Many-particle quantum hydrodynamics: Basic principles and fundamental applications

Abstract

Basic principles and technics of many-particle quantum hydrodynamics are described in full details for the spinless case developed in L. S. Kuz'menkov and S. G. Maksimov, Theor. Math. Phys. 118, 227 (1999). This method has been developed for different physical systems. Necessity of technical details of derivation are given in this paper.We present a detailed derivation of the continuity, Euler, and energy balance equations from many particle Schrodinger equation. Interparticle interaction is explicitly considered as the Coulomb interaction. We show the QHD equations in a form suitable for low dimensional systems. The QHD equations in the cylindrical coordinates are presented as well. We present a method of nonlinear Schrodinger equation derivation from the QHD equations. We describe a method of exchange interaction calculation for two-and three-dimensional quantum plasmas. We discuss explicit form and area of applicability of equations of state for 1D, 2D, and 3D degenerate electrons. We explicitly consider equations of state for degenerate electron gas in an external magnetic field. Considering QHD equations in cylindrical coordinates we focus our attention on the quantum part of the inertia forces and contribution of the quantum Bohm potential to the spectrum of quantum cylindrical waves.We apply the QHD equations to small amplitude collective excitations in linear regime of QHD equations. We consider the Langmuir waves in 1D, 2D, and 3D quantum plasmas including the exchange Coulomb interaction. Considering quantum plasmas located in an external magnetic field we focus our attention on longitudinal waves propagating perpendicular to the external magnetic field. We present this consideration for 1D, 2D, and 3D plasmas. We discuss dependence of the exchange interaction on rate of polarization of spins in the external magnetic field. We include dependence of the pressure of degenerate electrons on the strength of the external magnetic field via the change of occupation of quantum states at application of an external magnetic field.QHD equations in the spherical coordinates are derived. Corresponding inertia forces including the quantum part are obtained. Collective excitations of 2DEG on spherical surface are considered.Equation of state for degenerate 2DEG on a spherical surface is obtained. Spectrum of the collective excitations corresponding to Langmuir waves is calculated.Many-particle QHD in the quasi-classic approximation is also derived. Comparison of the quasiclassic limit with the full scheme of the many-particle QHD is presented.

Exchange Coulomb interaction in nanotubes: Dispersion of Langmuir waves

“…The explicit derivation of the QHD equations in the cylindric coordinates (ρ = x 2 + y 2 , ϕ = arctan(y/x), and z = z) has been recently performed in Ref. [21]. Nevertheless this form of the QHD equations had been applied earlier (see Refs.…”

“…The QHD equations in the cylindrical coordinates were explicitly derived in Ref. [21], where was discussed the contribution of the quantum part of the inertial forces, which, together with the quantum part of the momentum flux, forms the familiar form of the quantum Bohm potential presented in equations ( 5) and (6).…”

“…As mentioned above, the occupation numbers for states inside the sphere of radius p ↑↑ in the momentum states equal to one: n f = 1. Let us substitute wave functions (11) in formula (21). Sum over states appears as…”

“…This topic is reviewed in Ref. [21] as a part of a detailed description of the many-particle quantum hydrodynamic method. Application of the generalized Coulomb exchange interaction force field was applied to the threedimensional magnetized quantum plasmas in Ref.…”

“…Let us mention that the Fermi pressure for the electron gas on the spherical surface was recently derived in Ref. [21] (see formula (256)). We should also mention that application of the three-dimensional equations of state or the potentials of the exchange interaction, as it was presented in Refs.…”