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Annalen der Physik

A New Formulation of the Relativistic Wave Equations

David Lovelock

Published Jan 1, 1963• 2 citations
Publisher

Abstract

Abstract A process of quantisation is developed and applied to the equations of motion of those particles which may be regarded as the classical counterparts of the spinning particles of quantum mechanics. By invoking the requirement of uniqueness of rest mass (as exemplified by the Klein‐Gordon equation) it is possible to deduce the Dirac and Kemmer wave equations from a common approach.

DOI
10.1002/andp.19634670704

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Abstract A process of quantisation is developed and applied to the equations of motion of those particles which may be regarded as the classical counterparts of the spinning particles of quantum mechanics. By invoking the requirement of uniqueness of rest mass (as exemplified by the Klein‐Gordon equation) it is possible to deduce the Dirac and Kemmer wave equations from a common approach.
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The relativistic wave equations currently used in physics are the Dirac equation and the Klein-Gordon equation. Only recently, the Salpeter equation has stimulated the interest of the scientific community. In this paper a new relativistic wave equation for particles with non-zero rest mass, the Pearcey equation, is presented. There is great formal similarity between the Pearcey equation, the Schrödinger equation and the Salpeter equation. In particular, the Pearcey equation allows us to understand the onset of the relativistic behaviour in a similar way as the Salpeter equation. It is possibile also to keep the parallelism between optics and quantum mechanics at which we are used in non relativistic quantum mechanics.
Abstract This chapter is seeking relativistic wave equations, which are invariant under Lorentz transformations, in an attempt to obtain a quantum mechanical description of relativistic particles. First, the chapter starts with the Klein–Gordon equation and then it discusses the Dirac equation. It also takes the non-relativistic limit of the Dirac equation to derive the Schrödinger equation with two additional terms, the Zeeman term and the spin–orbit coupling term. These two terms emerge naturally from the Dirac equation, and thus the spin, as an internal quantum number which behaves like angular momentum, is clearly identified. Finally, the existence of antimatter is clearly supported by the nature of the solutions to the Dirac equation.
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.