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]}
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 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])
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}/∂r^{2} is the Laplace operator.
Further,
- D_{α} is a scale constant with physical dimension [D_{α}] = [energy]^{1 − α}·[length]^{α}[time]^{−α}, at α = 2, D_{2} =1/2m, where m is a particle mass,
- the operator (−ħ^{2}Δ)^{α/2} is the 3-dimensional fractional quantum Riesz derivative defined by (see, Refs.[3, 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.
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
- ^ 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: 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 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
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. 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