Dirac’s reduced radial equations and the problem of additional solutions

Abstract

We show that additional solutions must be ignored (in differences of the Schrödinger and Klein–Gordon equations) in the Dirac equation, where usually the second-order radial equation is passed, called the reduced equation, instead of a system. Analogously to the Schrödinger equation, in this process, the Dirac’s delta function appears, which was unnoted during the full history of quantum mechanics. This unphysical term we remove by a boundary condition at the origin. However, the distribution theory imposes on the radial function strong restriction and by this reason practically for all potentials, even regular, use of these reduced equations is not permissible. At the end, we include consideration in the framework of two-dimensional Dirac equation. We show that even here the additional solution does not survive as a result of usual physical requirements.