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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Self-consistent electron motion in the framework of alternative quantum theory

Andrey L. Sanin, Vera G. Ulianova

Published Nov 27, 2000• 1 citation
Publisher

Abstract

Quantum hydrodynamic equations jointly with Maxwell's equation for the electric field are applied to describe motion of single electron including self-field one. The set of equations listed here and including the self-field of an electron is fundamental and represents the alternative quantum theory. Solutions of these equations are self-consistent and determine corresponding motion. In this paper we investigated one- dimensional stationary motion under variable distributions of background positive charge. Numerical integration of equations was carried out with Runge-Kutta method of 4th order approximation. Fast Fourier-transformation was used to analyze the spatially inhomogeneous distributions of electron density. Self-consistent solutions are compared with that of quantum hydrodynamic equations without considering the self-field of an electron (as in ordinary quantum mechanics). It is shown that difference takes place between them.

DOI
10.1117/12.407669

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Quantum hydrodynamic equations jointly with Maxwell's equation for the electric field are applied to describe motion of single electron including self-field one. The set of equations listed here and including the self-field of an electron is fundamental and represents the alternative quantum theory. Solutions of these equations are self-consistent and determine corresponding motion. In this paper we investigated one- dimensional stationary motion under variable distributions of background positive charge. Numerical integration of equations was carried out with Runge-Kutta method of 4th order approximation. Fast Fourier-transformation was used to analyze the spatially inhomogeneous distributions of electron density. Self-consistent solutions are compared with that of quantum hydrodynamic equations without considering the self-field of an electron (as in ordinary quantum mechanics). It is shown that difference takes place between them.
Quantum hydrodynamic equations are used to describe motion of single electron. The initial and boundary conditions are formulated from the analytic formula for the electron packet in a free space. The numerical integration of equations for the free packet is performed. Then, we investigated the wave packet propagation through the back-ground positive charge, when the electric field is self-consistent. Moreover, we studied the influence of the external harmonic electric fields on the properties of the motion. The purpose of the present work consists in study of the spatial-temporal processes including packets when the quantum wave effects are essential.
The self-consistent set of equations that describes the interaction of an atom with the electromagnetic field is proposed. The set includes the equations for vector and scalar potentials of electromagnetic field vector A(vector r,t), (phi) (vector r,t) and electron current and charge density vector j(vector r,t), (rho) (vector r,t). The obtained equations are similar to the magnetic hydrodynamics equation and differ from them by the presence of the terms depending on the logarithmic gradient of charge density. The specific feature of the obtained equations is that the steady-state distribution of the electron charge and current density is determined by the real and imaginary parts of the Schorodinger equation. The obtained equations are generalized for the relativistic case. In the latter case the equations include the atomic variables of the two types. The atomic variables of the first type are the bilinear forms of the wave function and Dirac conjugated wave function. The atomic variables of second type are the Hermitian conjugated combinations of the wave function first derivative and the Dirac conjugated wave function. Another specific feature of the obtained equations is that the equation for the electron current density can be transformed to the classical Hamilton-Jacobi equation with the new Hamiltonian. The phase of the wave function plays the role of action and new Hamiltonian includes the term depending on the logarithmic gradient of the electron density and its divergence. The relativistic Hamilton-Jacobi equation includes the term depending on the electron spin as well as the terms depending on electron density logarithmic four- gradient and its four-dimensional derivative.
The orbital free density functional theory and the single density equation approach are formally equivalent. An orbital free density based quantum dynamical strategy is used to study the quantum-classical correspondence in both weakly and strongly coupled van der Pol and Duffing oscillators in the presence of an external electric field in one dimension. The resulting quantum hydrodynamic equations of motion are solved through an implicit Euler type real space method involving a moving weighted least square technique. The Lagrangian framework used here allows the numerical grid points to follow the wave packet trajectory. The associated classical equations of motion are solved using a sixth order Runge-Kutta method and the Ehrenfest dynamics is followed through the solution of the time dependent Schrodinger equation using a time dependent Fourier Grid Hamiltonian technique. Various diagnostics reveal a close parallelism between classical regular as well as chaotic dynamics and that obtained from the Bohmian mechanics.
Abstract In this article, we show that the wave equation for a free Dirac electron can be represented in a form that is analogous to Maxwell’s electrodynamics. The electron bispinor wavefunction is explicitly expressed in terms of its real and imaginary components. This leads us to incorporate into it appropriate scalar and pseudo-scalar fields in advance, so that a full symmetry may be accomplished. The Dirac equation then takes on a form similar to that of a set of inhomogeneous Maxwell’s equations involving a particular self-source. We relate plane wave solutions of these equations to waves corresponding to free Dirac electrons, identifying the longitudinal component of the electron motion, together with the corresponding Zitterbewegung (“trembling motion”).
The present paper is based upon equations obtained in an earlier paper by the author devoted to a new formulation of quantum electrodynamics. The equations describe the structure of the electron as well as its motion in external fields, interaction with a measuring apparatus inclusive, in a deterministic manner without any jumps. Quantum mechanics is an approximate theory because its equations follow from the above equations upon neglecting the self-field of the electron itself. Just this leads to paradoxes, seeming contradictions and jumps in quantum mechanics. The quantum mechanical wavefunction has a dual interpretation. In some problems the square of its modulus represents a real distribution of the electronic density while in others the same square determines the probability distribution of coordinates. It is shown why, given the different interpretations of the wavefunction, it satisfies one and the same Dirac or Schrödinger equation. Description of many-electron systems is also considered in the starting approach as well as in quantum mechanics. Neutrinos are discussed in brief.
Applying the concept of quantum hydrodynamics on a hydrogen atom, one can derive an equation of motion which describes the time derivative of the mass current density for the relative motion of the electron and the proton. Since this derivation can be performed with the Ehrenfest theorem, we call this equation the Ehrenfest equation of motion. As quantum hydrodynamics is consistent with Schrödinger's quantum mechanics, the stationary s-wave functions satisfy both the Schrödinger equation and the Ehrenfest equation of motion. It is the purpose of this paper to prove that the stationary s-wave functions satisfy the Ehrenfest equation of motion. From the quantum hydrodynamical point of view, this result for s-wave functions can be interpreted that the Coulomb force density that attracts the electron to the proton is compensated by a quantum force density that is related to the dispersion of the probability density.