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Fractional quantum mechanics
In physics, fractional quantum mechanics is a generalization of standard quantum mechanics. The term fractional quantum mechanics was coined by Nick Laskin.[1] Standard quantum mechanics can be approached in three different ways: the matrix mechanics, the Schrödinger equation and the Feynman path integral.
The Feynman path integral[2] is the path integral over Brownian-like quantum-mechanical paths. Fractional quantum mechanics has been discovered by Nick Laskin (1999) as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths. A path integral over the Lévy-like quantum-mechanical paths results in a generalization of quantum mechanics.[3] If the Feynman path integral leads to the well known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Schrödinger equation.[4] The Lévy process is characterized by the Lévy index α, 0 < α ≤ 2. At the special case when α = 2 the Lévy process becomes the process of Brownian motion. The fractional Schrödinger equation includes a space derivative of fractional order α instead of the second order (α = 2) space derivative in the standard Schrödinger equation. Thus, the fractional Schrödinger equation is a fractional differential equation in accordance with modern terminology.[5] This is the main point of the term fractional Schrödinger equation or a more general term fractional quantum mechanics. As mentioned above, at α = 2 the Lévy motion becomes Brownian motion. Thus, fractional quantum mechanics includes standard quantum mechanics as a particular case at α = 2. The quantum-mechanical path integral over the Lévy paths at α = 2 becomes the well-known Feynman path integral and the fractional Schrödinger equation becomes the well known Schrödinger equation.
See also
Quantum mechanics
Matrix mechanics
Schrödinger equation
Fractional Schrödinger equation
Path integral formulation
Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
Lévy process
Fractional calculus
Fractional dynamics
References
^ N. Laskin, (2000), Fractional Quantum Mechanics and Lévy Path Integrals. Physics Letters 268A, 298-304.
^ R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals ~McGraw-Hill, New York, 1965
^ N. Laskin, (2000), Fractional Quantum Mechanics, Physical Review E62, 3135-3145. (also available online: http://arxiv.org/abs/0811.1769)
^ N. Laskin, (2002), Fractional Schrödinger equation, Physical Review E66, 056108 7 pages. (also available online: http://arxiv.org/abs/quant-ph/0206098)
^ S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications ~Gordon and Breach, Amsterdam, 1993
Further reading
L.P.G. do Amaral, E.C. Marino, Canonical quantization of theories containing fractional powers of the d’Alembertian operator. J. Phys. A Math. Gen. 25 (1992) 5183-5261
Xing-Fei He, Fractional dimensionality and fractional derivative spectra of interband optical transitions. Phys. Rev. B, 42 (1990) 11751-11756.
A. Iomin, Fractional-time quantum dynamics. Phys. Rev. E 80, (2009) 022103.
A. Matos-Abiague, Deformation of quantum mechanics in fractional-dimensional space. J. Phys. A: Math. Gen. 34 (2001) 11059–11068.
N. Laskin, Fractals and quantum mechanics. Chaos 10(2000) 780-790
M. Naber, Time fractional Schrodinger equation. J. Math. Phys. 45 (2004) 3339-3352. arXiv:math-ph/0410028
V.E. Tarasov, Fractional Heisenberg equation. Phys. Lett. A 372 (2008) 2984-2988.
V.E. Tarasov, Weyl quantization of fractional derivatives. J. Math. Phys. 49 (2008) 102112.
S. Wang, M. Xu, Generalized fractional Schrödinger equation with space-time fractional derivatives J. Math. Phys. 48 (2007) 043502
E Capelas de Oliveira and Jayme Vaz Jr, "Tunneling in Fractional Quantum Mechanics" Journal of Physics A Volume 44 (2011) 185303.
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