Coulomb problem in NC quantum mechanics: Exact solution and non-perturbative aspects

TL;DR

This paper explores how non-commutative geometry modifies the quantum mechanical Coulomb problem, providing exact solutions and revealing novel bound states and scattering phenomena influenced by the non-commutativity parameter.

Contribution

It introduces a rotationally invariant non-commutative space framework and derives exact solutions for the Coulomb problem, including novel high-energy bound states and scattering effects.

Findings

Exact solutions for bound states with NC corrections

Analytic NC corrections for low-energy scattering

Singular NC corrections and new bound states at ultra-high energies

Abstract

The aim of this paper is to find out how would possible space non-commutativity (NC) alter the QM solution of the Coulomb problem. The NC parameter lambda is to be regarded as a measure of the non-commutativity - setting lambda = 0 means a return to the standard quantum mechanics. As the very first step a rotationaly invariant non-commutative 3D space, an analog of the Coulomb problem configuration space (3D space with origin extracted), is introduced. The non-commutative space in question is generated by NC coordinates realized as operators acting in an auxiliary Fock space. The properly weighted Hilbert-Schmidt operators in this Fock space form an NC analog of the Hilbert space of the wave functions. We will refer to them as "wave functions" also in the NC case. The definition of an NC analog of the hamiltonian as a hermitian operator is one of the key parts of this paper. The resulting problem is exactly solvable. The full solution is provided, including formulas for the bound states for negative energy E and low-energy scattering for positive E (both containig NC corrections analytic in lambda) and also formulas for high-energy scattering and unexpected bound states at ultra-high energy (both containing NC corrections singular in lambda). All the NC contributions to the known QM solutions either vanish or disappear in the limit of vanishing lambda.