q-Schrodinger Equations for V=u^2+ 1/u^2 and Morse Potentials in terms of the q-canonical Transformation

TL;DR

This paper introduces a q-deformed canonical transformation linking Schr"odinger equations for Morse and $V=u^2+1/u^2$ potentials, leading to new q-Laguerre polynomials and insights into their wave functions and eigenvalues.

Contribution

It presents a novel q-canonical transformation connecting two q-deformed Schr"odinger equations for different potentials, expanding the understanding of q-deformed quantum systems.

Findings

Derived a q-deformed Schr"odinger equation for the Morse potential.

Established a new definition of q-Laguerre polynomials.

Analyzed wave functions and eigenvalues of the q-deformed equations.

Abstract

The realizations of the Lie algebra corresponding to the dynamical symmetry group SO(2,1) of the Schr\"{o}dinger equations for the Morse and the $V=u^2+1/u^2$ potentials were known to be related by a canonical transformation. q-deformed analog of this transformation connecting two different realizations of the sl_q(2) algebra is presented. By the virtue of the q-canonical transformation a q-deformed Schr\"{o}dinger equation for the Morse potential is obtained from the q-deformed $V=u^2+ 1/u^2$ Schr\"{o}dinger equation. Wave functions and eigenvalues of the q-Schr\"{o}dinger equations yielding a new definition of the q-Laguerre polynomials are studied.