On the algebraic structure of rotationally invariant two-dimensional Hamiltonians on the noncommutative phase space

TL;DR

This paper explores the algebraic structure of two-dimensional rotationally invariant Hamiltonians on noncommutative phase space, revealing two possible quantum phases characterized by different Lie algebras and analyzing their spectra for key models.

Contribution

It introduces a classification of quantum phases based on Lie algebras in noncommutative phase space and analyzes spectra of models like the harmonic oscillator and Landau problem.

Findings

Two quantum phases characterized by $sl(2,\,\mathbb{R})$ or $su(2)$ algebras.

Rotation generator linked with the Casimir operator.

Spectral analysis of harmonic oscillator, Landau problem, and cylindrical well.

Abstract

We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant quadratic forms of noncommutative dynamical variables. We show that two quantum phases are possible, characterized by the Lie algebras $sl(2,\mathbb{R})$ or $su(2)$ according to the relation between the noncommutativity parameters, with the rotation generator related with the Casimir operator. From this algebraic perspective, we analyze the spectrum of some simple models with nonrelativistic rotationally invariant Hamiltonians in this noncommutative phase space, as the isotropic harmonic oscillator, the Landau problem and the cylindrical well potential. PACS: 03.65.-w; 03.65.Fd MSC: 81R05; 20C35; 22E70