Abstract
An operator analysis is presented that provides a unified treatment of the Schrödinger (S), Klein–Gordon (KG), and Dirac (D) equations with a Coulomb potential. The analysis uses energy shift operators that factorize an appropriate radial equation. This radial equation is based on standard results and a recent formulation of the Dirac–Coulomb problem [J. Y. Su, Phys. Rev. A 32, 3251 (1985)]. The shift operators yield energy eigenvalues and a formula that contains normalized, radial coordinate-space wave functions for the S, KG, and D equations. Formulas that contain expectation values for the S, KG, and D equations are obtained by applying the hypervirial theorem and the Hellmann–Feynman theorem to the radial equation.
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