Fractional Quantum Mechanics

Fractional quantum mechanics is a recently emerged and rapidly developing field of quantum physics.

This is the first monograph on fundamentals and physical applications of fractional quantum mechanics, written by its founder.

The fractional Schrödinger equation and the fractional path integral are new fundamental physical concepts introduced and elaborated in the book. The fractional Schrödinger equation is a manifestation of fractional quantum mechanics. The fractional path integral is a new mathematical tool based on integration over Lévy flights. The fractional path integral method enhances the well-known Feynman path integral framework.

Related topics covered in the text include time fractional quantum mechanics, fractional statistical mechanics, fractional classical mechanics and the α-stable Lévy random process.

The book is well-suited for theorists, pure and applied mathematicians, solid-state physicists, chemists, and others working with the Schrödinger equation, the path integral technique and applications of fractional calculus in various research areas. It is useful to skilled researchers as well as to graduate students looking for new ideas and advanced approaches.

Contents:

• Preface

• What is Fractional Quantum Mechanics?

• Fractals

• Fractional Schrödinger Equation

• Time-Independent Fractional Schrödinger Equation

• Fractional Uncertainty Relation

• Path Integral Over Lévy Flights

• A Free Particle Quantum Kernel

• Transforms of a Free Particle Kernel

• Fractional Oscillator

• Some Analytically Solvable Models of Fractional Quantum Mechanics

• Fractional Nonlinear Quantum Dynamics

• Time Fractional Quantum Mechanics

• Applications of Time Fractional Quantum Mechanics

• Fractional Statistical Mechanics

• Fractional Classical Mechanics

• Fractional Dynamics in Polar Coordinate System

• Afterword

• Appendices:

  • Fox H-Function
  • Fractional Calculus
  • Calculation of the Integral
  • Polylogarithm
  • Fractional Generalization of Trigonometric Functions cos z and sin z

• Bibliography

• Index

Readership: Graduate students and researches in quantum theory, stochastic processes, statistical mechanics, special functions.
Key Features:

• This is the first monograph on fractional quantum mechanics — a recently emerged and fast growing field of quantum theory
• It extends applications of fractional calculus to quantum theory
• It has been written by the founder of fractional quantum mechanics
• It presents recently developed fundamentals on the fractional Schrödinger equation and fractional path integral, and their applications to quantum mechanics and quantum statistical mechanics
• A new theoretical physics tool — the fractional path integral, presented for the first time by the book, significantly enhances the well-known Feynman's path integral approach to quantum mechanics and statistical physics
• Never before used in quantum mechanics, the Fox H-function and the Mittag-Leffler function have been introduced and used to obtain close form solutions of fractional quantum mechanical problems