Derivation of the differential equation for the Slater sum and of the differential virial theorem using the Wigner transform

Using the Wigner transform, we present an alternative derivation of the partial differential equation satisfied by the Slater sum, which is the diagonal element of the canonical Bloch density matrix. This is done in one dimension for the case of a general confining potential and also for the case of N independent fermions harmonically confined in d dimensions. We also present a simple proof of the so-called differential virial theorem for each of these cases.