LAPLACE-RUNGE-LENZ VECTOR IN QUANTUM MECHANICS IN NONCOMMUTATIVE SPACE

Peter Prešnajder,Veronika Gáliková,Samuel Kováčik

doi:10.14311/ap.2014.54.0149

Peter Prešnajder, Veronika Gáliková + Show 1 more

Open Access

https://doi.org/10.14311/ap.2014.54.0149

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Journal: Acta Polytechnica	Publication Date: Apr 30, 2014
License type: CC BY 4.0

Affiliation: Comenius University Bratislava

Abstract

The object under scrutiny is the dynamical symmetry connected with conservation of the Laplace-Runge-Lenz vector (LRL) in the hydrogen atom problem solved by means of noncommutative quantum mechanics (NCQM). The considered noncommutative configuration space has such a “fuzzy”<br />structure that the rotational invariance is not spoilt. An analogy with the LRL vector in the NCQM is brought to provide our results and also a comparison with the standard QM predictions.

Highlights

The main point of this paper is to examine a ”hidden” dynamical symmetry connected with the conservation of Laplace-Runge-Lenz vector (LRL) in the hydrogen atom problem solved by means of noncommutative quantum mechanics (NCQM)
The following few remarks on the subject have been well known since 1926, when Wolfgang Pauli published his paper on the subject, [1]. He used the LRL vector to find the spectrum of hydrogen atom using modern quantum mechanics and the hidden dynamical symmetry of the problem, without the knowledge of the explicit solution of Schrodinger equation
We have found the NC analog of the LRL vector; it’s components, together with those of the NC angular momentum operator, supply the algebra of generators of a symmetry group

Summary

Introduction

The main goal we are after is to investigate the existence of dynamical symmetry of the Coulomb-Kepler problem in the quantum mechanics in the non-commutative space and possibly to find the generalization of the LRL vector for this case. The following few remarks on the subject have been well known since 1926, when Wolfgang Pauli published his paper on the subject, [1] He used the LRL vector to find the spectrum of hydrogen atom using modern quantum mechanics and the hidden dynamical symmetry of the problem, without the knowledge of the explicit solution of Schrodinger equation. This is not an attempt to provide a full review; if there is a wish for deeper insight, the reader is invited to read Pauli’s original paper.

Basics of noncommutative quantum mechanics

Dynamical symmetry in NCQM

Auxiliary operators

NC operators revisited

Coulomb scattering

Conclusions