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Advances in Mathematical Physics

Volume 2013 (2013), Article ID 290216, 11 pages

http://dx.doi.org/10.1155/2013/290216

Research Article

## Time Fractional Schrodinger Equation Revisited

Physics Department, University of Memphis, Memphis, TN 38152, USA

Received 29 April 2013; Revised 1 July 2013; Accepted 2 July 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 B. N. Narahari Achar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- K. B. Oldham and J. Spanier,
*The Fractional Calculus*, Academic Press, New York, NY, USA, 1974. View at Zentralblatt MATH · View at MathSciNet - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993. View at Zentralblatt MATH · View at MathSciNet - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH · View at MathSciNet - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives: Theory and Applications*, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. View at Zentralblatt MATH · View at MathSciNet - A. Carpinteri and F. Mainardi,
*Fractals and Fractional Calculus in Continuum Mechanics*, vol. 378 of*CISM Courses and Lectures*, Springer, Vienna, Austria, 1997. View at Zentralblatt MATH · View at MathSciNet - R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific Publishing, River Edge, NJ, USA, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. J. West, M. Bologna, and P. Grigolini,
*Physics of Fractal Operators*, Institute for Nonlinear Science, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. L. Magin,
*Fractional Calculus in Bioengineering*, Begell House, Redding, Conn, USA, 2006. - L. Debnath and D. Bhatta,
*Integral Transforms and Their Applications*, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2nd edition, 2007. View at Zentralblatt MATH · View at MathSciNet - R. Hermann,
*Fractional Calculus*, World Scientific, Singapore, 2011. - A. B. Malinowska and D. F. M. Torres,
*Introduction to the Fractional Calculus of Variations*, Imperial College Press, London, UK, 2012. View at Zentralblatt MATH · View at MathSciNet - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus: Models and Numerical Methods*, vol. 3 of*Series on Complexity, Nonlinearity and Chaos*, World Scientific Publishing, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. K. GolmanKhaneh,
*Investigations in Dynamics:With Focus on Fractal Dynamics*, Lap Lambert Academic Publishing, 2012. - N. Laskin, “Fractional quantum mechanics,”
*Physical Review E*, vol. 62, no. 3, pp. 3135–3145, 2000. View at Publisher · View at Google Scholar · View at Scopus - N. Laskin, “Fractional quantum mechanics and Lévy path integrals,”
*Physics Letters A*, vol. 268, no. 4–6, pp. 298–305, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Laskin, “Fractional Schrödinger equation,”
*Physical Review E*, vol. 66, no. 5, Article ID 056108, 7 pages, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - N. Laskin, “Fractals and quantum mechanics,”
*Chaos*, vol. 10, no. 4, pp. 780–790, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Laskin, “Principles of fractional quantum mechanics,” http://arxiv.org/abs/1009.5533.
- M. Naber, “Time fractional Schrödinger equation,”
*Journal of Mathematical Physics*, vol. 45, no. 8, pp. 3339–3352, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. K. Saxena, R. Saxena, and S. L. Kalla, “Solution of space-time fractional Schrödinger equation occurring in quantum mechanics,”
*Fractional Calculus & Applied Analysis*, vol. 13, no. 2, pp. 177–190, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Wang and M. Xu, “Generalized fractional Schrödinger equation with space-time fractional derivatives,”
*Journal of Mathematical Physics*, vol. 48, no. 4, Article ID 043502, 10 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Dong and M. Xu, “Space-time fractional Schrödinger equation with time-independent potentials,”
*Journal of Mathematical Analysis and Applications*, vol. 344, no. 2, pp. 1005–1017, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Guo and M. Xu, “Some physical applications of fractional Schrödinger equation,”
*Journal of Mathematical Physics*, vol. 47, no. 8, Article ID 082104, 9 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Y. Jiang, “Time-space fractional Schrödinger like equation with a nonlocal term,”
*European Physical Journal: Special Topics*, vol. 193, no. 1, pp. 61–70, 2011. View at Publisher · View at Google Scholar · View at Scopus - J. Dong, “Fractional Green's function for the time-dependent scattering problem in the space-time-fractional quantum mechanics,” http://arxiv.org/abs/1301.3206.
- J. Dong and M. Xu, “Applications of continuity and discontinuity of a fractional derivative of the wave functions to fractional quantum mechanics,”
*Journal of Mathematical Physics*, vol. 49, no. 5, Article ID 052105, 16 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Scalas, D. Baleanu, F. Mainardi, and A. Mura, “Fractional calculus and the Schrodinger equation,” in
*Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications*, Porto, Portugal, July 2006. - M. Bhatti, “Fractional Schrödinger wave equation and fractional uncertainty principle,”
*International Journal of Contemporary Mathematical Sciences*, vol. 2, no. 17–20, pp. 943–950, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Wang, M. Xu, and X. Li, “Green's function of time fractional diffusion equation and its applications in fractional quantum mechanics,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 2, pp. 1081–1086, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Tofighi, “Probability structure of time fractional schrodinger equation,”
