Quantum Hydrodynamic Equations Research Articles

Quantum hydrodynamics approach to the formation of waves in polarized two-dimensional systems of charged and neutral particles

In this paper, we explicate a method of quantum hydrodynamics (QHD) for the study of the quantum evolution of a system of polarized particles. Although we focused primarily on the two-dimensional (2D) physical systems, the method is valid for three-dimensional (3D) and one-dimensional (1D) systems too. The presented method is based upon the Schr\"odinger equation. Fundamental QHD equations for charged and neutral particles were derived from the many-particle microscopic Schr\"odinger equation. The fact that particles possess the electric dipole moment (EDM) was taken into account. The explicated QHD approach was used to study dispersion characteristics of various physical systems. We analyzed dispersion of waves in a two-dimensional ion and hole gas placed into an external electric field, which is orthogonal to the gas plane. Elementary excitations in a system of neutral polarized particles were studied for 1D, 2D, and 3D cases. The polarization dynamics in systems of both neutral and charged particles is shown to cause formation of a new type of waves as well as changes in the dispersion characteristics of already known waves. We also analyzed wave dispersion in 2D exciton systems, in 2D electron-ion plasma, and in 2D electron-hole plasma. Generation of waves in 3D-system neutral particles with EDM by means of the beam of electrons and neutral polarized particles is investigated.

Quantum-hydrodynamic approach to the problem of electron exchange between atomic particles and nanosystems

The application of quantum-hydrodynamic methods for solving the problem of electron exchange between atomic particles and solid surfaces, and nanosystems has been examined. The derivation of a system of equations that is alternative to the nonstationary Schrodinger equation is given to describe the dynamics of electronic processes with variable charge and current densities. A comparison of results of solving the nonstationary Schrodinger equation and the quantum-hydrodynamic system of equations shows that both approaches give a good coincidence. The numerical solution to the system of quantum-hydrodynamic equations has a number of advantages, because it does not lead to oscillations at the boundary of the computational mesh and nor to the problem of exponential growth in numerical complexity for many-electron systems.

Nonlinear wake potential and stopping power for charged particles interacting with a one-dimensional electron gas

We examine the interaction of particles with a one-dimensional electron gas by employing the quantum hydrodynamic (QHD) theory, where the nonlinear wake potential and stopping power have been numerically calculated by solving the nonlinear QHD equations with flux corrected transport (FCT) numerical method. In our calculation, the nonlinear effects on the wake potential and stopping power are clearly observed and presented. In the moving coordinate, comparisons are made between the nonlinear and linear wake potentials, in which the maximum values are larger and more oscillations appear behind the projectile in a nonlinear case in contrast to that in a linear case. The nonlinear wake potentials show a clear dependence on time, that is, the FCT algorithm solves the nonlinear QHD equations by time integration starting from the initial time. It is shown that the nonlinear effects can enhance the wake potential and stopping power for particle velocities greater than a few Bohr velocities.

Extended hydrodynamic approach to quantum-classical nonequilibrium evolution. I. Theory

A mixed quantum-classical formulation is developed for a quantum subsystem in strong interaction with an N-particle environment, to be treated as classical in the framework of a hydrodynamic representation. Starting from the quantum Liouville equation for the N-particle distribution and the corresponding reduced single-particle distribution, exact quantum hydrodynamic equations are obtained for the momentum moments of the single-particle distribution coupled to a discretized quantum subsystem. The quantum-classical limit is subsequently taken and the resulting hierarchy of equations is further approximated by various closure schemes. These include, in particular, (i) a Grad-Hermite-type closure, (ii) a Gaussian closure at the level of a quantum-classical local Maxwellian distribution, and (iii) a dynamical density functional theory approximation by which the hydrodynamic pressure term is replaced by a free energy functional derivative. The latter limit yields a mixed quantum-classical formulation which has previously been introduced by I. Burghardt and B. Bagchi, Chem. Phys. 134, 343 (2006).

