IJPAM: Volume 15, No. 1 (2004)


Muhammad I. Bhatti$^1$, Lokenath Debnath$^2$
$^1$Department of Physics and Geology
University of Texas - Pan American
Edinburg, TX 78541, USA
e-mail: bhatti@panam.edu
$^2$Department of Mathematics
University of Texas - Pan American
1201 West University Drive, Edinburg, TX 78539, USA
e-mail: debnathl@panam.edu

Abstract.The free particle solutions of the fractional Schrödinger and Dirac equations are obtained. The solution of the Schrödinger equation is represented by the generalized three-dimension Gaussian type function whose graphs are presented in terms of several fractional values. Similarly, the solutions to the Dirac equation are presented and also graphs are given to show the behavior of the solution for a range of fractional values. It is observed that, when $\alpha $=1/2, the solution is completely damped. As the $\alpha $ approaches 1, the solutions start to behave normal. It is also shown that the solutions corresponding to the integral order Schrödinger and Dirac equations follow as special cases of those of the corresponding fractional partial differential equations. Using the joint Fourier and Laplace transforms, the solutions of the above equations are obtained in terms of Mittag-Leffler functions.

Received: February 17, 2004

AMS Subject Classification: 35J10, 35Q55, 81Q05

Key Words and Phrases: fractional Schrödinger equation, Dirac equation, Gaussian type function, Mittag-Leffler functions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 15
Issue: 1