The London Edinburgh and Dublin Philosophical Magazine and Journal of Science

XV. The forces on an electron, according to Dirac's wave equation

A. Lees

Published Feb 1, 1942• 0 citations

Abstract

Summary It is proved that the forces on an electron, according to Dirac's wave equation, consist of a purely internal system of stresses together with those forces which would be exerted upon it by the external electromagnetic field according to classical principles.

DOI: 10.1080/14786444208521355

Related papers

A consistency condition for electron wave functions

C. JAYARATNAM ELIEZER

Dirac's wave equation of an electron enables one to solve for the wave function of an electron moving in an electromagnetic field. The wave function ψ has 4 components ψ1, ψ2, ψ3, ψ4, and the electromagnetic field is described by a four-vector Aμ, consisting of a scalar potential φ and a vector potential A. In the usual problem, we solve for the wave function ψ when the potential Aμ is given. By doing so, we obtain the wave function of an electron which moves in a given electromagnetic field. It is of some interest to consider the reverse question: given the wave function ψ, what can we say about the electromagnetic potential Aμ, which is connected with ψ by Dirac's equation? Is Aμ uniquely determined, and if not, what is the extent to which it is arbitrary ?

Mathematical Proceedings of the Cambridge Philosophical SocietyApr 1, 195813 citations

An analog of the Feynman-Kac formula for Dirac's electron in electromagnetic field and the correspondence principle

A. A. Beilinson

The article describes a relation between the fundamental solutions of Dirac's equations for free electron and electron in given electromagnetic field viewed as functionals on bump functions. We explain how classical relativistic mechanics of a charged particle in given electromagnetic field arises from quantum mechanics of Dirac's electron in that field.

arXiv:1506.04034Jun 12, 20150 citations

The Wave Structure of the Electric Field

Michael Harney

Maxwell's equations describe the interactions of the electromagnetic field at a macroscopic level. In the 1920s, Louis DeBroglie demonstrated that every moving particle (including an electron) has a wave nature, and we know from Einstein that every wave has a particle nature, which we call the photon. Later in the 1930s, Paul Dirac's development of the famous Dirac equation showed the quantum nature of the electron at relativistic speeds. Then in 1948 Richard Feynman and Julian Schwinger extended these concepts in the development of quantum electrodynamics which gives a full accounting (although a very strange one) of how an electron can borrow energy from the vacuum of space and return it legally as long it does so within limits of the uncertainty principle.

Apr 1, 20101 citation

A First-Order Variational Principle for Classical Electrodynamics

Frederik Jozef Belinfante

The fact that the Schr\"odinger and the Dirac equations for the wave function of an electron are differential equations of the first order with respect to time, while in classical theories differential equations of the second order are common, has necessitated a slightly different setup of the canonical theory in both fields. Under influence of the classical methods, second-order equations are also often used in the quantum theory of particles of integer spin, thus causing a difference between the treatment of Fermi-Dirac and of Einstein-Bose particles.It is shown here that conformity between classical and quantum-mechanical methods can be achieved easily by use of first-order equations throughout, thus avoiding a superfluous distinction between integral and half-odd-integral spin fields. The classical theory of a point charge in an electromagnetic field of force is set up here from this point of view.

Physical ReviewOct 1, 19484 citations

The second internal degree of freedom of the Dirac particle and its optical-mechanical analogy

Nikolay Leonidovich Chuprikov

Dirac's equation with static external fields is considered as an equation of relativistic quantum theory, which describes the quantum dynamics of an electron in four-dimensional space-time. It is shown that the Dirac electron has only one intrinsic degree of freedom -- spin. The quantum ensemble of an electron with a given energy consists of two inseparable from each other subensembles of heavy and light spin-1/2 quasiparticles to have opposite (relative) intrinsic parities. In the case of electrical scalar potential nonuniform in the direction of a particle motion, these subensembles behave as analogs of the electrical and magnetic components of the plane monochromatic electromagnetic wave propagating in a dispersive medium, nonuniform in the direction of a wave motion. The role of states with negative energies and the properties of the velocity operator are explained.

arXiv:1702.02021Jan 30, 20170 citations

Description of the Electron in the Electromagnetic Field: The Dirac Type Equation and the Equation for the Wave Function in Spinor Coordinate Space

Pavel Gorev

Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by spinors - coordinate spinors, impulse spinors, spinor wave functions. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Schrödinger equation the wave function is represented by a single complex quantity, in the Dirac equation a step forward is taken and the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Using the example of the interaction between an electron and an electromagnetic field, we can see that the spinor equation contains more detailed information about the interaction than the vector equations. This is not new for quantum mechanics, since it describes interactions using complex wave functions, which cannot be observed directly, and only when measured goes to probabilities in the form of squares of the moduli of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors.In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In section 3 of the paper this quadratic form is expressed through momentum spinors, which makes it possible to obtain an equation for the spinor wave function in spinor coordinate space by replacing the momentum spinor components by partial derivative operators on the corresponding coordinate spinor component. At the end of the paper the question on a possibility of second quantization of the electron field in the spinor coordinate space is touch upon. The statement that the electron and positron have the same, and exactly positive energy, and have opposite signs of charge and opposite signs of mass is also justified. Accordingly, in the process of annihilation their total energy, momentum, charge and mass do not change. It is shown that if we take as an axiom the conservation at interaction of the total mass of the system of particles taking into account its sign, then it is possible to explain the difference between bosons and fermions in the statistics to which they obey.

Feb 27, 20240 citations