Generalización fraccional de la ecuación de Schrodinger relacionada a la Mecánica Cuántica
Palabras clave:
Funciòn de Mittag-Leffler, Funciòn-H, Transformada Sumudu, Transformada de Laplace, Derivada de CaputoResumen
El objeto de este trabajo es presentar una solución computacional de una generalización fraccional unidi- mensional de la ecuación de Schrodinger, relacionada a la Mecánica Cuántica. El método utiliza conjuntamente la transformada de Sumudu y la transformada de Fourier. La solución es obtenida en forma computacional y cerrada en términos de la función de Mittag-Leffler y la función H. El resultado principal obtenido aquí es general y, a partir de éste, se pueden deducir un gran número de casos especiales, hasta ahora dispersos en la literatura. Además, éste provee una extensión de un resultado dado anteriormente por Debnath, Saxena y Chaurasia. El resultado principal es presentado en forma de Teorema y se mencionan varios casos especiales.
Descargas
Referencias
Belgacem, F.B.M., Karaballi, A. A. and Kalla, S.L. Analytic investigations of the Sumudu transform and ap- plications to integral production equations, Mathematical Problems in Engineering 3(2003), 103-118.
Belgacem, F.B.M., Karaballi, A. A, Sumudu transform fundamental properties, investigations and applications, Intern. J. Appl. Math. Stoch. Anal. 2005(2005) 1-23.
Bhatti, M., Fractional Schrödinger wave and fractional uncertainty principle Int. J. Contemp. Math. Sciences, 2, No. 19 (2007), 943-950.
Caputo, M., Elasticitá e Dissipazione, Zanichelli, Bologna, 1969.
Chaurasia., V.B.L. and Singh, J, Application of Sumudu transform in Schrödinger equation occurring in quan- tum mechanics, Appl. Math. Sci., 4 (2010), no. 57, 2843-2850.
Debnath. L, Fractional integral and fractional differential equations in fluid Mechanics, Frac. Calc. Appl. Anal.
(2003) 119-155.
Debnath, L., Nonlinear Partial Differential Equations for Scientists and Engineers, Second Edition, Birkhäuser, Boston, Basel, Berlin, 2005.
Debnath, L. and Bhatti, M, On fractional Schrödinger and Dirac equations, Int. J. J. Pure Appl. Math. 15 (2004), 1-11.
Diethelm, The Analysis of Fractional Differential Equations. Springer, Berlin, 2010.
Doetsch, G, Anleitung zum Prakitischen Gebrauch der Laplace - Transformation. Oldenberg, Munich, 1956.
Erdélyi,A. (Ed.) Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York. 1955.
Feynman, R.P. and Hibbs, A. R, Quantum Mechanics and Path integrals, McGraw-Hill, New York, 1965.
Guo, X. and Xu, M, Some physical applications of Schrödinger equation, J. Math. Phys.. 47082104 (2006) doi: 10. 1063/1. 2235026 (9 pages).
Haubold, H.J, Mathai, A.M. and Saxena, R.K.(2007). Solution of reaction-diffusion equations in terms of the H-function, Bull.Astro. Soc.India., 35 (4), (2007), 681-689.
Henry,B.I. and Wearne, S.L. (2000). Fractional reaction-diffusion, Physica A, 276 (2000) 448-455
Hilfer, R. (editor) Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
Kilbas, A,A., Srivastava, H.M. and Trujillo, J.J, Theory and Applications of Fractional Differential equations, Elsevier, Amsterdam. 2006.
Laskin, N. Fractals and quantum mechanics, Chaos, 10, no.4, (2000) 780-790
Laskin, N. Fractional quantum mechanics and Levy path integrals, Physics Letters A 268 (2000), 298-305.
Laskin, N, Fractional Schrödinger equation, Physical Review E, 66, 056108 arXiv: quant-ph/0206098 v1 14 Jun 2002.
Mainardi,F, Fractional Calculus and Waves in Linear Viscoelacsticity, World Scientific, Singapore, 2010.
Mathai, A.M. Saxena, R.K. and Haubold, H.J. The H-function: Theory and Applications, Springer, New York, 2010.
Metzler, R. and Klafter, J. The random walk: A guide to anomalous diffusion: A fractional dynamics approach. Phy. Rep., 339 (2000)1-77.
Mittag-Leffler, G.M, Sur la nouvelle function E , C.R. Acad. Sci., Paris, 13 (1903) 554-558
Naber, M, Distributed order fractional sub-diffusion, Fractals 12, no.1, (2004) 23-32.
Oldham, K.B. and Spanier, J., The Fractional Calculus Theory and Appliations of Differentiation and Integration to Arbitrary Order, Academic Press,New York, 1974.
Orsingher, E and Beghin, L. Time- fractional telegraph equations and telegraph processes with brownian time, Probab.Theory Relat. Fields 128(2004) 141-160.
Prabhakar, T. R., A singular integral equation with generalized Mittag-Leffler function in the kernel, Yokohama Math. J, 19 (1971) 7-15.
Podlubny, I., Fractional Differential Equations, Academic Press, New York,1999.
Samko, S.G., Kilbas, A.A. and Marichev, O.I, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishing, Amsterdam, 1993.
Saxena, R.K., Mathai, A.M. and Haubold, H.J. (2006). Fractional reaction-diffusion equations, Astrophysics
and space science 305 (2006) 289-296.
Saxena, R.K., Mathai, A.M. and Haubold, H.J, Solution of generalized fractional reaction-diffusion equations, Astrophysics and space science 305, (2006) 305-313.
Saxena, R.K., Mathai, A.M. and Haubold, H.J., Reaction –diffusion systems and nonlinear waves, Astrophysics
and space science 305 (2006) 297-303.
Saxena, R.K. Saxena, R. and Kalla, S.L, Computational solution of a fractional generalization of the Schrödinger equation occurring in quantum Mechanics, Appl. Math. Comput. 216(2010), 1412-1417.
Saxena, R.K. Saxena, R. and Kalla, S.L., Solution of space-time fractional Schrödinger equation occurring in quantum Mechanics, Fract. Calc. Appl. Anal., 13, no.2 (2010), 177-190.
Tofight, A. Probability structure of time fractional Schrödinger equation, Acta Physica Polonica A 116, No. 2(2009), 111-118.
Wiman, A, Uber den fundamental satz in der theorie der funktionen Eα(x). Acta Math. 29 (1905), 191-201.
Descargas
Publicado
Número
Sección
Licencia
Derechos de autor 2011 ©S. L. Kalla ©R. K. Saxena ©Ravi Saxena
Esta obra está bajo una licencia Creative Commons Reconocimiento 3.0 Unported.
Avenida 2 “El Milagro”, entrada autónoma de la 