Applications Of Lévy Path Integral In Fractional Quantum Mechanics

Fractional quantum mechanics is a generalization of quantum mechanics.In the non-relativistic case,it is described by the space fractional Schr?dinger equation,which contains the fractional Riesz derivative operator.Most microscopic particles move in a finite space,so the local potential field problem is the basic problem in quantum mechanics.Because of the nonlocality of Riesz fractional operator,it is very difficult to solve the fractional Schr?dinger equation in local potential field,which limits the development of fractional quantum mechanics.In this paper,based on the origin of the fractional Schrodinger equation,a method for solving the fractional Schr?dinger equation under the local potential field by means of Lévy path integral is presented.The main work is given as follows:1.The Lévy path integrals and propagators of particles moving on a ring and in an infinite square potential well are established.By using the relationship between Lévy path integrals and solutions of fractional Schr?dinger equation,the wave functions and energy values of free particles moving on a circle and in an infinite square potential well are obtained.Laplace transformation,energy-time transformation and momentum representation of free particle propagator moving on a circle are given.2.In the case of infinite square potential well,considering the locality of potential field,we reconstruct the space fractional Schr?dinger equation by periodically extending the original Lévy path integral propagator,and use the fractional Schrodinger equation to solve its wave functions and energy values.The new equation overcomes the problem that the original fractional Schr?dinger equation is not applicable to the local potential field.The method used provides a new idea for the study of fractional quantum mechanics in the local potential field.