Time Fractional Schrodinger Equation Revisited. (28th July 2013)

• Record Type: Journal Article
• Authors: Achar, B. N. Narahari; Yale, Bradley T.; Hanneken, John W.
• Other Names: Baleanu Dumitru Academic Editor.
• Is Part Of: Advances in mathematical physics. Volume 2013(2013)
• Publication Date: 2013-07-28
• Subjects: Mathematical physics -- Periodicals; Mathematical physics; Periodicals; 530.15
• DOI: 10.1155/2013/290216
• Languages: English
• ISSNs: 1687-9120

Abstract

Abstract : The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a "free particle" obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order. The equations of motion contain quantities which have "fractional" dimensions, chosen such that the "energy" has the correct dimension [ ML 2 / T 2 ] . The action S is defined as a fractional time integral of the Lagrangian, and a "fractional Planck constant" is introduced. The TFSE corresponds to a "subdiffusion" equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order. The present work corrects a number of errors in Naber's work. The correct continuity equation for the probability density is derived and a Green function solution for the case of a "free particle" is obtained. The total probability for a "free" particle is shown to go to zero in the limit of infinite time, in contrast with Naber's result of a total probability greater than unity. A generalization to the case of a particle moving in a potential is also given.