Path Integral Approach to Time-Fractional Quantum Mechanics

Nikolai Laskin

Nick Laskin

"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."--Publisher's website

Nick Evans, Stephen F. King

Nick Laskin

Nick Laskin

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

Jianping Dong, Han Geng

Fractional quantum mechanics in the non-relativistic case is described by the space-fractional Schrödinger equation containing the fractional Riesz operator, which is a non-local operator. The non-locality makes it difficult to solve the fractional Schrödinger equation in the local potentials. In this paper, we study the solution of the fractional Schrödinger equation of a free particle moving on a circle by use of the Lévy path integrals approach. We present the Lévy path integrals propagator of a particle on circle and then use it to get the wave functions and energy eigenvalues of the free particle case. In addition, the Laplace transform, energy-time transform, and momentum representation of the free particle kernel are also obtained. The results of this paper contain the ones in standard quantum mechanics as special cases. The problem of a particle on circle is a fundamental problem of path integrals with topological constraints in quantum systems, and the method of Lévy path integrals can be generalized to study more complex local potentials.