*Acta Physica Polonica A*, vol. 116, no. 2, pp. 114–118, 2009. View at Google Scholar · View at Scopus - A. Iomin, “On fractional time quantum dynamics,”
*Physical Review E*, vol. 80, no. 2, Article ID 022103, 4 pages, 2009. View at Publisher · View at Google Scholar - A. Iomin, “Fractional-time Schrödinger equation: fractional dynamics on a comb,”
*Chaos, Solitons & Fractals*, vol. 44, no. 4-5, pp. 348–352, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. M. Shahin, E. Ahmed, and Y. A. Omar, “On fractional order quantum mechanics,”
*International Journal of Nonlinear Science*, vol. 8, no. 4, pp. 469–472, 2009. View at Google Scholar · View at MathSciNet - P. Rozmej and B. Bandrowski, “On fractional Schrödinger equation,”
*Computational Methods in Science and Technology*, vol. 16, pp. 191–194, 2010. View at Google Scholar - H. Ertik, D. Demirhan, H. Şirin, and F. Büyükkılıç, “Time fractional development of quantum systems,”
*Journal of Mathematical Physics*, vol. 51, no. 8, Article ID 082102, 15 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - S. I. Muslih, O. P. Agrawal, and D. Baleanu, “A fractional Schrödinger equation and its solution,”
*International Journal of Theoretical Physics*, vol. 49, no. 8, pp. 1746–1752, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. B. L. Chaurasia and D. Kumar, “Solutions of Unified fractional Schrödinger equation,”
*ISRN Mathematical Physics*, vol. 2012, Article ID 935365, 7 pages, 2012. View at Publisher · View at Google Scholar - A. K. Mahata, “On fractional Schrödinger equation and its application,”
*International Journal of Scientific Engineering Research*, vol. 4, pp. 1–5, 2013. View at Google Scholar - A. L. de Paoli and M. C. Rocca, “The fractionary Schrödinger equation, Green functions and ultradistributions,”
*Physica A*, vol. 392, no. 1, pp. 111–122, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - E. C. Oliveira, F. S. Costa, and J. Vaz Jr., “The fractional Schrödinger equation for delta potentials,”
*Journal of Mathematical Physics*, vol. 51, no. 12, Article ID 123517, 16 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - K. M. Kolwankar and A. D. Gangal, “Fractional differentiability of nowhere differentiable functions and dimensions,”
*Chaos*, vol. 6, no. 4, pp. 505–513, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis,”
*Boundary Value Problems*, vol. 2013, article 131, 2013. View at Publisher · View at Google Scholar - R. P. Feynman and A. R. Hibbs,
*Quantum Mechanics and Path Integrals*, McGraw-Hill, New York, NY, USA, 1965. - H. Kleinert,
*Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets*, World Scientific Publishing, Singapore, 2006. - L. S. Schulman,
*Techniques and Applications of Path Integration*, John Wiley & Sons, New York, NY, USA, 1981. View at MathSciNet - M. Kac,
*Probability and Related Topics in Physical Sciences*, Interscience, London, UK, 1957. - B. N. Narahari Achar, J. W. Hanneken, N. Becker, and M. Morgan, “Intrinsic dissipation in fractional order dynamics,” in
*Proceedings of the International Workshop on Fractional Differentiation and Its Applications (FDA '10)*, Badajoz, Spain, October 2010. - B. N. Narahari Achar and J. W. Hanneken, “Fractional order dynamics under gravity,” in
*Proceedings of the International Workshop on Fractional Differentiation and Its Applications (FDA '10)*, Badajoz, Spain, October 2010. - L. I. Schiff,
*Quantum Mechanics*, McGraw-Hill, New York, NY, USA, 1968. - F. Mainardi, P. Paradisi, and R. Gorenflo, “Probability distributions generated by fractional order diffusion equations,” in
*Econophysics: An Emerging Science*, J. Kertesz and I. Kondov, Eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998. View at Google Scholar - R. Gorenflo, A. Iskenderov, and Y. Luchko, “Mapping between solutions of fractional diffusion-wave equations,”
*Fractional Calculus & Applied Analysis*, vol. 3, no. 1, pp. 75–86, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,”
*Fractional Calculus & Applied Analysis*, vol. 4, no. 2, pp. 153–192, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Mainardi, “The time fractional diffusion-wave equation,”
*Radiofisica*, vol. 38, no. 1-2, pp. 20–36, 1996. View at Google Scholar - F. Mainardi, “The fundamental solutions for the fractional diffusion-wave equation,”
*Applied Mathematics Letters*, vol. 9, no. 6, pp. 23–28, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Mainardi and G. Pagnini, “The role of the Fox-Wright functions in fractional sub-diffusion of distributed order,”
*Journal of Computational and Applied Mathematics*, vol. 207, no. 2, pp. 245–257, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Mainardi, A. Mura, and G. Pagnini, “The M-Wright function in time-fractional diffusion processes: a tutorial survey,”
*International Journal of Differential Equations*, vol. 2010, Article ID 104505, 29 pages, 2010. View at Publisher · View at Google Scholar - R. Gorenflo, Y. Luchko, and F. Mainardi, “Analytical properties and applications of the Wright function,”
*Fractional Calculus & Applied Analysis*, vol. 2, no. 4, pp. 383–414, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Gorenflo, F. Mainardi, D. Moretti, G. Pagnini, and P. Paradisi, “Fractional diffusion: probability distributions and random walk models,”
*Physica A*, vol. 305, no. 1-2, pp. 106–112, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - F. Mainardi,
*Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models*, Imperial College Press, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet