Higher dimensional fractional time independent Schr\"{o}dinger equation via Jumarie fractional derivative with generalized pseudoharmonic potential

TL;DR

This paper derives approximate solutions for the N-dimensional fractional Schrödinger equation with a generalized pseudoharmonic potential using Jumarie derivatives, expressing results via Mittag-Leffler functions and exploring applications to molecular and quarkonium systems.

Contribution

It introduces a fractional approach to the Schrödinger equation with a generalized potential, extending previous models to arbitrary dimensions and fractional parameters using Jumarie derivatives.

Findings

Derived approximate bound state solutions in fractional space.

Numerical energy eigenvalues for diatomic molecules.

Predicted mass spectra of quarkonia with fractional potential.

Abstract

In this paper we obtain approximate bound state solutions of $N$-dimensional time independent fractional Schr\"{o}dinger equation for generalised pseudoharmonic potential which has the form $V(r^{\alpha})=a_1r^{2\alpha}+\frac{a_2}{r^{2\alpha}}+a_3$. Here $\alpha(0<\alpha<1)$ acts like a fractional parameter for the space variable $r$. The entire study is composed with the Jumarie type derivative and the elegance of Laplace transform. As a result we successfully able to express the approximate bound state solution in terms of Mittag-Leffler function and fractionally defined confluent hypergeometric function. Our study may be treated as a generalization of all previous works carried out on this topic when $\alpha=1$ and $N$ arbitrary. We provide numerical result of energy eigenvalues and eigenfunctions for a typical diatomic molecule for different $\alpha$ close to unity. Finally, we try to correlate our work with Cornell potential model which corresponds to $\alpha=\frac{1}{2}$ with $a_3=0$ and predict the approximate mass spectra of quarkonia.