Propagation and interaction of ion—acoustic solitary waves in a quantum electron—positron—ion plasma

This paper discusses the existence of ion-acoustic solitary waves and their interaction in a dense quantum electronpositron—ion plasma by using the quantum hydrodynamic equations. The extended Poincaré—Lighthill—Kuo perturbation method is used to derive the Korteweg-de Vries equations for quantum ion—acoustic solitary waves in this plasma. The effects of the ratio of positrons to ions unperturbation number density p and the quantum diffraction parameter He (Hp) on the newly formed wave during interaction, and the phase shift of the colliding solitary waves are studied. It is found that the interaction between two solitary waves fits linear superposition principle and these plasma parameters have significantly influence on the newly formed wave and phase shift of the colliding solitary waves. The investigations should be useful for understanding the propagation and interaction of ion-acoustic solitary waves in dense astrophysical plasmas (such as white dwarfs) as well as in intense laser-solid matter interaction experiments.

Quasineutral limit of bipolar quantum hydrodynamic model for semiconductors

This paper is concerned with the quasineutral limit of the bipolar quantum hydrodynamic model for semiconductors. It is rigorously proved that the strong solutions of the bipolar quantum hydrodynamic model converge to the strong solution of the so-called quantum hydrodynamic equations as the Debye length goes to zero. Moreover, we obtain the convergence of the strong solutions of bipolar quantum hydrodynamic model to the strong solution of the compressible Euler equations with damping if both the Debye length and the Planck constant go to zero simultaneously.

Quantum effects on the propagation of surface waves in magnetized plasma

The quantum effects on the propagation of surface waves is investigated in semi-bounded magnetized quantum plasmas. The results show that surface oscillations can be propagated due to quantum effects. It is also shown that these quantum effects enhance the propagation velocity. For this purpose, the quantum hydrodynamic (QHD) equations with a magnetic field, using Maxwell’s equations, are used to derive the dispersion relations of transverse electromagnetic surface modes (E-modes). The QHD includes quantum forces involving Fermi electron temperature and quantum Bohm potential. It is found that the external magnetic field which is parallel to the propagation waves has a stronger effect on wave decay in our model than in the classical regime. Beside that, the quantum effects enhance the propagation velocity of these waves.

Instability of plasma waves caused by incoherent photons in dense plasmas

AbstractWe consider the nonlinear instability of modified Langmuir and ion–sound waves caused by partially coherent photons in dense quantum plasmas. In our model, the dynamics of the photons is governed by a wave kinetic equation. The evolution equations for the Langmuir and ion–sound waves are deduced from the quantum hydrodynamic equations accounting for the incoherent photon pressure, the quantum statistical electron pressure, and the quantum Bohm force acting on the degenerate electrons. The governing equations are Fourier analyzed to obtain nonlinear dispersion relations. The latter are analyzed to predict instability of the modified Langmuir and ion–sound waves in the presence of partially coherent photons. Possible applications of our investigation to the next generation of intense laser–solid dense plasma experiments and compact dense astrophysical bodies are mentioned.

Propagation of TE-Surface Waves on Semi-Bounded Quantum Plasma

The propagation of the TE-surface waves on a semibounded quantum plasma is investigated by using the system of generalized quantum hydrodynamic (QHD) model and Maxwell's equations. The dispersion relations for these surface waves on quantum electron plasma in the presence of external magnetic field which is parallel to the wave propagation are derived. The perturbation of electron density and the electric fields of the TE-surface waves are also obtained. However, it was found that quantum effects (Bohm potential and statistical) have no remarkable action on the electric and magnetic field components in the case of unmagnetized plasma. But, it was found that the dispersion relation of surface modes depends significantly on these effects in the case of electrostatic or unmagnetized plasma.

Approximation method for a spherical bound system in the quantum plasma

A system of quantum hydrodynamic equations has been used for investigating the dielectric tensor and dispersion equation of a semiconductor as a quantum magnetized plasma. Dispersion relations and their modifications due to quantum effects are derived for both longitudinal and transverse waves. The number of states and energy levels are analytically estimated for a spherical bound system embedded in a semiconductor quantum plasma. The results show that longitudinal waves decay rapidly and do not interact with the spherical bound system. The energy shifts caused by the spin-orbit interaction and the Zeeman effect are calculated.

