using the exact rho (r) and show that it locally reproduces the ex act kinetic energy density tau (r). including the shell oscillations, surpr isingly well everywhere except near the classical turning point. Fur the sp ecial case of two dimensions (2D), we obtain the unexpected analytical resu lt that the integral of tau (TF)[rho (r)] yields the exact total kinetic en ergy.