Laplace-Runge-Lenz vector for arbitrary spin

TL;DR

This paper introduces a class of superintegrable quantum systems with arbitrary spin, extending the Laplace-Runge-Lenz vector to generate an so(4) symmetry, allowing algebraic spectrum determination for particles with multipole moments.

Contribution

It generalizes the Laplace-Runge-Lenz vector to arbitrary spin, providing new superintegrable systems with algebraic spectra and explicit solutions for specific spins.

Findings

Systems exhibit so(4) symmetry with generalized Laplace-Runge-Lenz vector.

Spectra can be determined algebraically similar to hydrogen atom.

Explicit solutions provided for spins 1/2 and 1; spin 3/2 solutions linked to differential equations.

Abstract

A countable set of superintegrable quantum mechanical systems is presented which admit the dynamical symmetry with respect to algebra so(4). This algebra is generated by the Laplace-Runge-Lenz vector generalized to the case of arbitrary spin. The presented systems describe neutral particles with non-trivial multipole momenta. Their spectra can be found algebraically like in the case of Hydrogen atom. Solutions for the systems with spins 1/2 and 1 are presented explicitly, solutions for spin 3/2 are expressed via solutions of an ordinary differential equation of first order..