Electromagnetic wave equations for relativistically degenerate quantum magnetoplasmas

A generalized set of nonlinear electromagnetic quantum hydrodynamic (QHD) equations is derived for a magnetized quantum plasma, including collisional, electron spin- 1/2, and relativistically degenerate electron pressure effects that are relevant for dense astrophysical systems, such as white dwarfs. For illustrative purposes, linear dispersion relations are derived for one-dimensional magnetoacoustic waves for a collisionless nonrelativistic degenerate gas in the presence of the electron spin- 1/2 contribution and for magnetoacoustic waves in a plasma containing relativistically degenerate electrons. It is found that both the spin and relativistic degeneracy at high densities tend to slow down the magnetoacoustic wave due to the Pauli paramagnetic effect and relativistic electron mass increase. The present study outlines the theoretical framework for the investigation of linear and nonlinear behaviors of electromagnetic waves in dense astrophysical systems. The results are applied to calculate the magnetoacoustic speeds for both the nonrelativistic and relativistic electron degeneracy cases typical for white dwarf stars.

Unitarity Schrödinger Equation and Ground State Properties of the Finite Trapped Superfluid Fermi Gases in a BCS-BEC Crossover

On the basis of quantum hydrodynamical equations we derive a unitarity Schrödinger equation of a finite trapped superfluid Fermi gas valid in the whole interaction regime from BCS superfluid to BEC. This equation is just the Ginzburg–Laudau-type equation for the fermionic Cooper pairs in the BCS side, the Gross–Pitaevskii-type equation for the bosonic dimers in the BEC side, and a unitarity equation for a strongly interacting Fermi superfluid in the unitarity limit. By taking a modified Gauss-like trial wave function, we solve the unitarity Schrödinger equation, calculate the energy, chemical potential, sizes and profiles of the ground-state condensate, and discuss the properties of the ground state in the entire BCS-BEC crossover regimes.

Relativistic quantum corrections to laser wakefield acceleration

The influence of quantum effects on the interaction of intense laser fields with plasmas is investigated by using a hydrodynamic model based on the framework of the relativistic quantum theory. Starting from the covariant Wigner function and Dirac equation, the hydrodynamic equations for relativistic quantum plasmas are derived. Based on the relativistic quantum hydrodynamic equations and Poisson equation, the perturbations of electron number densities and the electric field of the laser wakefield containing quantum effects are deduced. It is found that the corrections generated by the quantum effects to the perturbations of electron number densities and the accelerating field of the laser wakefield cannot be neglected. Quantum effects will suppress laser wakefields, which is a classical manifestation of quantum decoherence effects, however, the contribution of quantum effects for the laser wakefield correction will been partially counteracted by the relativistic effects. The analysis also reveals that quantum effects enlarge the effective frequencies of plasmas, and the quantum behavior appears a screening effect for plasma electrons.

The combined semi-classical and relaxation limit in a quantum hydrodynamic semiconductor model

We discuss the combined semi-classical and relaxation limit of a one-dimensional isentropic quantum hydrodynamical model for semiconductors. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density, including the quantum potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. With the help of the Maxwell-type iteration, we prove that, as the relaxation time and Planck constant tend to zero, periodic initial-value problems of a scaled one-dimensional isentropic quantum hydrodynamic model have unique smooth solutions existing in the time interval where the classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the classical drift-diffusion model from the quantum hydrodynamic model.

Quantum effect on modulational instability of laser radiation in a semiconductor plasma

Modulational instability of a high power laser radiation in a homogeneous unmagnetized piezoelectric semiconductor plasma has been investigated analytically. The fluid equations of quantum hydrodynamics coupled with the Maxwell’s equations have been employed to find the nonlinear response of electrons in the piezoelectric semiconductor. The analysis is carried out through the derivation of the nonlinear dispersion relation for the four-wave modulational instability. An expression for the growth rate of the instability including the quantum effect due to Bohm potential has been obtained from the nonlinear dispersion relation. The quantum effect is observed to play a vital role in the four-wave scattering process. For a particular set of parameters, the quantum effect enhances the growth rate of the modulational instability by 37% compared to the growth rate predicted by the classical theory.

