Exact local density method for linear harmonic oscillator

Using an exact form for the particle density for N particles moving independently in a linear harmonic oscillator potential, a differential equation is derived for the particle density. This is then shown to be exactly that which is obtained from a local density assumption for the kinetic energy density t, in the form −1/2Σψi(δ2/δx2) ψi. The form of the kinetic energy density functional is then discussed in this example. The implications for wider classes of potential are briefly referred to.