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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]


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])

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

using the standard definitions:

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.


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]);

\( (-\hbar ^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)=\frac 1{(2\pi \hbar )^3}\int d^3pe^{i \mathbf{p}\cdot \mathbf{r}/\hbar}|\mathbf{p}|^\alpha \varphi ( \mathbf{p},t), \)

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

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

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.
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


Laskin, Nikolai (2000). "Fractional quantum mechanics and Lévy path integrals". Physics Letters A. 268 (4–6): 298–305.arXiv:hep-ph/9910419. doi:10.1016/S0375-9601(00)00201-2.
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals ~McGraw-Hill, New York, 1965
Laskin, Nick (1 August 2000). "Fractional quantum mechanics". Physical Review E. American Physical Society (APS). 62 (3): 3135–3145.arXiv:0811.1769. Bibcode:2000PhRvE..62.3135L. doi:10.1103/physreve.62.3135. ISSN 1063-651X.
Laskin, Nick (18 November 2002). "Fractional Schrödinger equation". Physical Review E. American Physical Society (APS). 66 (5): 056108.arXiv:quant-ph/0206098. Bibcode:2002PhRvE..66e6108L. doi:10.1103/physreve.66.056108. ISSN 1063-651X. PMID 12513557.

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. CiteSeerX 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.arXiv:1508.03621. 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.arXiv:0909.1183. 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.arXiv:quant-ph/0107062. 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.arXiv:math-ph/0410028. doi:10.1063/1.1769611. ISSN 0022-2488.
Tarasov, Vasily E. (2008). "Fractional Heisenberg equation". Physics Letters A. Elsevier BV. 372 (17): 2984–2988.arXiv:0804.0586. 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.arXiv:0907.2699. 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.arXiv:1011.1948. 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. ISSN 1867-8440.
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. ISSN 1867-8440.


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