An iterative finite difference method for solving the quantum hydrodynamic equations of motion

A new stable computational method is presented for the direct numerical solution of the quantum hydrodynamic equations of motion associated with the de Broglie–Bohm formulation of quantum mechanics. The methodology is based on an iterative solution of the coupled non-linear hydrodynamic equations using a finite difference approximation for evaluating the numerical derivatives. The method is 2nd-order accurate in both space and time and exhibits exponential convergence with respect to the iteration count. The method is applied to a one-dimensional Gaussian wave packet: freely moving and scattering from an Eckart barrier. Excellent agreement is obtained between the numerical and analytic solutions for the free Gaussian wave packet. Accurate transmission probabilities are computed for the Eckart scattering problem which are in excellent agreement with two other computational methods. This method represents the first successful implementation of a stable finite difference approach for directly solving the quantum hydrodynamic equations of motion.

Fluid moment hierarchy equations derived from quantum kinetic theory

A set of quantum hydrodynamic equations are derived from the moments of the electrostatic mean-field Wigner kinetic equation. No assumptions are made on the particular local equilibrium or on the statistical ensemble wave functions. Quantum diffraction effects appear explicitly only in the transport equation for the heat flux triad, which is the third-order moment of the Wigner pseudo-distribution. The general linear dispersion relation is derived, from which a quantum modified Bohm–Gross relation is recovered in the long wave-length limit. Nonlinear, traveling wave solutions are numerically found in the one-dimensional case. The results shed light on the relation between quantum kinetic theory, the Bohm–de Broglie–Madelung eikonal approach, and quantum fluid transport around given equilibrium distribution functions.

Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations

The existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynamic equations is proved. The model consists of the conservation laws for the particle density and particle current density, including quantum corrections from the Bohm potential and viscous stabilizations arising from quantum Fokker–Planck interaction terms in the Wigner equation. The model equations are coupled self-consistently to the Poisson equation for the electric potential and are supplemented with periodic boundary and initial conditions. When a diffusion term linearly proportional to the velocity is introduced in the momentum equation, the positivity of the particle density is proved. This term, which introduces a strong regularizing effect, may be viewed as a classical conservative friction term due to particle interactions with the background temperature. Without this regularizing viscous term, only the nonnegativity of the density can be shown. The existence proof relies on the Faedo–Galerkin method together with a priori estimates from the energy functional.

Quantum hydrodynamics with trajectories: The nonlinear conservation form mixed/discontinuous Galerkin method with applications in chemistry

We present a solution to the conservation form (Eulerian form) of the quantum hydrodynamic equations which arise in chemical dynamics by implementing a mixed/discontinuous Galerkin (MDG) finite element numerical scheme. We show that this methodology is stable, showing good accuracy and a remarkable scale invariance in its solution space. In addition the MDG method is robust, adapting well to various initial-boundary value problems of particular significance in a range of physical and chemical applications. We further show explicitly how to recover the Lagrangian frame (or pathline) solutions.

Erratum: “Nonlinear electromagnetic wave equations for superdense magnetized plasmas” [Phys. Plasmas 16, 072114 (2009)

By using the quantum hydrodynamic and Maxwell equations, we derive nonlinear electron-magnetohydrodynamic (MHD), Hall-MHD, and dust Hall-MHD equations for dense quantum magnetoplasmas. The nonlinear equations include the electromagnetic, the electron pressure gradient, as well as the quantum electron tunneling and electron spin forces. They are useful for investigating a number of wave phenomena including linear and nonlinear electromagnetic waves, as well as three-dimensional electromagnetic wave turbulence spectra arising from the mode coupling processes in dense magnetoplasmas.