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Pramana

, 93:76 | Cite as

Higher-dimensional fractional time-independent Schrödinger equation via fractional derivative with generalised pseudoharmonic potential

  • Tapas Das
  • Uttam GhoshEmail author
  • Susmita Sarkar
  • Shantanu Das
Article
  • 6 Downloads

Abstract

In this paper, we obtain approximate bound-state solutions of N-dimensional time-independent fractional Schrödinger equation for the generalised pseudoharmonic potential which has the form \(V(r^{\alpha })=a_1r^{2\alpha } + ({a_2}/{r^{2\alpha }})+a_3\). Here \(\alpha \;(0<\alpha <1)\) acts like a fractional parameter for the space variable r. The entire study consists of the Jumarie-type fractional derivative and the elegance of Laplace transform. As a result, we can successfully 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 generalisation 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 a Cornell potential model which corresponds to \(\alpha = {1}\) \(/\) \({2}\) with \(a_3=0\) and predicts the approximate mass spectra of quarkonia.

Keywords

Fractional radial Schrödinger equation generalised pseudoharmonic potential bound-state solutions Mittag–Leffler function 

PACS Nos

02.30.−f 03.65.Db 03.65.Ge 02.30.Rz 

Notes

Acknowledgements

The authors would like to thank the anonymous reviewers for their careful reading, useful comments and constructive suggestions for the improvement of the manuscript of the present research work.

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Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Kodalia Prasanna Banga High School (HS)KolkataIndia
  2. 2.Department of Applied MathematicsUniversity of CalcuttaKolkataIndia
  3. 3.Reactor Control System Design Section (E & I Group)Bhabha Atomic Research CentreMumbaiIndia

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