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 A→=k(r→×S→)/r2 together with a Coulomb-type scalar potential φ=κ/r. The model is governed by the Schrödinger-type Hamiltonian HS=Π→2/(2M)+qφ in nonrelativistic quantum mechanics and by the Dirac-type Hamiltonian HD=cα→·Π→+βMc2+qφ in relativistic quantum mechanics, where Π→=p→−(q/c)A→ is the canonical momentum. We demonstrate two main results: (i) Just as the Coulomb-type scalar potential SMaxwell={A→=0,φ=κ/r} is a local exact solution of Maxwell’s equations on r≠0, the gauge potential SYM={A→=k(r→×S→)/r2,φ=κ/r} constitutes a local exact solution of the Yang–Mills equations on the punctured region r≠0. (ii) Both Hamiltonians HS and HD 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.