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. This concept was discovered by Nick Laskin who coined the term fractional quantum mechanics.
The Feynman path integral 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. 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. 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. 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
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]);
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.
- 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
- N. Laskin, (2000), Fractional Quantum Mechanics, Physical Review E62, 3135-3145. (also available online: https://arxiv.org/abs/0811.1769)
- N. Laskin, (2002), Fractional Schrödinger equation, Physical Review E66, 056108 7 pages. (also available online: https://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
- 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 10.1.1.247.5449. doi:10.1142/10541. ISBN 978-981-322-379-0.
- F. Pinsker, W. Bao, Y. Zhang, H. Ohadi, A. Dreismann and J.J. Baumberg (2015) https://arxiv.org/abs/1508.03621
- 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. arXiv:0804.0586v1
- V.E. Tarasov, Weyl quantization of fractional derivatives. J. Math. Phys. 49 (2008) 102112. arXiv:0907.2699
- 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.
- V.E. Tarasov, Fractional Dynamics of Open Quantum Systems. in Fractional Dynamics, 2010, pp. 467-490. DOI: 10.1007/978-3-642-14003-7_20
- V.E. Tarasov, Fractional Dynamics of Hamiltonian Quantum Systems. in Fractional Dynamics, 2010, pp. 457-466. DOI: 10.1007/978-3-642-14003-7_20