General solution of the Dirac equation with the Coulomb potential

TL;DR

This paper derives a comprehensive solution to the Dirac equation with Coulomb potential, revealing a new invariant and analyzing how different invariants influence electron probability and spin polarization in hydrogen-like atoms.

Contribution

It introduces a new invariant for the Dirac-Coulomb problem and provides explicit solutions and physical interpretations based on this invariant.

Findings

Existence of a new invariant beyond known ones.

Explicit solutions for bispinors and quantum numbers.

Dependence of electron distributions on invariant sets.

Abstract

The Dirac equation with the Coulomb potential is studied. It is shown that there exists a new invariant in addition to the known Dirac and Johnson-Lippman ones. The solution of the Dirac equation, using the generalized invariant, and explicit expressions for the bispinors corresponding to the three sets of the invariants, their eigenvalues and quantum numbers are obtained. The general solution of the Dirac equation with the Coulomb potential is shown to contain free parameters, whose variation transforms one particular solution into any other and controls spatial electron probability amplitude and spin polarization. The electron probability densities and spin polarizations are obtained in the general form and calculated explicitly for some electron states in the hydrogen-like energy spectrum. The spatial distributions of these characteristics are shown to depend essentially on the invariant set, demonstrating physical difference of the states corresponding to different invariants.