acta physica slovaca

Acta Physica Slovaca 65, No.3, 153-234 (2015) (82 pages)


Veronika Gáliková , Samuel Kováčik , Peter Prešnajder
    Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava Mlynská dolina, Bratislava, 842 48, Slovakia

Full text: ::pdf :: (Received 14 October 2015, accepted 19 October 2015)

Abstract: This paper provides an examination of how are prediction of standard quantum mechanics (QM) affected by introducing a non-commutative (NC) structure into the configuration space of the considered system (electron in the Coulomb potential in the present case). The parameter controlling the extent of the modification is denoted as λ. The coordinates in the NC space are realized via creation and annihilation operators acting in an auxiliary Fock space, this one being chosen in such a way that the rotational invariance of the system remains intact also in NCQM. Analog of the Schroedinger equation for hydrogen atom is found and analytically solved, both for bound and scattering states. The exact formulas for NC corrections are given. None of the NC predictions contradicts experimentally verified QM results, since in the correspondence limit λ -> 0 both QM and NCQM coincide. Highly surprising feature of the NC version is the existence of bound states for repulsive potential at ultra-high energies. However, these disappear from the Hilbert space in the mentioned limit. The whole problem is solved also using a Pauli method. Besides rotational invariance, the dynamical symmetry related to the conservation of NC analog of Laplace-Runge-Lenz vector is being used and the results obtained this way are in full agreement with those given by Schroedinger-like approach. The presented NC deformation of QM preserves all those mysterious properties of the Coulomb system that made it a distinguished cornerstone of the modern physics.

PACS: 02.40.Gh, 03.65.Ge, 03.65.Nk
Keywords: Non-commutative space, Hydrogen atom, Quantum mechanics
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