On the path integral simulation of space-time fractional Schroedinger equation with time independent potentials

Abstract

In this work a Feynman-Kac path integral method based on Levy measure has been proposed for solving the Cauchy problems associated with the space-time fractional Schroedinger equations arising in interacting systems in fractional quantum mechanics. The Continuous Time Random Walk(CTRW) model is used to simulate the underlying Levy process-a generalized Wiener process. Since we are interested to capture the lowest energy state of quantum systems, we use Pareto distribution as opposed to Mittag-Leffler random variables, which are more suitable for finite time. Adopting the CTRW model we have been able to simulate the space-time fractional diffusion process with comparable simplicity and convergence rate as in the case of a standard diffusion. We hope this paves an elegant way to solve space-time diffusion equations numerically through Fractional Feynman-Kac path integral technique as an alternative to fractional calculus.