zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 t α t α 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 α=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 t α t α in the Schrödinger equation leads to the loss of most of the information about quantum dynamics.
MSC:
81Q05Closed and approximate solutions to quantum-mechanical equations
35R11Fractional partial differential equations
26A33Fractional 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: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: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: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: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: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: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: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:10.1016/0960-0779(95)00125-5