Superintegrable systems with spin invariant with respect to the rotation group

TL;DR

This paper classifies quantum systems with spin that are invariant under rotations and possess second order integrals of motion, enabling variable separation and explicit solutions for certain cases.

Contribution

It identifies all rotationally invariant 2x2 matrix potential systems with second order integrals of motion, advancing understanding of their symmetries and solvability.

Findings

Systems can be separated into decoupled differential equations.

Explicit solutions are provided for two specific problems.

Integrals of motion facilitate the analysis of these quantum systems.

Abstract

Quantum nonrelativistic systems with $2\times2$ matrix potentials are investigated. Physically, they simulate charged or neutral fermions with non-trivial dipole momenta, interacting with an external electric field. Assuming rotationally invariance of the Hamiltonian all such systems allowing second order integrals of motion are identified. It is shown that the integrals of motion can be effectively used to separate variables and to reduce the systems to decoupled ordinary differential equations. Solutions for two of the discussed problems are presented explicitly.