An "Accidental" Symmetry Operator for the Dirac Equation in the Coulomb Potential

TL;DR

This paper introduces a new symmetry operator for the Dirac equation in Coulomb potential, linking it to known operators and the Laplace-Runge-Lenz vector, enhancing understanding of symmetries in quantum systems.

Contribution

The paper generalizes K-odd operators to construct a new operator that commutes with the Dirac Hamiltonian in Coulomb fields, connecting it to the Johnson-Lippmann operator and the Laplace-Runge-Lenz vector.

Findings

Derived a new symmetry operator for the Dirac equation in Coulomb potential.

Established the operator's equivalence to the Johnson-Lippmann operator.

Linked the operator to the Laplace-Runge-Lenz vector.

Abstract

On the basis of the generalization of the theorem about K-odd operators (K is the Dirac's operator), certain linear combination is constructed, which appears to commute with the Dirac Hamiltonian for Coulomb field. This operator coincides with the Johnson and Lippmann operator and is intimately connected to the familiar Laplace-Runge-Lenz vector. Our approach guarantees not only derivation of Johnson-Lippmann operator, but simultaneously commutativity with the Dirac Hamiltonian follows.