Quantization of the Bateman damping system with conformable derivative

TL;DR

This paper introduces a quantization approach for the damped harmonic oscillator using conformable derivatives, deriving energy spectra and probability densities that depend on the fractional order, thus extending classical quantum mechanics to fractional systems.

Contribution

It proposes a novel conformable derivative-based Lagrangian and Hamiltonian for the Bateman system, deriving the corresponding quantum equations and analyzing their solutions.

Findings

Energy eigenvalues are real for the conformable system.

Probability densities show gradual ordering with fractional order .

The conformable Schr46dinger equation extends quantum mechanics to fractional derivatives.

Abstract

In this work, the conformable Bateman Lagrangian for the damped harmonic oscillator system is proposed using the conformable derivative concept. In other words, the integer derivatives are replaced by conformable derivatives of order $\alpha$ with $0<\alpha\leq 1$. The corresponding conformable Euler-Lagrange equations of motion and fractional Hamiltonian are then obtained. The system is then canonically quantized and the conformable Schrodinger equation is constructed. The fractional-order dependence of the energy eigenvalues $E_n ^\alpha$ and eigenfunctions $\psi_n ^\alpha$ are obtained using using suitable transformations and the extended fractional Nikiforov-Uvarov method. The corresponding conformable continuity equation is also derived and the probability density and probability current are thus suitably defined. The probability density evolution as well as its dependence on $\alpha$ is plotted and analyzed for various situations. It is found that the energy eigenvalues are real and there are sort of gradual ordering in the behavior of the probability densities.