Fractional Quantum Mechanics

TL;DR

This paper introduces fractional quantum mechanics using a path integral approach based on Levy flights, leading to a fractional Schrödinger equation and generalizations of quantum principles, with potential implications for understanding fractal structures in quantum systems.

Contribution

It develops a fractional path integral framework and derives a fractional Schrödinger equation, extending quantum mechanics to fractal Levy paths and establishing new relationships between energy, momentum, and uncertainty.

Findings

Derived fractional Schrödinger equation

Expressed free particle kernel with Fox's H function

Established fractional uncertainty relation

Abstract

A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the L\'evy flights are defined. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the L\'evy paths leads to fractional quantum mechanics and fractional statistical mechanics. The fractional quantum and statistical mechanics have been developed via our fractional path integral approach. A fractional generalization of the Schr\"odinger equation has been found. A relationship between the energy and the momentum of the nonrelativistic quantum-mechanical particle has been established. The equation for the fractional plane wave function has been obtained. We have derived a free particle quantum-mechanical kernel using Fox's H function. A fractional generalization of the Heisenberg uncertainty relation has been established. Fractional statistical mechanics has been developed via the path integral approach. A fractional generalization of the motion equation for the density matrix has been found. The density matrix of a free particle has been expressed in terms of the Fox's H function. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum and statistical mechanics.