Generalized electromagnetic fields associated with the Coulomb-like atom problem

Abstract

It is well known that the principle of minimal coupling in quantum mechanics establishes a unique interaction form for a charged particle. By properly redefining the canonical commutation relations between (canonical) conjugate components of position and momentum of the particle, e.g., a $\pi ^{-}$ meson, we restate the Klein–Gordon equation for the Coulomb-like problem incorporating a generalized minimal electromagnetic replacement. The corresponding interaction keeps the $1/\vert \mathbf {q}\vert$ dependence in both the scalar potential $V(\vert \mathbf {q} \vert )$ and the vector potential $\mathbf {A}(\mathbf {q})(\vert \mathbf {A}(\mathbf {q})\vert \sim 1/\vert \mathbf {q}\vert )$. This equation can be exactly solved in closed form. Thus, we present a novel relativistic quantum-mechanical model which can be further explored.