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 \(\vec{\mathcal{A}} = k (\vec{r} \times \vec{S}) / r^2\) together with a Coulomb-type scalar potential \(\varphi = \kappa / r\). The model is governed by the Schr\"odinger-type Hamiltonian \(\mathcal{H}_{\rm S} = \vec{\Pi}^2 / (2M) + q \varphi\) in nonrelativistic quantum mechanics and by the Dirac-type Hamiltonian \(\mathcal{H}_{\rm D} = c \vec{\alpha} \cdot \vec{\Pi} + \beta M c^2 + q \varphi\) in relativistic quantum mechanics, where \(\vec{\Pi} = \vec{p} - (q/c)\vec{\mathcal{A}}\) is the canonical momentum. We demonstrate two main results: (i) Just as the Coulomb-type scalar potential \(\mathcal{S}_{\rm Maxwell} = \{\vec{\mathcal{A}} = 0,\ \varphi = \kappa / r\}\) is an exact solution of Maxwell's equations, the gauge potential \(\mathcal{S}_{\rm YM} = \{\vec{\mathcal{A}} = k (\vec{r} \times \vec{S}) / r^2,\ \varphi = \kappa / r\}\) constitutes an exact solution of the Yang--Mills equations. (ii) Both Hamiltonians \(\mathcal{H}_{\rm S}\) and \(\mathcal{H}_{\rm D}\) 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.