Algebra of the spinor invariants and the relativistic hydrogen atom

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 transformati...

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Bibliographic Details
Published inarXiv.org
Main Authors Eremko, A A, Brizhik, L S, Loktev, V M
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 03.11.2022
Subjects
Algebra
Coulomb potential
Dirac equation
Eigenvectors
Group theory
Hydrogen atoms
Invariants
Physics - Other Condensed Matter
Physics - Quantum Physics
Quantum mechanics
Quantum numbers
Relativistic effects
Rotation
Symmetry
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2211.01857