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