Fractional quantum mechanics

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Quantum mechanics
iℏ∂∂t|ψ(t)⟩=H^|ψ(t)⟩

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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. This concept was discovered by Nick Laskin who coined the term fractional quantum mechanics.[1]

## Contents

* 1 Fundamentals
* 1.1 Fractional Schrödinger equation
* 1.2 Fractional quantum mechanics in solid state systems
* 2 See also
* 3 References
* 4 Further reading

## 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])

iℏ∂ψ(𝐫,t)∂t=Dα(−ℏ2Δ)α/2ψ(𝐫,t)+V(𝐫,t)ψ(𝐫,t)

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 function that determines the 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/∂r2 is the Laplace operator.

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]);

(−ℏ2Δ)α/2ψ(𝐫,t)=1(2πℏ)3∫d3pei𝐩⋅𝐫/ℏ|𝐩|αφ(𝐩,t),

Here, the wave functions in the position and momentum spaces; ψ(𝐫,t) and φ(𝐩,t) are related each other by the 3-dimensional Fourier transforms:

ψ(𝐫,t)=1(2πℏ)3∫d3pei𝐩⋅𝐫/ℏφ(𝐩,t),φ(𝐩,t)=∫d3re−i𝐩⋅𝐫/ℏψ(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.

## 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

1. ↑ Laskin, Nikolai (2000). "Fractional quantum mechanics and Lévy path integrals". Physics Letters A 268 (4–6): 298–305. doi:10.1016/S0375-9601(00)00201-2.
2. ↑ R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals ~McGraw-Hill, New York, 1965
3. ↑ Laskin, Nick (1 August 2000). "Fractional quantum mechanics". Physical Review E (American Physical Society (APS)) 62 (3): 3135–3145. doi:10.1103/physreve.62.3135. ISSN 1063-651X. Bibcode: 2000PhRvE..62.3135L.
4. ↑ Laskin, Nick (18 November 2002). "Fractional Schrödinger equation". Physical Review E (American Physical Society (APS)) 66 (5): 056108. doi:10.1103/physreve.66.056108. ISSN 1063-651X. PMID 12513557. Bibcode: 2002PhRvE..66e6108L.
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 978-2-88124-864-1.

* Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier. ISBN 978-0-444-51832-3.

* Herrmann, R. (2014). Fractional Calculus - An Introduction for Physicists. Singapore: World Scientific. doi:10.1142/8934. ISBN 978-981-4551-07-6.

* Laskin, N. (2018). Fractional Quantum Mechanics. World Scientific. doi:10.1142/10541. ISBN 978-981-322-379-0.

* Pinsker, F.; Bao, W.; Zhang, Y.; Ohadi, H.; Dreismann, A.; Baumberg, J. J. (25 November 2015). "Fractional quantum mechanics in polariton condensates with velocity-dependent mass". Physical Review B (American Physical Society (APS)) 92 (19): 195310. doi:10.1103/physrevb.92.195310. ISSN 1098-0121.

## Further reading

* Amaral, R L P G do; Marino, E C (7 October 1992). "Canonical quantization of theories containing fractional powers of the d'Alembertian operator". Journal of Physics A: Mathematical and General (IOP Publishing) 25 (19): 5183–5200. doi:10.1088/0305-4470/25/19/026. ISSN 0305-4470.
* He, Xing-Fei (15 December 1990). "Fractional dimensionality and fractional derivative spectra of interband optical transitions". Physical Review B (American Physical Society (APS)) 42 (18): 11751–11756. doi:10.1103/physrevb.42.11751. ISSN 0163-1829.
* Iomin, Alexander (28 August 2009). "Fractional-time quantum dynamics". Physical Review E (American Physical Society (APS)) 80 (2): 022103. doi:10.1103/physreve.80.022103. ISSN 1539-3755.
* Matos-Abiague, A (5 December 2001). "Deformation of quantum mechanics in fractional-dimensional space". Journal of Physics A: Mathematical and General (IOP Publishing) 34 (49): 11059–11068. doi:10.1088/0305-4470/34/49/321. ISSN 0305-4470.
* Laskin, Nick (2000). "Fractals and quantum mechanics". Chaos: An Interdisciplinary Journal of Nonlinear Science (AIP Publishing) 10 (4): 780. doi:10.1063/1.1050284. ISSN 1054-1500.
* Naber, Mark (2004). "Time fractional Schrödinger equation". Journal of Mathematical Physics (AIP Publishing) 45 (8): 3339–3352. doi:10.1063/1.1769611. ISSN 0022-2488.
* Tarasov, Vasily E. (2008). "Fractional Heisenberg equation". Physics Letters A (Elsevier BV) 372 (17): 2984–2988. doi:10.1016/j.physleta.2008.01.037. ISSN 0375-9601.
* Tarasov, Vasily E. (2008). "Weyl quantization of fractional derivatives". Journal of Mathematical Physics (AIP Publishing) 49 (10): 102112. doi:10.1063/1.3009533. ISSN 0022-2488.
* Wang, Shaowei; Xu, Mingyu (2007). "Generalized fractional Schrödinger equation with space-time fractional derivatives". Journal of Mathematical Physics (AIP Publishing) 48 (4): 043502. doi:10.1063/1.2716203. ISSN 0022-2488.
* de Oliveira, E Capelas; Vaz, Jayme (5 April 2011). "Tunneling in fractional quantum mechanics". Journal of Physics A: Mathematical and Theoretical (IOP Publishing) 44 (18): 185303. doi:10.1088/1751-8113/44/18/185303. ISSN 1751-8113.
* Tarasov, Vasily E. (2010). "Fractional Dynamics of Open Quantum Systems". Nonlinear Physical Science. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 467–490. doi:10.1007/978-3-642-14003-7_20. ISBN 978-3-642-14002-0.
* Tarasov, Vasily E. (2010). "Fractional Dynamics of Hamiltonian Quantum Systems". Nonlinear Physical Science. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 457–466. doi:10.1007/978-3-642-14003-7_19. ISBN 978-3-642-14002-0.

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Quantum mechanics

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