# Fractional quantum mechanics

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]

## 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 key point to launch the term fractional Schrödinger equation and 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])

${\displaystyle i\hbar {\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}=D_{\alpha }(-\hbar ^{2}\Delta )^{\alpha /2}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)\,}$

using the standard definitions:

Further,

• Dα is a scale constant with physical dimension [Dα] = [energy]1 − α·[length]α[time]α, at α = 2, D2 =1/2m, where m is a particle mass,
• the operator (−ħ2Δ)α/2 is the 3-dimensional fractional quantum Riesz derivative defined by (see, Refs.[3, 4]);
${\displaystyle (-\hbar ^{2}\Delta )^{\alpha /2}\psi (\mathbf {r} ,t)={\frac {1}{(2\pi \hbar )^{3}}}\int d^{3}pe^{i\mathbf {p} \cdot \mathbf {r} /\hbar }|\mathbf {p} |^{\alpha }\varphi (\mathbf {p} ,t),}$

Here, the wave functions in the position and momentum spaces; ${\displaystyle \psi (\mathbf {r} ,t)}$ and ${\displaystyle \varphi (\mathbf {p} ,t)}$ are related each other by the 3-dimensional Fourier transforms:

${\displaystyle \psi (\mathbf {r} ,t)={\frac {1}{(2\pi \hbar )^{3}}}\int d^{3}pe^{i\mathbf {p} \cdot \mathbf {r} /\hbar }\varphi (\mathbf {p} ,t),\qquad \varphi (\mathbf {p} ,t)=\int d^{3}re^{-i\mathbf {p} \cdot \mathbf {r} /\hbar }\psi (\mathbf {r} ,t).}$

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

### Fractional quantum mechanics in solid state systems

The effective mass of states in solid state systems can depend on the wave vector k, i.e. formally one considers m=m(k). Polariton Bose-Einstein condensate modes are examples of states in solid state systems with mass sensitive to variations and locally in k fractional quantum mechanics is experimentally feasible.

## References

1. ^ N. Laskin, (2000), Fractional Quantum Mechanics and Lévy Path Integrals. Physics Letters 268A, 298-304.
2. ^ R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals ~McGraw-Hill, New York, 1965
3. ^ N. Laskin, (2000), Fractional Quantum Mechanics, Physical Review E62, 3135-3145. (also available online: https://arxiv.org/abs/0811.1769)
4. ^ N. Laskin, (2002), Fractional Schrödinger equation, Physical Review E66, 056108 7 pages. (also available online: https://arxiv.org/abs/quant-ph/0206098)
5. ^ S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications ~Gordon and Breach, Amsterdam, 1993
• Samko, S.; Kilbas, A.A.; Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications. Taylor & Francis Books. ISBN 2-88124-864-0.
• Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier. ISBN 0-444-51832-0.
• Herrmann, R. (2014). Fractional Calculus - An Introduction for Physicists. Singapore: World Scientific.
• Laskin, N. (2018). Fractional Quantum Mechanics. World Scientific.
• F. Pinsker, W. Bao, Y. Zhang, H. Ohadi, A. Dreismann and J.J. Baumberg (2015) https://arxiv.org/abs/1508.03621

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