Abstract
In this paper, a nonlocal complexified Schrödinger equation is introduced based on the notion of complexified backward-forward wave functions. Some of its implications in quantum mechanics are discussed, where a number of properties are revealed and discussed, in particular, the discretization of space-time nonlocal infinitesimal coordinates.
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Suykens, J.A.K.: Extending Newton’s law from nonlocal-in-time kinetic energy. Phys. Lett. A 373, 1201–1211 (2009)
El-Nabulsi, R.A.: Non-standard non-local-in-time Lagrangians in classical mechanics. Qual. Theor. Dyn. Sys. 13, 149–160 (2014)
El-Nabulsi, R.A.: Complex backward-forward derivative operator in non-local-in-time Lagrangians mechanics. Qual. Theor. Dyn. Sys. (2016). doi:10.1007/s12346-016-0187-y
Li, Z.-Y., Fu, J.-L., Chen, L.-Q.: Euler–Lagrange equation from nonlocal-in-time kinetic energy of nonconservative system. Phys. Lett. A 374, 106–109 (2009)
Stecki, J.: On the kinetic equation nonlocal in time for the generalized self-diffusion process. J. Comp. Phys. 7, 547–553 (1971)
Gomis, J., Kamimura, K., Llosa, J.: Hamiltonian formalism for space-time noncommutative theories. Phys. Rev. D63(4), 045003 (6 pages) (2001)
Gordeziani, D.G.: On some initial conditions for parabolic equations. Reports of the Enlarged Session of the Seminar of I. Vekua Inst. Appl. Math. 4, 57–60 (1989)
Gordeziani, D.G.: On one problem for the Navier–Stokes equation, Abstracts, Contin. Mech. Related Probl. Anal., Tbilisi, 83 (1991)
Gordeziani, D.G.: On solution of in time nonlocal problems for some equations of mathematical physics, ICM-94, Abstracts, Short Comm, pp. 240 (1994)
Gordeziani, D.G., Grigalashvili, Z.: Non-local problems in time for some equations of mathematical physics. Dokl. Semin. Inst. Prikl. Mat. im. I. N. Vekua. 22, 108–114 (1993)
Feynman, R.P.: Space-time approach to relativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948)
Nelson, E.: Derivation of the Schrödinger equation from newtonian mechanics. Phys. Rev. 150, 1079–1085 (1966)
Nottale, L.: Fractal space-time and microphysics: towards a theory of scale relativity. World Scientific (1993)
Valchev, T.: On a Nonlocal Nonlinear Schrödinger Equation. In: Slavova, A (ed.) Mathematics in Industry, pp. 36–52. Cambridge Scholars Publishing (2014)
Wu, X.Y., Zhang, B.J., Liu, X.J., Xiao, Li, Wu, Y.H., Wang, Y., Wang, Q.C., Cheng, S.: Derivation of nonlinear Schrödinger equation. Int. J. Theor. Phys. 49, 2437–2445 (2010)
Doebner, H.-D., Goldin, G.A.: On a general nonlinear Schrödinger equation admitting diffusion currents. Phys. Lett. A 162, 397–401 (1992)
Doebner, H.-D., Goldin, G.A.: Properties of nonlinear Schrödinger equations associated with diffeomorphism groups representations. J. Phys. A 27, 1771–1780 (1994)
Nattermann, P., Zhdanov, R.: On Integrable Doebner–Goldin equations. J. Phys. A 29, 2869–2886 (1996)
Choudhuri, A., Porsezian, K.: Higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms: a model for sub-10-fs-pulse propagation. Phys. Rev. A88, 033808 (5 pages) (2013)
Wang, L.H., Porsezian, K., He, J.S.: Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation. Phys. Rev. E87, 053202 (20 pages) (2013)
Griffiths, D.J.: Introduction to Quantum Mechanics, 2\(^{nd}\) Edition. Prentice Hall (2004)
Itzykson C., Zuber J.B.: Quantum Field Theory. McGraw-Hill Book Co, Singapore (1985)
El-Nabulsi, R.A.: Generalized Klein-Gordon and Dirac equations from nonlocal kinetic approach. Zeitschrift für Naturforschung A. doi:10.1515/zna-2016-0226
Polyanin, A.D.: Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman & Hall/CRC Press, Boca Raton (2002)
Bender, C.M., Mead, L.R., Milton, K.A.: Discrete time quantum mechanics. Comp. Math. Appl. 28(1–2), 279–317 (1994)
Stovicek, P., Tolar, J.: Quantum mechanics in a discrete space-time. Rep. Math. Phys. 20(2), 157–170 (1984)
Khorrami, M.: A general formulation of discrete-time quantum mechanics, restrictions on the action and the relation of unitarity to the existence theorem for initial-value problems. Ann. Phys. 244, 101–111 (1995)
Elze, H.-T.: Quantum mechanics and discrete time from “timeless” classical dynamics. Lect. Notes Phys. 633, 196–220 (2004)
Walleczek, J., Groessing, G.: Is the world local or nonlocal? Towards an emergent quantum mechanics in the 21\(^{st}\) century. J. Phys.: Conf. Ser. 701, 012001 (10 pages) (2016)
Berberan-Santos, M.N., Bodunov, E.N., Pogliani, L.: Classical and quantum study of the motion of a particle in a gravitational field. J. Math. Chem. 37(2), 101–115 (2005)
Liemert, A., Kienle, A.: Fractional Schrödinger equation in the presence of the linear potential. Mathematics 4(31), 1–14 (2016)
Farhang Martin L., Hasan Bouzari, H., Ahmadi, F.: Solving Schrödinger equation specializing to the Stark effect in linear potential by the canonical function method. J. Theor. Appl. Phys. 8(140), 1–6 (2014)
Johnson, R.P.: Solution to the Schrödinger Equation for a Linear 1-D Potential, Lecture Given at Santa Cruz Institute for Particle Physics. University of California at Santa Cruz, Autumn (2011)
Lorente, M.: Quantum mechanics on discrete space and time. In: Proceedings: M. Ferrero, A. van der Merwe, eds. New Developments on Fundamental Problems in Quantum Physics (Kluwer, N.Y. 1997) pp. 213–224
Vaidman, L.: Tracing the past of a quantum particle. Phys. Rev. A89, 024102 (3 pages) (2014)
Vaidman, L.: The past of a quantum particle. Phys. Rev. A87, 052104 (5 pages) (2013)
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El-Nabulsi, R.A. On nonlocal complexified Schrödinger equation and emergence of discrete quantum mechanics. Quantum Stud.: Math. Found. 3, 327–335 (2016). https://doi.org/10.1007/s40509-016-0080-z
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DOI: https://doi.org/10.1007/s40509-016-0080-z