Hermite polynomials and Fibonacci Oscillators

TL;DR

This paper introduces ($q_1,q_2$)-deformed Hermite polynomials derived from Fibonacci oscillators, revealing how deformation affects energy spectra and potentially aids in modeling disorder in quantum systems.

Contribution

It presents a novel ($q_1,q_2$)-deformation of Hermite polynomials based on Fibonacci oscillators and explores their impact on quantum energy spectra.

Findings

Deformation influences higher excited states more significantly.

Deformed energy spectrum expressed in terms of ($q_1,q_2$) parameters.

Classical quantum mechanics recovered when $q_1=1$ and $q_2 o1$.

Abstract

We compute the ($q_1,q_2$)-deformed Hermite polynomials by replacing the quantum harmonic oscillator problem to Fibonacci oscillators. We do this by applying the ($q_1, q_2$)-extension of Jackson derivative. The deformed energy spectrum is also found in terms of these parameters. We conclude that the deformation is more effective in higher excited states. We conjecture that this achievement may find applications in the inclusion of disorder and impurity in quantum systems. The ordinary quantum mechanics is easily recovered as $q_1 = 1$ and $q_2\to1$ or vice versa.