Fractional quantum mechanics

By Nick Laskin. Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2008

784 citations recorded by the source corpus.

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.

Subfields: Economics and Econometrics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a102682999f8539b6c841c84b0d43e93bffdf8c8

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Citations in the corpus

The higher order fractional Calderón problem for linear local operators: Uniqueness (2020, background): “y and ecology just to mention a few [2, 5, 12, 27, 33]. Another place where the fractional counterpart of the classical Laplacian naturally shows up is the formulation of fractional quantum mechanics [22, 23]. For more applications of fractional mathematical models, see [5] and the references therei”
Strichartz Estimates in Spherical Coordinates (2012, neutral): “Typical examples of ω are ρ(0 < a 6= 1) , √ 1 + ρ2, ρ √ 1 + ρ2, and ρ √ 1+ρ2 which describe Schrödinger type equation [22], Klein-Gordon or semirelativistic equation[15], iBq, and imBq (see [10] and references therein).”
Fractional powers of second order partial differential operators: extension problem and regularity theory (2010, methodology): “S olo por mencionar algunos m as, adem as de los modelos en Matem atica Financiera [29] y en mec anica de uidos [56, 27], hay aplicaciones en la F sica Moderna, por ejemplo, cuando se consideran cin eticas fraccionarias y transporte an omalo [90, 70], cin eticas raras [64], mec anica cu antica fracc”
Existence and stability of standing waves for nonlinear fractional Schr\"odinger equations with Hartree type nonlinearity (2012, background): “Laskin [14][15][16] as a result of extending the Feynman path integral from the Brownian-like to Lévy-like quantum mechanical paths.”
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, background): “and nonlocal diusion, and also in the formulation of fractional quantum mechanics where the fractional Schrodinger equation arises naturally as a generalization of the ordinary Schrodinger equation [3, 6, 17, 27, 48, 49, 53, 66]. Since the fractional Laplacian is a non-local operator, it is more nat”
Concentrating standing waves for the fractional nonlinear Schr\"odinger equation (2013, background): “1) was introduced by Laskin [19] as an extension of the classical nonlinear Schrödinger equation s = 1 in which the Brownian motion of the quantum paths is replaced by a Lévy flight.”