New Solutions of the Dirac Equation for Central Fields

Abstract

A modified form of Hestenes’s space-time version of Dirac’s equation is separated in spherical polar coordinates. The angular part of the solution spinor is derived in its most general form in terms of Gegenbauer polynomials and exponentials. One- and two-valued (multi-valued) spinors are obtained as a generalization of Weyl’s spherical harmonics with spin. The radial spinor equation defines a reduced real space-time algebra with one time and two space dimensions and displays symmetries which are hidden in the conventional matrix form. The superiority of this real valued Clifford algebraic formulation of the radial problem is demonstrated for the bound states of hydrogen-like atoms. The single-valued solutions turn out to be the ones of Darwin in a different representation.

One- and two-valued solutions together imply the unexpected discovery that the hydrogen atom in its states of lowest energy may realize the following values of its magnetic moment: μ = μ _D and μ = μ _D /$\pi ,where {\mu _D} = \mu {}_{Darwin} = {1 \over 3}(1 + 2\gamma ){\mu _B},$ $\gamma = \sqrt {1 - {\alpha ^2},}$ $\alpha = {{{q^2}} \over {nc}},$ ${\mu _B} = {\mu _{Bohr}} = {{\mid q\mid n} \over {2mc}}.$ Presumably it is an effect of the selfinteraction which seems to prefer the single-valued Darwin solution of the Dirac equation in the normal state of the hydrogen atom.