Superintegrability in the interaction of two particles with spin

TL;DR

This paper systematically investigates quantum superintegrable systems involving two spin-1/2 particles in three-dimensional space, identifying new systems with scalar integrals of motion that extend understanding of such interactions.

Contribution

It introduces a method to classify superintegrable systems with spin, deriving 30 new systems with explicit scalar integrals of motion.

Findings

Identified 30 new superintegrable systems with spin.

Derived explicit first-order scalar integrals of motion.

Excluded certain potentials via gauge transformations.

Abstract

We initiate a research program for the systematic investigation of quantum superintegrable systems involving the interaction of two non-relativistic particles with spin $1/2$ moving in the three-dimensional Euclidean space. In this paper, we focus specifically on such superintegrable systems that allow additional scalar integrals of motion, linear in the momenta. We first identify specific potentials appearing in the Hamiltonian that should be excluded from the analysis, as they can be immediately derived through gauge transformations of the natural Hamiltonian. Next, we construct the most general symmetric scalar operator and derive the determining equations from its commutation with the Hamiltonian. Solving these equations, we obtain 30 new superintegrable systems with spin, along with their corresponding first-order scalar integrals of motion.