Exactly Solvable Quantum Model with Spin-Dependent Coulomb Interaction

Abstract

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 \(φ= κ/ r\) . The model is governed by the Schrödinger-type Hamiltonian \(\mathcal{H}_{\rm S} = \vecΠ^2 / (2M) + q φ\) in nonrelativistic quantum mechanics and by the Dirac-type Hamiltonian \(\mathcal{H}_{\rm D} = c \vecα \cdot \vecΠ+ βM c^2 + q φ\) in relativistic quantum mechanics, where \(\vecΠ= \vec{p} - (q/c)\vec{\mathcal{A}}\) is the canonical momentum. We demonstrate two main results: (i) Just as the Coulomb-type scalar potential \(\mathcal{S}_{\rm Maxwell} = \{\vec{\mathcal{A}} = 0,\ φ= κ/ r\}\) is a local exact solution of Maxwell's equations on $r\ne 0$, the gauge potential \(\mathcal{S}_{\rm YM} = \{\vec{\mathcal{A}} = k (\vec{r} \times \vec{S}) / r^2,\ φ= κ/ r\}\) constitutes a local exact solution of the Yang--Mills equations on the punctured region $r\ne 0$. (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 clarify the quantization conditions and the fixed-background interpretation of the model.

Main 17 pages + SM 85 pages. 1 figure. Revised version. Accepted for publication in Symmetry (2026)