Spectrum of the non-commutative spherical well

TL;DR

This paper defines and analyzes the non-commutative spherical well in two dimensions, revealing how non-commutativity affects bound states, scattering, and symmetry properties, and how classical limits restore familiar features.

Contribution

It provides a rigorous formulation of piecewise constant potentials in non-commutative quantum mechanics and solves for eigenvalues and eigenfunctions of the non-commutative spherical well.

Findings

Time reversal symmetry is broken by non-commutativity.

Eigenstates and eigenfunctions recover their commutative forms in the classical limit.

Bound states and scattering are well-defined in the non-commutative setting.

Abstract

We give precise meaning to piecewise constant potentials in non-commutative quantum mechanics. In particular we discuss the infinite and finite non-commutative spherical well in two dimensions. Using this, bound-states and scattering can be discussed unambiguously. Here we focus on the infinite well and solve for the eigenvalues and eigenfunctions. We find that time reversal symmetry is broken by the non-commutativity. We show that in the commutative and thermodynamic limits the eigenstates and eigenfunctions of the commutative spherical well are recovered and time reversal symmetry is restored.