Principles of Fractional Quantum Mechanics

TL;DR

This paper reviews the fundamentals and diverse physical applications of fractional quantum mechanics, including its mathematical foundations and specific quantum systems modeled with fractional derivatives.

Contribution

It provides a comprehensive overview of fractional quantum mechanics, highlighting new representations and applications that extend traditional quantum theory.

Findings

Fractional Schrödinger equation and Riesz fractional derivative are fundamental.

Applications include fractional models of potential wells, atoms, and oscillators.

Lévy path integral approach links fractional quantum mechanics with statistical mechanics.

Abstract

A review of fundamentals and physical applications of fractional quantum mechanics has been presented. Fundamentals cover fractional Schr\"odinger equation, quantum Riesz fractional derivative, path integral approach to fractional quantum mechanics, hermiticity of the Hamilton operator, parity conservation law and the current density. Applications of fractional quantum mechanics cover dynamics of a free particle, new representation for a free particle quantum mechanical kernel, infinite potential well, bound state in {\delta}-potential well, linear potential, fractional Bohr atom and fractional oscillator. We also review fundamentals of the L\'evy path integral approach to fractional statistical mechanics.