Langmuir waves dispersion in semi-relativistic spinless quantum plasmas

TL;DR

This paper investigates the effects of semi-relativistic corrections, derived from the Darwin Hamiltonian, on Langmuir wave dispersion in spinless quantum plasmas, highlighting new contributions to wave behavior.

Contribution

It introduces a semi-relativistic quantum hydrodynamics framework based on the Darwin Hamiltonian, detailing new terms affecting Langmuir wave dispersion in quantum plasmas.

Findings

Relativistic corrections modify the Langmuir wave spectrum.

New terms analogous to Darwin contributions appear in the Euler equation.

The semi-relativistic quantum Bohm potential is derived.

Abstract

Many particle quantum hydrodynamics based on the Darwin Hamiltonian (the Hamiltonian corresponding to the Darwin Lagrangian) is considered. A force field appearing in corresponding Euler equation is considered in details. Contributions from different terms of the Darwin Hamiltonian in the Euler equation are traced. For example, the relativistic correction to the kinetic energy of particles leads to several terms in the Euler equation, these terms have different form. One of them has a form similar to a term appearing from the Darwin term. Hence, the two different mechanisms give analogous contributions in wave dispersion. Microscopic analog of the Biot-Savart law, called the current-current interaction and describing an interaction of moving charges via the magnetic field, is also included in our description. The semi-relativistic generalization of the quantum Bohm potential is obtained. Contribution of the relativistic effects in the spectrum of plasma collective excitations is considered.