Integrable and superintegrable quantum systems in a magnetic field

TL;DR

This paper constructs integrable and superintegrable quantum systems with magnetic fields in 2D, highlighting differences from scalar potential cases and exploring quantum-classical distinctions.

Contribution

It introduces new quantum integrable systems with magnetic fields, analyzing their properties and differences from classical counterparts.

Findings

Quadratic integrability does not imply separation of variables.

Quantum and classical integrable systems can differ significantly.

Hamiltonians may depend nontrivially on Planck's constant.

Abstract

Integrable quantum mechanical systems with magnetic fields are constructed in two-dimensional Euclidean space. The integral of motion is assumed to be a first or second order Hermitian operator. Contrary to the case of purely scalar potentials, quadratic integrability does not imply separation of variables in the Schroedinger equation. Moreover, quantum and classical integrable systems do not necessarily coincide: the Hamiltonian can depend on the Planck constant in a nontrivial manner.