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2013

Laplace-Runge-Lenz vector in quantum mechanics in noncommutative space

Galikova, Veronika, Kovacik, Samuel, Presnajder, Peter

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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 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.

Laplace-Runge-Lenz vector in quantum mechanics in noncommutative space · Around

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W. Lenz, Zeitschrift für Physik 24 (1924) 197, Über den Bewegungsverlauf und Quantenzustände der gestörten Keplerbewegung
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W. Pauli, Zeitschrift für Physik 36 (1926) 336, Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik
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F. A. Berezin, Commun. Math. Phys. 40
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Jakob Hermann, Giornale de Letterati D’Italia 2 (1710) 447; Jakob Hermann, Extrait d’une lettre de M. Herman a M. Bernoulli dat’ee de Padoüe le 12. Juillet 1710
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Johann Bernoulli, Extrait de la Réponse de M. Bernoulli a M. Herman datée de Basle le 7. Octobre 1710
Cited in the paper.
P. S. Laplace, Traité de mécanique celeste
Cited in the paper.
S. Kováčik and P. Prešnajder, The velocity operator in quantum mechanics in noncommutative space
Cited in the paper.

Then

H. Grosse and P. Prešnajder, Lett. Math. Phys. 28
1993
Later among the works it cites.
A. Connes, Publ. IHES 62
1994
Later among the works it cites.
H. Grosse, C. Klimčík and P. Prešnajder, Int. J. Theor. Phys. 35
1996
Later among the works it cites.
Veronika Gáliková and Peter Prešnajder, J. Math. Phys. 54, 052102 (2013);
2013
Closest in time.

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