On Separable Solutions of Dirac's Equation for the Electron

A. H. Cook
1982 Proceedings of the Royal Society A  
The properties of coordinate systems th a t adm it separation of the Laplacian and H am ilton-Jacobi operators have been thoroughly explored so th a t the nature of solutions in separable form of Laplace's equation, the wave equation, Schrodinger's equation and the H am ilton-Jacobi equation are well understood. The corresponding problems for the Dirac operator in flat space-tim e have been less completely examined and this paper con tains studies intended to produce a more systematic account
more » ... ystematic account of possible solutions of D irac's equation. Because the Dirac operator differs from the Laplacian in being a firstdegree differential operator and in having m atrix coefficients, it is not possible to discuss possible solutions in as general a way and the separable solutions are far less rich than for equations with the Laplacian. In particular, the forms of the potentials for which separable solutions are possible are not for the most p art of physical interest. Although the discussion is confined to coordinates in flat space-tim e, some of the procedures are derived from those developed to solve Dirac's equation in coordinates with a K err metric.
doi:10.1098/rspa.1982.0130 fatcat:6pxtnbn45bbj7g6dbdg3wfa2xa