On Fractional Schrödinger Equation
Rozmej Piotr 1, Bandrowski Bartosz 2
1Institute of Physics, 2Faculty of Mathematics Computer Science and Econometrics
University of Zielona Góra,
ul. Szafrana 4a, 65-516 Zielona Góra, Poland
e-mail: P.Rozmej@if.uz.zgora.pl, B.Bandrowski@gmail.com
Received:
Received: 23 March 2010; accepted: 7 July 2010; published online: 13 September 2010
DOI: 10.12921/cmst.2010.16.02.191-194
OAI: oai:lib.psnc.pl:730
Abstract:
In the note, recent efforts to derive fractional quantum mechanics are recalled. Some applications of a fractional approach to the Schrödinger equation are discussed as well.
Key words:
Fractional calculus, fractional quantum mechanics, fractional Schrödinger equation
References:
[1] I. Podlubny, Fractional Differential Equations. Academic Press, 1999.
[2] Fractional Calculus Modeling, http://www.fracalmo.org/
[3] N. Laskin, Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298 (2000); Fractional quantum mechanics. Phys. Rev. E 62, 3135 (2000).
[4] N. Laskin, Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002).
[5] M. Naber, Time fractional Schrödinger equation. J. Math. Phys. 45, 3339 (2004).
[6] F. Ben Adda, J. Cresson, Fractional differential equations and the Schrödinger equation. Appl. Math. Comp. 161, 323 (2005).
[7] R. Herrmann, Properties of fractional derivative Schrödinger type wave equation and a new interpretation of the charmonium spectrum. arXiv:math-ph/05100099 (2006).
[8] R. Herrmann, The fractional symmetric rigid rotor. J. Phys. G 34, 607 (2007).
[9] R. Herrmann, q-deformed Lie algebras and fractional calculus. arXiv:0711:3701 (2007).
[10] R. Herrmann, Gauge invariance in fractional field theories. Phys. Lett. A 372, 5515 (2008).
[11] R. Herrmann, Fractional dynamic symmetries and the ground properties of nuclei. arXiv:0806.2300 (2008).
[12] R. Herrmann, Fractional phase transition in medium size metal cluster and some remarks on magic numners in gravitationally and weakly interacting clusters. arXiv:0907.1953 (2009).
[13] B. Bandrowski, A. Karczewska, P. Rozmej, Numerical solutions to integral equations equivalent to differential equations with fractional time derivative. Int. J. Appl. Math. Comp. Sci. 20 (2), 261-269 (2010). (http://www.uz.zgora.pl/ prozmej/amcs2.pdf)
In the note, recent efforts to derive fractional quantum mechanics are recalled. Some applications of a fractional approach to the Schrödinger equation are discussed as well.
Key words:
Fractional calculus, fractional quantum mechanics, fractional Schrödinger equation
References:
[1] I. Podlubny, Fractional Differential Equations. Academic Press, 1999.
[2] Fractional Calculus Modeling, http://www.fracalmo.org/
[3] N. Laskin, Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298 (2000); Fractional quantum mechanics. Phys. Rev. E 62, 3135 (2000).
[4] N. Laskin, Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002).
[5] M. Naber, Time fractional Schrödinger equation. J. Math. Phys. 45, 3339 (2004).
[6] F. Ben Adda, J. Cresson, Fractional differential equations and the Schrödinger equation. Appl. Math. Comp. 161, 323 (2005).
[7] R. Herrmann, Properties of fractional derivative Schrödinger type wave equation and a new interpretation of the charmonium spectrum. arXiv:math-ph/05100099 (2006).
[8] R. Herrmann, The fractional symmetric rigid rotor. J. Phys. G 34, 607 (2007).
[9] R. Herrmann, q-deformed Lie algebras and fractional calculus. arXiv:0711:3701 (2007).
[10] R. Herrmann, Gauge invariance in fractional field theories. Phys. Lett. A 372, 5515 (2008).
[11] R. Herrmann, Fractional dynamic symmetries and the ground properties of nuclei. arXiv:0806.2300 (2008).
[12] R. Herrmann, Fractional phase transition in medium size metal cluster and some remarks on magic numners in gravitationally and weakly interacting clusters. arXiv:0907.1953 (2009).
[13] B. Bandrowski, A. Karczewska, P. Rozmej, Numerical solutions to integral equations equivalent to differential equations with fractional time derivative. Int. J. Appl. Math. Comp. Sci. 20 (2), 261-269 (2010). (http://www.uz.zgora.pl/ prozmej/amcs2.pdf)
