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arXiv:q-alg/9703026

q-Deformed Schrödinger Equation

Mircea Micu

Published Mar 13, 1997• 1 citation
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Abstract

We found hermitian realizations of the position vector $\vec{r}$, angular momentum $\vecΛ$ and linear momentum $\vec{p}$ behaving like vectors with respect to the $SU_q(2)$ algebra, generated by $L_0$ and $L_\pm$. They are used to write the $q$ deformed Schrödinger equation, whose solution for Coulomb and oscillator potential are briefly discussed.

DOI
10.48550/arxiv.q-alg/9703026

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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.
Transparent closed forms of the radial SOq{\mathrm{SO}}_{\mathit{q}}SOq​(N) Schr\"odinger equation acting on the noncommutative quantum Euclidean space are written both for l=0 and l\ensuremath{\ne}0. Starting from the s-wave equation leads to the derivation of an alternative q analog of the radial Schr\"odinger equation in N space dimensions exhibiting a modified symmetry, now in terms of wave functions depending on the radial coordinate only. The harmonic oscillator and the Coulomb system are treated as concrete examples. These results open the way to the derivation of q-deformed (1/N)-energy formulas for arbitrary spherically symmetrical potentials. Comparisons with exact SOq{\mathrm{SO}}_{\mathit{q}}SOq​(N) results, as well as further interpretations, have also been made.