Fractional Quantum Mechanics

Abstract

A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the Lévy 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 Lévy 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 Schrödinger 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.

Fractional Schrödinger Equation in the Presence of the Linear Potential

“…Several years ago, the classical Schrödinger equation has been generalized to a fractional partial differential equation that takes into account the Riesz space-fractional derivative instead of the conventional Laplacian [1,2]. Apart from quantum mechanics, there are many other equations occurring in science that have been reconsidered in terms of fractional derivatives such as the diffusion-wave equation [3][4][5][6], the Langevin equation [7] or the radiative transport equation [8].…”

On the Well-Posedness for a Class of Pseudo-Differential Parabolic Equations

“…On the other hand, fractional diffusion equations have a wide variety of applications e.g. fluid mechanics [6,22], mathematical finance [7] and fractional dynamics [18,19,26].…”

Nontrivial Solutions for Time Fractional Nonlinear Schrödinger-Kirchhoff Type Equations

“…For further details and applications, we refer the reader to [18,19] and the references cited therein.…”