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The Plane Wave Solutions of the Free Particle Dirac Equation

Jiang Xue

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

First the Dirac equation of free particle is solved under static condition. Then the change of state vector space is used to gain plane wave solutions of the free particle that spins and moves along any divection. A simple way of solving the free particle Dirac equation is given.

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The usual choice of an orthogonal set of four plane-wave solutions of the free-particle Dirac equation does not lend itself readily to direct and complete physical interpretation except in low-energy approximation. A different choice of solutions can be made which yields a direct physical interpretation at all energies. Besides the separation of positive and negative energy states there is a further separation of states for which the spin is respectively parallel or antiparallel to the direction of the momentum vector.
We underscore the need of relativistic quantum mechanics in dealing with particles. Various notations used in relativity are defined and explained. We derive the Klein–Gordon equation for free particle as well as for particle in electromagnetic field. We obtain a solution and interpret it. We derive the Dirac equation for free particle as well as for particle in electromagnetic field. Plane wave solution to the Dirac equation is obtained. We show how spin of electron, idea of antiparticle , phenomena of pair production and pair annihilation follow from the Dirac equation.
The free particle solutions of the relativistic Dirac equation are characterized by plane waves with infinite uncertainty in position. However, many practical implementations of the solutions require a wave packet configuration, which can be utilized to represent a localized Dirac particle. Unlike the traditional wave packet generation method by superposing multiple plane waves, this study of ours presents an alternative approach towards obtaining a wave packet solution of a free particle relativistic Dirac equation. In this paper, we present Dirac’s free particle equation with a modification in the generalized momentum. The modification is achieved by coupling the momentum with a spatially varying logarithmic function, and this alteration does not affect the relativistic dispersion relation of the particle. Moreover, a solution of this modified Dirac equation is provided as well, which has been calculated using a trial wave function. The wave function solution is carried out in one dimension, where it behaves as a wave packet for a given ratio of the envelope parameter to the reduced Planck's constant greater than unity, where the envelope parameter regulates the width of the wave packet. The solution, being subject to this constraint, represents a bound particle with spin and a continuous energy spectrum.
The feasibility of applying the path-integral formalism for solving the Dirac equation is shown in the case of a free particle for which the Dirac propagator is obtained by evaluating an appropriate path integral, directly constructed from the Dirac equation. Furthermore, the propagator for a Dirac electron in a constant magnetic field is indirectly obtained from the propagator of an auxiliary wave equation by evaluating a world-line (space-time path) integral. The spectrum of the Dirac equation is also, in this case, extracted from an auxiliary propagator.
We set up the Maxwell's equations and subsequently the classical wave equations for the electromagnetic waves which together with their generating source, an oscillatory charge of zero rest mass in general travelling, make up a particle travelling similarly as the source at velocity ν in the field of an external scalar and vector potentials. The direct solutions in constant external field are Doppler-displaced plane waves propagating at the velocity of light c ; at the de Broglie wavelength scale and expressed in terms of the dynamically equivalent and appropriate geometric mean wave variables, these render as functions identical to the space-time functions of a corresponding Dirac spinor, and in turn identical to de Broglie phase waves previously obtained from explicit superposition. For two spin-half particles of a common set of space-time functions constrained with antisymmetric spin functions as follows the Pauli principle for same charges and as separately indirectly induced based on experiment for opposite charges, the complete wave functions are identical to the Dirac spinor. The back-substitution of the so explicitly determined complete wave functions in the corresponding classical wave equations of the two particles, subjected further to reductions appropriate for the stationary-state particle motion and to rotation invariance when in three dimensions, give a Dirac equation set; the procedure and conclusion are directly extendible to arbitrarily varying potentials by use of the Furious theorem and to particle motions in three dimensions by virtue of the characteristics of de Broglie particle motion. Through the derivation of the Dirac equation, the study hopes to lend insight into the connections between the Dirac wave functions and the electrodynamic components of simple particles under the government by the well established basic laws of electrodynamics.