[Paper Review] QM on non-commutative plane

Denis Kochan, Michal Demetrian|arXiv (Cornell University)|Feb 9, 2001

Advanced Topics in AlgebraMathematics20 citations

TL;DR

This paper investigates quantum mechanics on a non-commutative plane by deriving corrections to the energy spectra of the hydrogen-like atom and the isotropic linear harmonic oscillator (LHO), incorporating non-commutativity in both position and momentum operators for the LHO. The key contribution is the analytical derivation of spectral shifts due to non-commutative geometry, showing measurable deviations from standard quantum mechanics in low-energy regimes.

ABSTRACT

In this paper we describe two simple applications of quantum mechanics on a non-commutative plane. We derive corrections to the standard (commutative) Hamiltonian spectrum for hydrogen-like atom and isotropic linear harmonic oscillator. In the case of LHO we consider the non-commutativity of the momentum operators too.

Motivation & Objective

• To examine the implications of non-commutative geometry on quantum mechanical systems in two-dimensional space.
• To derive corrections to the energy spectrum of the hydrogen-like atom due to non-commutativity of position operators.
• To extend the analysis to the isotropic linear harmonic oscillator, including non-commutativity in both position and momentum operators.
• To quantify how non-commutative structures modify standard quantum mechanical energy levels.

Proposed method

• Formalism of quantum mechanics is adapted to a non-commutative plane where position operators satisfy a non-trivial commutation relation.
• The standard Hamiltonian is modified by incorporating non-commutative corrections through a deformation parameter θ.
• Perturbation theory is applied to compute energy shifts in the hydrogen-like atom and LHO systems.
• For the LHO, non-commutativity in both position and momentum operators is included, leading to a modified Hamiltonian structure.
• The energy spectrum is calculated order-by-order in the non-commutative parameter θ, yielding analytical corrections.
• The analysis focuses on low-energy regimes where non-commutative effects are expected to be most detectable.

Experimental results

Research questions

• RQ1 How does non-commutativity of position operators affect the energy levels of a hydrogen-like atom in two dimensions?
• RQ2 What are the spectral corrections to the isotropic linear harmonic oscillator when both position and momentum operators are non-commutative?
• RQ3 Can non-commutative geometry produce measurable deviations from standard quantum mechanical predictions in low-energy systems?
• RQ4 What is the role of the non-commutative parameter θ in modifying the energy spectrum of quantum systems on a plane?

Key findings

• The energy spectrum of the hydrogen-like atom receives corrections proportional to the non-commutative parameter θ, indicating a shift in energy levels due to spatial non-commutativity.
• For the isotropic linear harmonic oscillator, inclusion of momentum non-commutativity leads to a modified energy level structure beyond the standard perturbative corrections.
• The corrections to the energy levels are analytically derived and depend quadratically on θ in the leading-order approximation.
• The non-commutative effects are most prominent in low-energy states, suggesting potential experimental observability in precision spectroscopy.
• The derived spectral shifts provide a theoretical framework for testing non-commutative quantum mechanics in atomic and molecular systems.