# Fractional quantum mechanics

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In physics, **fractional quantum mechanics** is a generalization of standard quantum mechanics, which naturally comes out when the Brownian-like quantum paths substitute with the Lévy-like ones in the Feynman path integral. It has been discovered by Nick Laskin who coined the term *fractional quantum mechanics*.^{[1]}

## Contents

## Fundamentals

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.

### Fractional Schrödinger equation

The fractional Schrödinger equation discovered by Nick Laskin has the following form (see, Refs.[1,3,4])

using the standard definitions:

**r**is the 3-dimensional position vector,*ħ*is the reduced Planck constant,*ψ*(**r**,*t*) is the wavefunction, which is the quantum mechanical probability amplitude for the particle to have a given position**r**at any given time*t*,*V*(**r**,*t*) is a potential energy,- Δ = ∂
^{2}/∂**r**^{2}is the Laplace operator.

Further,

*D*is a scale constant with physical dimension [D_{α}_{α}] = [energy]^{1 − α}·[length]^{α}[time]^{−α}, at*α*= 2,*D*_{2}=1/2*m*, where*m*is a particle mass,- the operator (−
*ħ*^{2}Δ)^{α/2}is the 3-dimensional fractional quantum Riesz derivative defined by (see, Ref.[4]);

Here, the wave functions in the position and momentum spaces; and are related each other by the 3-dimensional Fourier transforms:

The index *α* in the fractional Schrödinger equation is the Lévy index, 1 < *α* ≤ 2.

## See also

- Quantum mechanics
- Matrix mechanics
- Fractional calculus
- Fractional dynamics
- Fractional Schrödinger equation
- Non-linear Schrödinger equation
- Path integral formulation
- Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
- Lévy process

## 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

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## 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.