Exactly Solvable Quantum Model with Spin-Dependent Coulomb Interaction

In this work, we report an exactly solvable quantum model featuring a spin-dependent Coulomb interaction, described by the spin vector potential $\vec{\mathcal{A}}=k(\vec{r}\times \vec{S})\mathrm{/}{r}^2$ together with a Coulomb-type scalar potential $\phi =\kappa \mathrm{/}r$. The model is governed by the Schrödinger-type Hamiltonian ${\mathcal{H}}_{\mathrm{S}}={\vec{\mathrm{\Pi }}}^2\mathrm{/}(2M)+q\phi$ in nonrelativistic quantum mechanics and by the Dirac-type Hamiltonian ${\mathcal{H}}_{\mathrm{D}}=c\vec{\alpha }\cdot \vec{\mathrm{\Pi }}+\beta M{c}^2+q\phi$ in relativistic quantum mechanics, where $\vec{\mathrm{\Pi }}=\vec{p}-(q\mathrm{/}c)\vec{\mathcal{A}}$ is the canonical momentum. We demonstrate two main results: (i) Just as the Coulomb-type scalar potential ${\mathcal{S}}_{\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}}=\{\vec{\mathcal{A}}=0,\phi =\kappa \mathrm{/}r\}$ is an exact solution of Maxwell’s equations, the gauge potential ${\mathcal{S}}_{\mathrm{Y}\mathrm{M}}=\{\vec{\mathcal{A}}=k(\vec{r}\times \vec{S})\mathrm{/}{r}^2,\phi =\kappa \mathrm{/}r\}$ constitutes an exact solution of the Yang–Mills equations. (ii) Both Hamiltonians ${\mathcal{H}}_{\mathrm{S}}$ and ${\mathcal{H}}_{\mathrm{D}}$ can be solved exactly in the presence of this spin-dependent Coulomb interaction. The resulting energy spectra are derived, and they naturally reduce to those of the ordinary hydrogen atom when the spin-dependent terms are neglected. Finally, we provide concluding remarks and discuss potential implications.