Balance equations in semi-relativistic quantum hydrodynamics

Abstract

Аннотация. Method of the quantum hydrodynamics has been applied in quantum plasmas studies. As the first step in our consideration, derivation of classical semi-relativistic (i. e. described by the Darwin Lagrangian on microscopic level) hydrodynamical equations is given after a brief review of method development. It provides better distinguishing between classic and quantum semi-relativistic effects. Derivation of the classical equations is interesting since it is made by a natural, but not very widespread method. This derivation contains explicit averaging of the microscopic dynamics. Derivation of corresponding quantum hydrodynamic equations is presented further. Equations are obtained in the five-momentum approximation including the continuity equation, Euler and energy balance equations. It is shown that relativistic corrections lead to presence of new quantum terms in expressions for a force field, a work field etc. The semi-relativistic generalization of the quantum Bohm potential is obtained. Quantum part of the energy current, which is an analog of the quantum Bohm potential for the energy evolution equation, is derived. The Langmuir wave dispersion in semi-relativistic quantum plasmas, corresponding to the Darwin Lagrangian, is also considered to demonstrate contribution of semi-relativistic effects on basic plasma phenomenon.

Separated spin evolution quantum hydrodynamics of degenerate electrons with spin–orbit interaction and extraordinary wave spectrum

“…This method allows us to include the spin–orbit interaction (Andreev & Kuzmenkov 2011; Andreev & Kuz’menkov 2012) and other relativistic effects (Andreev & Kuz’menkov 2007; Ivanov et al. 2014; Andreev 2015 a ; Ivanov et al. 2015).…”

“…The spin-1/2 QHD of plasmas can be directly derived from the many-particle Pauli equation (Kuz'menkov et al 2001a;Kuz'menkov, Maksimov & Fedoseev 2001b). This method allows us to include the spin-orbit interaction (Andreev & Kuzmenkov 2011;Andreev & Kuz'menkov 2012) and other relativistic effects (Andreev & Kuz'menkov 2007;Ivanov et al 2014;Andreev 2015a;Ivanov et al 2015). Next, the QHD equations can be represented in the form of the nonlinear Schrödinger equation or the nonlinear Pauli equation (Kuz'menkov & Maksimov 1999;Kuz'menkov et al 2001a;Andreev & Kuz'menkov 2012;Andreev 2014Andreev , 2015a.…”

“…(2015). There is another quantum-relativistic force which appears via the simultaneous consideration of the Coulomb interaction and the semi-relativistic part of the kinetic energy (Ivanov, Andreev & Kuz’menkov 2014; Ivanov et al. 2015).…”

“…The semi-relativistic part of the quantum Bohm potential is derived in Ivanov et al. (2014, 2015) along with the current–current interaction (CCI). The CCI leads to three kinds of terms in the Euler equation: the first of them is , the classic semi-relativistic forces, and the quantum semi-relativistic forces (Ivanov et al.…”

“…The CCI leads to three kinds of terms in the Euler equation: the first of them is , the classic semi-relativistic forces, and the quantum semi-relativistic forces (Ivanov et al. 2014). Some works (see for instance Asenjo et al.…”