Dispersion of waves in a weakly relativistic quantum plasma and beam of particles

A brief review of works devoted to quantum hydrodynamics is given. A system of quantum hydrodynamics equations is presented for a weakly relativistic system of charged particles described by the Darwin Hamiltonian. This system of equations involves the continuity equation and the momentum balance equation for such system of particles. The role of the quantum terms in the momentum balance equation is analyzed. As an illustration of principally new effects arising in the examined approximation, longitudinal waves in electron quantum plasma and waves in a beam of charged particles are considered.