Superintegrable systems with spin and second-order integrals of motion

TL;DR

This paper classifies superintegrable quantum systems involving spin-1/2 and spin-0 particles with second-order integrals of motion, extending known potentials like Coulomb with new scalar and spin-orbit terms.

Contribution

It identifies all rotationally invariant, parity-conserving quantum systems with spin that admit second-order integrals of motion, generalizing classical superintegrable potentials.

Findings

Discovered a generalized Coulomb potential with additional scalar and spin-orbit terms.

Classified all systems with second-order integrals of motion under specified symmetries.

Provided explicit forms of integrals of motion for these systems.

Abstract

We investigate a quantum nonrelativistic system describing the interaction of two particles with spin 1/2 and spin 0, respectively. We assume that the Hamiltonian is rotationally invariant and parity conserving and identify all such systems which allow additional integrals of motion that are second order matrix polynomials in the momenta. These integrals are assumed to be scalars, pseudoscalars, vectors or axial vectors. Among the superintegrable systems obtained, we mention a generalization of the Coulomb potential with scalar potential $V_0=\frac{\alpha}{r}+\frac{3\hbar^2}{8r^2}$ and spin orbital one $V_1=\frac{\hbar}{2r^2}$.