Mean–Field Approximation. Quantum Vlasov Equation. Collective Effects

Abstract

This chapter discusses the simplest many-body approximation—the mean field approximation which amounts to the complete neglect of correlations. Alternative names are Vlasov approximation (in the classical limit) or time-dependent Hartree (or Hartree-Fock), in the quantum literature. Performing a linearization with respect to the perturbing field we derive linear response quantities such as the longitudinal polarization and the dielectric function \(\epsilon (\mathbf{k},\omega )\). In the following, collective plasma oscillations and instabilities are discussed for 1D, 2D and 3D quantum systems. The chapter concludes with extensions beyond the linear regime, in particular with the formulation of a quasi-linear theory for quantum systems.