Algebra of the spinor invariants and the relativistic hydrogen atom

TL;DR

This paper demonstrates an algebraic method to solve the Dirac equation for the hydrogen atom using spinor invariants, revealing hidden symmetries and clarifying the role of quantum numbers without supersymmetric quantum mechanics.

Contribution

It introduces an algebraic approach based on spinor invariants to solve the Dirac equation, avoiding supersymmetric methods and elucidating the symmetry structure of the relativistic hydrogen atom.

Findings

Eigenstates and eigenenergies can be calculated algebraically.

The Dirac Hamiltonian exhibits $SU(2)$ dynamical symmetry.

The principal quantum number is fundamental and independent.

Abstract

It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry $ SU(2) $ of the Dirac equation. The total symmetry of the Dirac equation is the symmetry $ SO(3) \otimes SU(2) $. The generator of the $ SO(3) $ symmetry group is given by the total momentum operator, and the generator of $ SU(2) $ group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson-Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.