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Lecture notes in physics

Construction of the dynamical symmetry group of the relativistic harmonic oscillator by the Infeld-Hull factorization method

Natig M. Atakishiyev

Published Jan 1, 1983• 10 citations
Publisher
DOI
10.1007/3-540-12291-5_61

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Factorization method for Schrödinger equation in relativistic configuration space and q-deformations

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Review paper is devoted to the relativistic configuration space (RCS) concept, a version of the relativistic Quantum Mechanics in RCS, the generalization of the Dirac-Infeld-Hall factorization method in the framework of the noncommutative differential calculus natural for RCS, different versions of the deformed oscillators, emerging as the generalization of the harmonic oscillator for RCS. In the formulation of the Newton-Wigner postulates for the relativistic localized states the hypothesis of commutativity of the position operator components is silently accepted as an evident fact. In the present work it is shown that commutativity is not necessary condition and the alternative (noncommutative) approach to the relativistic position operator and localization concept can be realized in a framework of the physically as well as mathematically comprehensive scheme. The different generalizations of the Dirac-Infeld-Hall factorization method for this case are constructed. This method enables us to find out all possible generalizations of the most important nonrelativistic integrable case—the harmonic oscillator. It is shown also that the relativistic oscillator = q —oscillator.

Physics of Particles and Nuclei•May 1, 2013•2 citations
Abstract For a three‐dimensional model of the harmonic oscillator in the relativistic configurational r‐representation wave functions in the spherical coordinates r = (r, Θ, ϕ) are found. The generating function, orthogonality and various recurrence relations for the radial part of the wave function are obtained. The radial and orbital quantum numbers raising and lowering operators are defined and the dynamical symmetry group is constructed by means of the Infeld‐Hull factorization method.
Review paper is devoted to the relativistic configuration space (RCS) concept, a version of the relativistic Quantum Mechanics in RCS, the generalization of the Dirac-Infeld-Hall factorization method in the framework of the noncommutative differential calculus natural for RCS, different versions of the deformed oscillators, emerging as the generalization of the harmonic oscillator for RCS. In the formulation of the Newton-Wigner postulates for the relativistic localized states the hypothesis of commutativity of the position operator components is silently accepted as an evident fact. In the present work it is shown that commutativity is not necessary condition and the alternative (noncommutative) approach to the relativistic position operator and localization concept can be realized in a framework of the physically as well as mathematically comprehensive scheme. The different generalizations of the Dirac-Infeld-Hall factorization method for this case are constructed. This method enables us to find out all possible generalizations of the most important nonrelativistic integrable case—the harmonic oscillator. It is shown also that the relativistic oscillator = q —oscillator.