General solution vs spin invariant eigenstates of the Dirac equation with the Coulomb potential

TL;DR

This paper derives exact solutions to the Dirac equation in a Coulomb potential using operator invariants, revealing how these invariants influence electron probability and spin polarization despite energy degeneracy.

Contribution

It introduces a generalized invariant for the Dirac equation, constructs its solutions, and demonstrates how invariants determine electron spin and spatial distributions.

Findings

Exact solutions of the Dirac equation with Coulomb potential are obtained.

Electron spin polarization and probability densities depend on the chosen invariants.

Different invariants lead to physically distinct states despite energy degeneracy.

Abstract

Solutions of the Dirac equation for an electron in the Coulomb potential are obtained using operator invariants of the equation, namely the Dirac, Johnson-Lippmann and recently found new invariant. It is demonstrated that these operators are the spin invariants. The generalized invariant is constructed and the exact general solution of the Dirac equation are found. In particular, the explicit expressions of the bispinors corresponding to the three complete sets of the invariants, their eigenvalues and quantum numbers are calculated. It is shown that the general solution of one center Coulomb Dirac equation contains free parameters. Changing one or more of these parameters, one can transform one solution of the Dirac equation into any other. It is shown for the first time that these invariants determine electron spatial probability amplitude and spin polarization in each quantum state. Electron probability densities and spin polarizations are explicitly calculated in the general form for several electron states in the hydrogen-like energy spectrum. Spatial distributions of these characteristics are shown to depend essentially on the invariant set, demonstrating, in spite of the accidental degeneracy of energy levels, physical difference of the states corresponding to different spin invariants.