Algebra of the spinor invariants and the relativistic hydrogen atom

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.

arXiv comment16 pages, no figures

References in corpus (5)

• Algebraic Solution of the Supersymmetric Hydrogen Atom in d Dimensions
• Dirac Hamiltonian in a supersymmetric framework
• Exact supersymmetry in the relativistic hydrogen atom in general dimensions -- supercharge and the generalized Johnson-Lippmann operator
• Defect in the Joint Spectrum of Hydrogen due to Monodromy
• Generalized spin-orbit interaction in two-dimensional electron systems