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An energy and determinist approach of quantum mechanics

Patrick Vaudon

Published Jan 1, 2017• 0 citations
Publisher

Abstract

This document presents a work based on the analysis of the properties of exact solutions to the DIRAC equation in relativistic condition of energy conservation. Inspired by the classic methods of electromagnetism, these solutions can be interpreted as exchanges of energy between the stationary modes that settled within a cavity. Exact solutions are established as a first step in Cartesian coordinates, then in a second step in spherical coordinates. It is shown that an interpretation of this nature leads in fine to an energy and deterministic approach of quantum mechanics in complete coherence with current theories.

DOI
10.13140/rg.2.2.24670.74565

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This document presents a work based on the analysis of the properties of exact solutions to the DIRAC equation in relativistic condition of energy conservation. Inspired by the classic methods of electromagnetism, these solutions can be interpreted as exchanges of energy between the stationary modes that settled within a cavity. Exact solutions are established as a first step in Cartesian coordinates, then in a second step in spherical coordinates. It is shown that an interpretation of this nature leads in fine to an energy and deterministic approach of quantum mechanics in complete coherence with current theories.
Ce document présente un travail basé sur l'analyse des propriétés de solutions exactes à l'équation de DIRAC sous la condition relativiste de conservation de l'énergie. Inspirées des méthodes classiques de l'électromagnétisme, ces solutions peuvent être interprétées comme des échanges d'énergie entre les modes stationnaires qui s'établissent au sein d'une cavité. Les solutions exactes sont établies dans un premier temps en coordonnées cartésiennes, puis dans un second temps en coordonnées sphériques. Il est montré qu'une interprétation de cette nature conduit in fine à une approche énergétique et déterministe de la mécanique quantique en complète cohérence avec les théories actuelles.
Solutions of Dirac equation are well known in terms of bi-spinors, but the complete determination of these spinors is always derived using the assumption that the wave function is a plane wave. We investigate in this paper the search of general solutions under the form of standing wave modes, using methods issued from electromagnetism. It is shown that such solutions exist as solutions of a system of 64 equations and 64 unknowns, and that they must be associated with a relativistic relation of energy conservation. Complete exact solutions are presented. It is shown that these solutions highlight in a new manner the duality wave-particle, and the uncertainty principle.
Ce document presente un travail base sur l'analyse des proprietes de solutions exactes a l'equation de DIRAC sous la condition relativiste de conservation de l'energie. Inspirees des methodes classiques de l'electromagnetisme, ces solutions peuvent etre interpretees comme des echanges d'energie entre les modes stationnaires qui s'etablissent au sein d'une cavite. Les solutions exactes sont etablis dans un premier temps en coordonnees cartesiennes, puis dans un second temps en coordonnees spheriques. Il est montre qu'une interpretation de cette nature conduit infine a une approche energetique et deterministe de la mecanique quantique en complete coherence avec les theories actuelles.
The chapter covers some of the essential aspects of relativistic quantum mechanics. The Klein-Gordon equation and its solution in coulombic fields, how to interpret probability density and probability current derived from the Klein-Gordon equation, derivation of the Dirac equation and its covariant form, probability density and probability current for Dirac equation, positive and negative energy eigenvalues in the case of a free particle, constants of motion for relativistic treatment, presence of spin magnetic moment and spin-orbit interaction energy in the Dirac equation, and the solution of the Dirac equation for central potentials and for the hydrogen atom are presented with elegant mathematical treatment and physical interpretations. A modern and reader friendly approach is adopted for derivation of the Dirac equation, constants of motion, and for the solution of the Dirac equation for central potentials. Solved examples and unsolved exercises relating to fundamental concepts of relativistic quantum mechanics are added towards the end of chapter.
The paper shows the relationship between the major wave equations in quantum mechanics and electromagnetism, such as Schroedinger's equation, Dirac's equation and the Maxwell equations. It is shown that they can be derived in a striking simple way from a common root. This root is the relativistic fourvector formulation of the momentum conservation law. This is shown to be a more attractive starting-point than Einstein's energy relationship for moving particles, which is commonly used for the purpose. The theory developed gives a new interpretation for the origin of antiparticles.