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Space-time fractional Schrödinger equation with time-independent potentials. (English) Zbl 1140.81357
Summary: We develop a space-time fractional Schrödinger equation containing Caputo fractional derivative and the quantum Riesz fractional operator from a space fractional Schrödinger equation in this paper. By use of the new equation we study the time evolution behaviors of the space-time fractional quantum system in the time-independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively. The space-time fractional Schrödinger equation with time-independent potentials is divided into a space equation and a time one. A general solution, which is composed of oscillatory terms and decay ones, is obtained. We investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative, but also on the sign (positive or negative) of the eigenvalues of the space equation. We also find that the limit value of the total probability can be greater or less than one, which means the space-time fractional Schrödinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials. Additionally, the non-Markovian time evolution laws of the space-time fractional quantum system are discussed. The formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space-time fractional quantum system. We also get a Mittag-Leffler type of time evolution operator of wave functions and then establish a Heisenberg equation containing fractional operators.
81Q05Closed and approximate solutions to quantum-mechanical equations
26A33Fractional derivatives and integrals (real functions)
47B06Riesz operators; eigenvalue distributions; approximation numbers, s-numbers etc.of operators
[1]Oldham, K. B.; Spanier, J.: The fractional calculus: theory and applications, differentiation and integration to arbitrary order, Math. sci. Eng. (1974)
[2]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[3]Podlubny, I.: Fractional differential equations, (1999)
[4]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[5]Mandelbrot, B. B.: The fractal geometry of nature, (1982) · Zbl 0504.28001
[6]Zaslavsky, G. M.: Chaos, fractional kinetics, and anomalous transport, Phys. rep. 371, 461-580 (2002) · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9
[7]Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. rep. 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[8]Mainardi, F.; Luchko, Yu.; Pagnini, G.: The fundamental solution of the space – time fractional diffusion equation, Fract. calc. Appl. anal. 4, 153-192 (2001) · Zbl 1054.35156
[9]Zaslavsky, G. M.: Hamiltonian chaos and fractional dynamics, (2005)
[10]Carpinteri, A.; Mainardi, F.: Fractals and fractional calculus in continuum mechanics, (1997) · Zbl 0917.73004
[11]Stanislavsky, A. A.: Hamiltonian formalism of fractional systems, Eur. phys. J. B 49, 93-101 (2006)
[12]Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics, Phys. rev. E 53, 1890 (1996)
[13]Riewe, F.: Mechanics with fractional derivatives, Phys. rev. E 55, 3581 (1997)
[14]Baleanu, D.; Muslih, S.; Tas, K.: Fractional Hamiltonian analysis of higher order derivatives systems, J. math. Phys. 47, No. 10, 103503 (2006) · Zbl 1112.81074 · doi:10.1063/1.2356797
[15]Rabei, E. M.; Almayteh, I.; Muslih, S. I.; Baleanu, D.: Hamilton – jaccobi formulation of systems with Caputo’s fractional derivative, Phys. scr. 77, No. 1, 015101 (2008) · Zbl 1145.70011 · doi:10.1088/0031-8949/77/01/015101
[16]Baleanu, D.; Agrawal, O. P.: Fractional Hamilton formalism within Caputo’s derivative, Czechoslovak J. Phys. 56, No. 10 – 11, 1087-1092 (2006) · Zbl 1111.37304 · doi:10.1007/s10582-006-0406-x
[17]Agrawal, O. P.: Formulation of Euler – Lagrange equations for fractional variational problems, J. math. Anal. appl. 272, 368-379 (2002) · Zbl 1070.49013 · doi:10.1016/S0022-247X(02)00180-4
[18]Agrawal, O. P.: Fractional variational calculus and the transversality conditions, J. phys. A 39, 10375-10384 (2006) · Zbl 1097.49021 · doi:10.1088/0305-4470/39/33/008
[19]Muslih, S.; Baleanu, D.; Rabei, E.: Hamiltonian formulation of classical fields within Riemann – Liouville fractional derivatives, Phys. scr. 73, 436-438 (2006) · Zbl 1165.70310 · doi:10.1088/0031-8949/73/5/003
[20]Feynman, R. P.; Hibbs, A. R.: Quantum mechanics and path integrals, (1965) · Zbl 0176.54902
[21]Naber, M.: Time fractional Schrödinger equation, J. math. Phys. 45, No. 8, 3339-3352 (2004) · Zbl 1071.81035 · doi:10.1063/1.1769611
[22]Wang, S. W.; Xu, M. Y.: Generalized fractional Schrödinger equation with space – time fractional derivatives, J. math. Phys. 48, 043502 (2007) · Zbl 1137.81328 · doi:10.1063/1.2716203
[23]Laskin, N.: Fractional Schrödinger equation, Phys. rev. E 66, 056108 (2002)
[24]Laskin, N.: Fractional quantum mechanics and Lévy path integrals, Phys. lett. A 298, 298-305 (2000) · Zbl 0948.81595 · doi:10.1016/S0375-9601(00)00201-2
[25]Laskin, N.: Fractional quantum mechanics, Phys. rev. E 62, 3135-3145 (2000)
[26]Laskin, N.: Fractals and quantum mechanics, Chaos 10, 780-790 (2000) · Zbl 1071.81513 · doi:10.1063/1.1050284
[27]Guo, X. Y.; Xu, M. Y.: Some physical applications of fractional Schrödinger equation, J. math. Phys. 47, 082104 (2006) · Zbl 1112.81028 · doi:10.1063/1.2235026
[28]Dong, J. P.; Xu, M. Y.: Some solutions to the space fractional Schrödinger equation using momentum representation method, J. math. Phys. 48, 072105 (2007) · Zbl 1144.81341 · doi:10.1063/1.2749172
[29]Hatfield, B.: Quantum field theory of point particles and strings, (1992)
[30]Xu, M. Y.; Tan, W. C.: Representation of the constitutive equation of viscoelastic materials by the generalized fractional element networks and its generalized solutions, Sci. China ser. G 46, No. 2, 145-157 (2003)
[31]Kwok, Yue Kuen: Applied complex variables for scientists and engineers, (2002)
[32]Mainardi, F.; Gorenflo, R.: On Mittag – Leffler-type functions in fractional evolution processes, J. comput. Appl. math. 118, 283-299 (2000) · Zbl 0970.45005 · doi:10.1016/S0377-0427(00)00294-6
[33]Landau, L. D.; Lifshitz, E. M.: Quantum mechanics: non-relativistic theory, (1977)
[34]Griffiths, D. J.: Introduction to quantum mechanics, (2005)
[35]Mathai, A. M.; Saxena, R. K.: The H-function with applications in statistics and other disciplines, (1978) · Zbl 0382.33001
[36]Schiff, L. I.: Quantum mechanics, (1968)
[37]Fayyazuddin; Riazuddin: Quantum mechanics, (1990)