Chapter 1 Quantum Hydrodynamics

Abstract Quantum plasma physics is a rapidly evolving research field with a very inter-disciplinary scope of potential applications, ranging from nano-scale science in condensed matter to the vast scales of astrophysical objects. The theoretical de-scription of quantum plasmas relies on various approaches, microscopic or macro-scopic, some of which have obvious relation to classical plasma models. The ap-propriate model should, in principle, incorporate the quantum mechanical effects such as diffraction, spin statistics and correlations, operative on the relevant scales. However, first-principle approaches such as quantum Monte Carlo and density func-tional theory or quantum-statistical methods such as quantum kinetic theory or non-equilibrium Green’s functions require substantial theoretical and computational ef-forts. Therefore, for selected problems, alternative simpler methods have been put forward. In particular, the collective behavior of many-body systems is usually de-scribed within a self-consistent scheme of particles and fields on the mean-field level. In classical plasmas, further simplifications are achieved by a transition to hydrodynamic equations. Similar fluid-type descriptions for quantum plasmas have been proposed and widely used in the recent decade. This chapter is devoted to an overview of the main concepts of quantum hydrodynamics (QHD), thereby criti-cally analyzing its validity range and its main limitations. Furthermore, the results of the linearized QHD in unmagnetized and magnetized plasmas and a few nonlin-ear solutions are examined with illustrations. The basic concepts and formulation of particle-particle interactions are also reviewed at the end, indicating their possible consequences in quantum many-body problems