Infinite wall in the fractional quantum mechanics

Abstract

The space-fractional Schrödinger equation for a local potential is difficult to solve because the fractional Riesz operator in it is nonlocal. In fractional quantum mechanics, the infinite wall is a basic problem and has not been solved yet. In this paper, we consider a free particle in an infinite wall region. Making use of the Lévy path integral method, we derive the Lévy path integral amplitude of the particle and take advantage of it to get the solution of the fractional Schrödinger equation for an infinite wall. Then, the infinite wall potential with a delta-function perturbation is also studied. According to Fox’s H-function, we get an equation of bound state energies of a free particle moving in an infinite wall region perturbed by the δ-function for two situations, E < 0 and E > 0. We give a transcendental equation that determines energy levels when E < 0 and prove that there is no bound states when E > 0. The asymptotic behaviors of the center of the delta function approaching to the origin and the infinite are also shown. The results of this paper include special cases in standard quantum mechanics.