ISRN Mathematical PhysicsVolume 2012 (2012), Article ID 935365, 7 pagesdoi:10.5402/2012/935365

Research Article

Solutions of Unified Fractional Schrödinger Equations

V. B. L. Chaurasia¹ and Devendra Kumar²

¹Department of Mathematics, University of Rajasthan, Jaipur 302055, Rajasthan, India²Department of Mathematics, JaganNath Gupta Institute of Engineering and Technology, Jaipur 302022, Rajasthan, India

Received 15 October 2011; Accepted 13 November 2011

Academic Editors: D. Gepner, M. Montesinos, D. Singleton, and F. Sugino

Copyright © 2012 V. B. L. Chaurasia and Devendra Kumar. 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.

Abstract.

We obtain the solution of a unified fractional Schrödinger equation. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the Mittag-Leffler function. The result obtained here is quite general in nature and capable of yielding a very large number of results (new and known) hitherto scattered in the literature. Most of results obtained are in a form suitable for numerical computation.

1. Introduction

Recent applications of fractional differential equations to number of systems such as those exhibiting enormously slow diffusion or subdiffusion have given opportunity for physicists to study even more complicated systems. These systems include charge transport in amorphous semiconductors: the relaxation in polymer systems, fluid mechanics, viscoelasticity, and Hall effect. The generalized diffusion equation is studied to describe complex systems with anomalous behavior in much the same way as simpler systems. Fractional calculus is now considered as practical techniques in many branches of applied sciences and engineering. Several authors notably Hilfer [1], Beyer and Kempfle [2], Kempfle and Gaul [3], Schneider and Wyss [4], and Debnath [5–7] have discussed many examples of homogeneous fractional ordinary differential equations and homogeneous fractional diffusion and wave equations.

Laskin [8–10] constructed space fractional quantum mechanics by using Feynman path integrals, the only difference being that Lévy distributions are employed instead of Gaussians distributions for the set of possible paths. The Schrödinger equation thus obtained contains fractional derivatives. Naber [11] has investigated certain properties of time fractional Schrödinger equation by writing the Schrödinger equation in terms of fractional derivatives as dimensionless objects. In recent work, solutions of fractional Schrödinger equations are investigated by Bhatti [12], Chaurasia and Singh [13], Saxena et al. [14], among others.

In the present paper, we obtain the solution of a unified fractional Schrödinger equation. The result obtained here provides an elegant extension of the results given earlier by Bhatti [12], Chaurasia and Singh [13], Saxena et al. [14], and Debnath [5].

The Riemann-Liouville fractional integral of order is defined by (Miller and Ross [15, page 45]; Kilbas et al. [16]), where .

The following fractional derivative of order is introduced by Caputo [17]; see also Kilbas et al. [16] in the form where is the th partial derivative of with respect to .

The Laplace transform of the Caputo derivative is given by Caputo [17]; see also Kilbas et al. [16] in the form We also need the Weyl fractional operator, defined by where is an integral part of .

Its Fourier transform is (Metzler and Klafter [18, page 59, A.11]) where is the Fourier transform of with respect to the variable of .

Following the convention initiated by Compte [19], we suppress the imaginary unit in Fourier space by adopting the slightly modified form of above result in our investigations (Metzler and Klafter, [18, p.59, A.12]) instead of (1.5).

2. Unified Fractional Schrödinger Equation

In this section, we will derive the solution of the unified fractional Schrödinger equation (2.1). The result is as follows.

Theorem 2.1. Consider the following unified fractional Schrödinger equation: with the initial conditions: where is Planck’s constant divided by 2π, is the mass, and is a wave function of the particle. Then, for the solution of (2.1), subject to the initial conditions (2.2), there holds the formula where and is the generalized Mittag-Leffler function [20].

Proof. Applying the Laplace transform with respect to the time variable on both the sides of (2.1) and using the initial conditions (2.2), we get where . If we apply the Fourier transform with respect to variable and use the formula (1.6), it yields Solving for , it gives On taking the inverse Laplace transform of (2.6) and applying the formula (Saxena et al. [21]), it is seen that Finally, the required solution (2.3) is obtained by taking inverse Fourier transform of (2.7).

3. Special Cases

If we take in (2.1), then we obtain the following result.

