We present a derivation of the dispersion relation for electrostatic oscillations in a zero-temperature quantum plasma, in which degenerate electrons are governed by the Wigner equation, while non-degenerate ions follow the classical fluid equations. The Poisson equation determines the electrostatic wave potential. We consider parameters ranging from semiconductor plasmas to metallic plasmas and electron densities of compressed matter such as in laser compression schemes and dense astrophysical objects. Owing to the wave diffraction caused by overlapping electron wave function because of the Heisenberg uncertainty principle in dense plasmas, we have the possibility of Landau damping of the high-frequency electron plasma oscillations at large enough wavenumbers. The exact dispersion relations for the electron plasma oscillations are solved numerically and compared with the ones obtained by using approximate formulas for the electron susceptibility in the high- and low-frequency cases.