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arXiv:math-ph/9904011 · Journal of Physics A Mathematical and General

Aq-deformed Schrödinger equation

M Micu

Published Oct 21, 1999• 17 citations
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Abstract

We found Hermitian realizations of the position vector , the angular momentum and the linear momentum , all behaving like vectors under the su q (2) algebra, generated by L 0 and L ± . They are used to introduce a q -deformed Schrödinger equation. Its solutions for the particular cases of the Coulomb and the harmonic oscillator potentials are given and briefly discussed.

DOI
10.1088/0305-4470/32/44/312

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We found Hermitian realizations of the position vector , the angular momentum and the linear momentum , all behaving like vectors under the su q (2) algebra, generated by L 0 and L ± . They are used to introduce a q -deformed Schrödinger equation. Its solutions for the particular cases of the Coulomb and the harmonic oscillator potentials are given and briefly discussed.
We found hermitian realizations of the position vector r⃗\vec{r}r, angular momentum Λ⃗\vecΛΛ and linear momentum p⃗\vec{p}p​ behaving like vectors with respect to the SUq(2)SU_q(2)SUq​(2) algebra, generated by L0L_0L0​ and L±L_\pmL±​. They are used to write the qqq deformed Schrödinger equation, whose solution for Coulomb and oscillator potential are briefly discussed.
From the algebraic treatment of the quasi-solvable systems, and a q-deformation of the associated su(2)su(2)su(2) algebra, we obtain exact solutions for the q-deformed Schrodinger equation with a 3-dimensional q-deformed harmonic oscillator potential.
Based on the tensor method, a q -analogue of the spin-orbit coupling is introduced in a q -deformed Schrödinger equation, previously derived for a central potential. Analytic expressions for the matrix elements of the representations j = ℓ±½ are derived. The spectra of the harmonic oscillator and the Coulomb potential are calculated numerically as a function of the deformation parameter, without and with the spin-orbit coupling. The harmonic oscillator spectrum presents strong analogies with the bound spectrum of a Woods-Saxon potential customarily used in nuclear physics. The Coulomb spectrum simulates relativistic effects. The addition of the spin-orbit coupling reinforces this picture.
In an earlier paper a q -Schrödinger equation was obtained based on a particular quantization procedure, called Borel quantization, and a related q -deformation of the Witt algebra. This q -deformation is a deformation in the category of Lie algebras and hence the corresponding q -Witt algebra has a trivial Hopf algebra structure. In this paper, we extend the above algebra by the addition of a set of shift-type generators, which appear in the expression for the quantum mechanical position operator and hence lead to a new type of quantum kinematics. The latter gives rise to a new kind of evolution equation and it is shown that in the limit q 1 a specific class of Schrödinger equations is obtained from it. This specification of a particular class is a new phenomenon, because in earlier references, where a different q -deformation has been implemented or no deformation has been used at all, such a class could not be determined uniquely. The extended algebra used here has a nontrivial Hopf structure. The appearance of the shift-type generator in the q -deformed picture hence leads to a selection of a particular type of dynamics and delivers in the limit q 1 new information for the characterization of the corresponding dynamics in the undeformed situation.