Two-dimensional relativistic hydrogenic atoms: A complete set of constants of motion

TL;DR

This paper provides a complete set of constants of motion for two-dimensional relativistic hydrogenic atoms, offering exact solutions and a new classification scheme without relying on non-relativistic limits.

Contribution

It introduces a full set of commuting operators and quantum numbers for 2D Dirac hydrogen atoms, enabling a comprehensive state classification and analysis of the Paschen-Back effect.

Findings

Exact analytic solutions of the Dirac equation for 2D Coulomb potential.

New classification of states using full constants of motion.

Analysis of the linear Paschen-Back effect with exact wave-functions.

Abstract

The complete set of operators commuting with the Dirac Hamiltonian and exact analytic solution of the Dirac equation for the two-dimensional Coulomb potential is presented. Beyond the eigenvalue $\mu$ of the operator $j_{z}$, two quantum numbers $\eta$ and $\kappa$ are introduced as eigenvalues of hermitian operators $P=\beta\sigma_{z}'$ and $K=\beta(\sigma_{z}'l_{z}+1/2)$, respectively. The classification of states according to the full set of constants of motion without referring to the non-relativistic limit is proposed. The linear Paschen-Back effect is analyzed using exact field-free wave-functions as a zero-order approximation.