Time independent fractional Schrodinger equation for generalized Mie-type potential in higher dimension framed with Jumarie type fractional derivative

TL;DR

This paper derives approximate solutions for the fractional Schrödinger equation with a generalized Mie-type potential in higher dimensions using Jumarie fractional derivatives, expressing results through Mittag-Leffler functions and validating with known cases.

Contribution

It introduces a fractional parameter into the Mie-type potential and provides approximate bound state solutions using Mittag-Leffler functions, extending quantum models to fractional dimensions.

Findings

Solutions reduce to classical form when ta=1

Eigenvalues and eigenfunctions computed numerically for diatomic molecules

Validation against previous results confirms accuracy

Abstract

In this paper we obtain approximate bound state solutions of $N$-dimensional fractional time independent Schr\"{o}dinger equation for generalised Mie-type potential, namely $V(r^{\alpha})=\frac{A}{r^{2\alpha}}+\frac{B}{r^{\alpha}}+C$. Here $\alpha(0<\alpha<1)$ acts like a fractional parameter for the space variable $r$. When $\alpha=1$ the potential converts into the original form of Mie-type of potential that is generally studied in molecular and chemical physics. The entire study is composed with Jumarie type fractional derivative approach. The solution is expressed via Mittag-Leffler function and fractionally defined confluent hypergeometric function. To ensure the validity of the present work, obtained results are verified with the previous works for different potential parameter configurations, specially for $\alpha=1$. At the end, few numerical calculations for energy eigenvalue and bound states eigenfunctions are furnished for a typical diatomic molecule.