Time and Space Fractional Schrödinger Equation with Fractional Factor**Supported by National Natural Science Foundation of China under Grant Nos. 11472247 and 11872335

Abstract

In this paper, we introduce a new definition of fractional derivative which contains a fractional factor, and its physical meanings are given. When studying the fractional Schrödinger equation (FSE) with this form of fractional derivative, the result shows that under the description of time FSE with fractional factor, the probability of finding a particle in the whole space is still conserved. By using this new definition to construct space FSE, we achieve a continuous transition from standard Schrödinger equation to the fractional one. When applying this form of Schrödinger equation to a particle in an infinite symmetrical square potential well, we find that the probability density distribution loses spatial symmetry and shows a kind of attenuation property. For the situation of a one-dimensional infinite δ potential well, the first derivative of time-independent wave function Φ to space coordinate x can be continuous everywhere when the particle is at some special discrete energy levels, which is much different from the standard Schrödinger equation.