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Fractional-time Schrödinger equation: fractional dynamics on a comb. (English) Zbl 1225.81053
Summary: The physical relevance of the fractional time derivative in quantum mechanics is discussed. It is shown that the introduction of the fractional time Schrödinger equation (FTSE) in quantum mechanics by analogy with the fractional diffusion $\frac{\partial }{\partial t}\to \frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}$ can lead to an essential deficiency in the quantum mechanical description, and needs special care. To shed light on this situation, a quantum comb model is introduced. It is shown that for $\alpha =1/2$, the FTSE is a particular case of the quantum comb model. This exact example shows that the FTSE is insufficient to describe a quantum process, and the appearance of the fractional time derivative by a simple change $\frac{\partial }{\partial t}\to \frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}$ in the Schrödinger equation leads to the loss of most of the information about quantum dynamics.
##### MSC:
 81Q05 Closed and approximate solutions to quantum-mechanical equations 35R11 Fractional partial differential equations 26A33 Fractional derivatives and integrals (real functions)
##### References:
 [1] Feynman, R. P.; Hibbs, A. R.: Quantum mechanics and path integrals, (1965) · Zbl 0176.54902 [2] Kac, M.: Probability and related topics in physical sciences, (1959) · Zbl 0087.33003 [3] Laskin, N.: Fractals and quantum mechanics, Chaos 10, No. 4, 780-790 (2000) · Zbl 1071.81513 · doi:doi:10.1063/1.1050284 [4] West, B. J.: Quantum Lévy propagators, J phys chem B 104, No. 16, 3830-3832 (2000) [5] Naber, M.: Time fractional Schrödinger equation, J math phys 45, No. 8, 3339-3352 (2004) · Zbl 1071.81035 · doi:doi:10.1063/1.1769611 [6] Stone, M. H.: On one-parameter unitary groups in Hilbert space, Ann math 33, 643 (1932) · Zbl 0005.16403 · doi:doi:10.2307/1968538 [7] Dong, J.; Xu, M.: Space-time fractional Schrödinger equation with time-independent potentials, J math anal appl 344, 1005-1017 (2008) · Zbl 1140.81357 · doi:doi:10.1016/j.jmaa.2008.03.061 [8] Wang, S.; Xu, M.: Generalized fractional Schrödinger equation with space-time fractional derivatives, J math phys 48, No. 4, 043502 (2007) · Zbl 1137.81328 · doi:doi:10.1063/1.2716203 [9] Bhatti, M.: Fractional Schrödinger wave equation and fractional uncertainty principle, Int J contemp math sci 2, No. 17 – 20, 943-950 (2007) · Zbl 1146.35082 [10] Iomin, A.: Fractional-time quantum dynamics, Phys rev E 80, 022103 (2009) [11] Weiss, G. H.; Havlin, S.: Some properties of a random-walk on a comb structure, Physica A 134, No. 2, 474-482 (1986) [12] Baskin, E.; Iomin, A.: Superdiffusion on a comb structure, Phys rev lett 93, No. 2, 120603 (2004) [13] Montroll, E. W.; Shlesinger, M. F.: The wonderful world of random walks, Studies in statistical mechanics 11 (1984) · Zbl 0556.60027 [14] Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys rep 339 (2000) · Zbl 0984.82032 · doi:doi:10.1016/S0370-1573(00)00070-3 [15] Arkhincheev, V. E.; Baskin, E. M.: Anomalous diffusion and drift in a comb model of percolation clusters, Sov phys JETP 73, No. 1, 161-300 (1991) [16] Iomin, A.; Baskin, E.: Negative superdiffusion due to inhomogeneous convection, Phys rev E 71, No. 6, 061101 (2005) [17] Schulman, L.: Path integrals from mev to mev, (1986) [18] Schulman, L.: Techniques and applications of path integration, (1981) · Zbl 0587.28010 [19] Gaveau, B.; Schulman, L.: Explicit time-dependent Schrödinger propagators, J phys A – math gen 19, No. 10, 1833-1846 (1986) · Zbl 0621.35082 · doi:doi:10.1088/0305-4470/19/10/024 [20] Podlubny, I.: Fractional differential equations, (1999) [21] Oldham, K. B.; Spanier, J.: The fractional calculus, (1974) [22] Mainardi, F.: Fractional relaxation – oscillation and fractional diffusion-wave phenomena, Chaos solitons fract 7, No. 9, 1461-1477 (1996) · Zbl 1080.26505 · doi:doi:10.1016/0960-0779(95)00125-5