Corollary 3.1. Consider the following fractional Schrödinger equation with the initial conditions: where is Planck’s constant divided by 2π, is the mass, and is the wave function of the particle. Then, for the solution of (3.1), subject to the initial conditions (3.2), there holds the formula If we set in (2.1), then we arrive at the following result recently obtained by Saxena et al. [14].

Corollary 3.2. The solution of the following fractional Schrödinger equation: with the initial conditions where is the Planck’s constant divided by 2π, is the mass, and is a wave function of the particle, is given by Finally, on taking and μ = 2 in (2.1), then we arrive at the following result given by Bhatti [12].

Corollary 3.3. Consider the following fractional Schrödinger equation: with the initial conditions: where is Planck’s constant divided by 2π, is the mass, and is a wave function of the particle. Then, for the solution of (3.7), under the initial conditions (3.8), there holds the relation:

4. Conclusion

In this paper, we have introduced a unified fractional Schrödinger equation and established solution for the same. The solution has been developed in terms of the generalized Mittag-Leffler function in a compact and elegant form with the help of Laplace and Fourier transforms and their inverses. All the results derived in this paper are in a form suitable for numerical computation. The fractional Schrödinger equation discussed in the present article contains a number of known (may be new also) fractional Schrödinger equations. The results obtained in the present paper provide an extension of the results given by Bhatti [12], Chaurasia and Singh [13], Debnath [5], and Saxena et al. [14].

Acknowledgment

The authors are grateful to Professor H. M. Srivastava, University of Victoria, Canada, for his kind help and valuable suggestions in the preparation of this paper.

References

R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000.

H. Beyer and S. Kempfle, “Definition of physical consistent damping laws with fractional derivatives,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 75, pp. 623–635, 1995.

S. Kempfle and L. Gaul, “Global solutions of fractional linear differential equations,” ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik, vol. 76, no. 2, pp. 571–572, 1996.

W. R. Schneider and W. Wyss, “Fractional diffusion and wave equations,” Journal of Mathematical Physics, vol. 30, no. 1, pp. 134–144, 1989.

L. Debnath, “Fractional integrals and fractional differential equations in fluid mechanics,” Fractional Calculus and Applied Analysis, vol. 6, pp. 119–155, 2003.

L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäuser, Basel, Switzerland, 2nd edition, 2005.

L. Debnath, “Recent applications of fractional calculus to science and engineering,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 54, pp. 3413–3442, 2003.

N. Laskin, “Fractals and quantum mechanics,” Chaos, vol. 10, no. 4, pp. 780–790, 2000.

N. Laskin, “Fractional quantum mechanics and Levy path integrals,” Physics Letters, Section A, vol. 268, no. 4–6, pp. 298–305, 2000.

N. Laskin, “Fractional Schrödinger equation,” Physical Review E, vol. 66, no. 5, Article ID 056108, 7 pages, 2002.

M. Naber, “Distributed order fractional sub-diffusion,” Fractals, vol. 12, no. 1, pp. 23–32, 2004.

M. Bhatti, “Fractional Schrödinger wave equation and fractional uncertainty principle,” International Journal of Contemporary Mathematical Sciences, vol. 2, pp. 943–952, 2007.

V. B. L. Chaurasia and J. Singh, “Application of sumudu transform in schödinger equation occurring in quantum mechanics,” Applied Mathematical Sciences, vol. 4, no. 57, pp. 2843–2850, 2010.

R. K. Saxena, R. Saxena, and S. L. Kalla, “Computational solution of a fractional generalization of the Schrödinger equation occurring in quantum mechanics,” Applied Mathematics and Computation, vol. 216, no. 5, pp. 1412–1417, 2010.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.

M. Caputo, Elasticitá e Dissipazione, Zanichelli, Bologna, Italy, 1969.

R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Report, vol. 339, no. 1, pp. 1–77, 2000.

A. Compte, “Stochastic foundations of fractional dynamics,” Physical Review E, vol. 53, no. 4, pp. 4191–4193, 1996.

T. R. Prabhakar, “A singular integral equation with a generalized Mittag-Leffler function in the kernel,” Yokohama Journal of Mathematics, vol. 19, pp. 7–15, 1971.

R. K. Saxena, A. M. Mathai, and H. J. Haubold, “Solultions of fractional reaction-diffusion equations in terms of Mittag-Leffler functions,” International Journal of Scientific Research, vol. 15, pp. 1–17, 2006.