Abstract
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 non-commutative quantum mechanics (NCQM). The basic features of NCQM will be introduced to the reader, the key one being the fact that the notion of a point, or a zero distance in the considered configuration space, is abandoned and replaced with a “fuzzy” structure in such a way that the rotational invariance is preserved. The main facts about the conservation of LRL vector in both classical and quantum theory will be reviewed. Finally, we will search for an analogy in the NCQM, provide our results and their comparison with the QM predictions. The key notions we are going to deal with are non-commutative space, Coulomb-Kepler problem, and symmetry.
Highlights
Our main goal we are after is to investigate the existence of dynamical symmetry of the Coulomb-Kepler problem in the quantum mechanics in noncommutative space, and possibly to find the generalization of the so-called Laplace-Runge-Lenz (LRL) vector for this case
This paper deals with the Coulomb-Kepler problem in noncommutative space
We have found the NC analog of the LRL vector; its components, together with those of the NC angular momentum operator, supply the algebra of generators of a symmetry group
Summary
Our main goal we are after is to investigate the existence of dynamical symmetry of the Coulomb-Kepler problem in the quantum mechanics in noncommutative space, and possibly to find the generalization of the so-called Laplace-Runge-Lenz (LRL) vector for this case. When the motion of a planet around the Sun is considered, the conservation of the given quantity has to do with the constant eccentricity of the orbit and the position of the perihelion Another well-known system characterized by Coulomb potential is the hydrogen atom. In 1926 Wolfgang Pauli published his paper on the subject [6] He used the LRL vector to find the spectrum of a hydrogen atom using modern quantum mechanics and the hidden dynamical symmetry of the problem, without knowledge of the explicit solution of the Schrödinger equation. We skip all the detailed and lengthy calculations that can be found in our recently published paper [7]
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