On maximal acceleration and quantum acceleratum operator in quantum mechanics

Regular Paper


We introduce the notion of quantum acceleratum based on nonlocal-in-time kinetic energy approach and we discuss its implication on quantum mechanics. A modified higher-order Schrodinger equation is obtained and our approach has the advantage of methodically producing higher-order derivatives terms and to calculate small relativistic corrections to results obtained from the standard level of approximation.


Nonlocal-in-time kinetic energy Quantum acceleratum operator Quantum mechanics Higher-order modified Schrödinger equation 



I would like to thank the anonymous referee for his useful suggestions and valuable comments.


  1. 1.
    Suykens, J.A.K.: Extending Newton’s law from nonlocal-in-time kinetic energy. Phys. Lett. A 373, 1201–1211 (2009)CrossRefMATHGoogle Scholar
  2. 2.
    Feynman, R.P.: Space-time approach to relativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    El-Nabulsi, R.A.: Non-standard non-local-in-time Lagrangians in classical mechanics. Qual. Theor. Dyn. Syst. 13, 149–160 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    El-Nabulsi, R.A.: Complex backward-forward derivative operator in non-local-in-time Lagrangians mechanics. Qual. Theor. Dyn. Syst. (2016). doi: 10.1007/s12346-016-0187-y Google Scholar
  5. 5.
    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)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Stecki, J.: On the kinetic equation nonlocal in time for the generalized self-diffusion process. J. Comput. Phys. 7, 547–553 (1971)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gomis, J., Kamimura, K., Llosa, J.: Hamiltonian formalism for space-time noncommutative theories. Phys. Rev. D 63(4), 045003 (2001). (6 pages)MathSciNetCrossRefGoogle Scholar
  8. 8.
    El-Nabulsi, R.A.: On nonlocal complexified Schrödinger equation and emergence of discrete quantum mechanics. Quant. Stud. Math. Found. 3, 327–335 (2016)CrossRefMATHGoogle Scholar
  9. 9.
    El-Nabulsi, R.A.: Generalized Klein–Gordon and Dirac equations from nonlocal kinetic energy approach. Zeitsch. Natur. A71(9), 817–821 (2016)Google Scholar
  10. 10.
    Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150, 1079–1085 (1966)CrossRefGoogle Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Gordeziani, D.G.: On nonlocal in time problems for Navier-Stokes equations. Rep. Sem. I. Vekua Inst. Appl. Math. 22, 22–27 (1993)Google Scholar
  14. 14.
    Hu, H., Hu, M.: Evidence of non-local physical, chemical and biological effects supports quantum. Neuro Quant 4(4), 291–306 (2006)Google Scholar
  15. 15.
    Cushman, J., Ginn, T.: Nonlocal dispersion in media with continuously evolving scales of heterogeneity. Transp. Por. Media 13, 123–128 (1993)CrossRefGoogle Scholar
  16. 16.
    Carmichael, H.: An Open Systems Approach to Quantum Optics. Springer, New York (1991)MATHGoogle Scholar
  17. 17.
    El-Nabulsi, R.: Nonlocal-in-time kinetic energy in nonconservative fractional systems, disordered dynamics, Jerk and Snap and oscillatory motions in the rotating fluid tube. Int. J. Nonlinear Mech. 93, 65–81 (2017)CrossRefGoogle Scholar
  18. 18.
    Nottale, L.: Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity. World Scientific, Singapore (1993)CrossRefMATHGoogle Scholar
  19. 19.
    Pecknold, S., Lovejoy, S., Schertzer, D., Hooge, C., Malouin, J.F.: The simulation of universal multifractals. In: Perdang, J.M., Lejeune, A. (Eds.), Cellular automata: prospects in astronomy and astrophysics, pp. 228–267. World Scientific, Singapore (1993)Google Scholar
  20. 20.
    Grosse-Knetter, C.: Effective Lagrangians with higher-order derivatives. Phys. Rev. D 49, 6709–6719 (1994)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Deriglazov, A., Nersessian, A.: Rigid particle revisited: extrinsic curvature yields the Dirac equation. Phys. Lett. A 378, 1224–1227 (2014)CrossRefMATHGoogle Scholar
  22. 22.
    Chen, T.-J., Fasiello, M., Lim, E.A., Tolley, A.J.: Higher derivative theories with constraints: exorcising Ostrogradiski’s ghost. J. Cosmo. Astropart. Phys. 1302, 042–059 (2013)CrossRefGoogle Scholar
  23. 23.
    Kamalov, T.: Classical and quantum-mechanical axioms with the higher time derivative formalism. J. Phys. Conf. Ser. 442, 012051 (2013). (6pages)CrossRefGoogle Scholar
  24. 24.
    Kamalov, T.: Quantum corrections for Newton’s law of motion. arXiv:1612.06712
  25. 25.
    Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Benjamin, Reading, MA (1978)MATHGoogle Scholar
  26. 26.
    Rashid, M.S., Khalil, S.S.: Hamiltonian Description of Higher Order Lagrangians, IC/93/420. Miramare, Trieste (1993)Google Scholar
  27. 27.
    Sakurai, J.: Modern Quantum Mechanics. Addison-Wesley, Boston (1994)Google Scholar
  28. 28.
    Dirac, P.A.M.: Classical theory of radiating electrons. Proc. R. Soc. Lond. A167, 148–169 (1938)CrossRefMATHGoogle Scholar
  29. 29.
    Caianiello, E.R.: Is there a maximal acceleration. Lett. Nuovo Cimento 32, 65–70 (1981)CrossRefGoogle Scholar
  30. 30.
    Pati, A.K.: A note on maximal acceleration. Europhys. Lett. 18(4), 285–289 (1992)CrossRefGoogle Scholar
  31. 31.
    Papini, G.: Revisiting Caianiello’s maximal acceleration. Nuovo Cimento B117, 1325–1331 (2003)Google Scholar
  32. 32.
    Pati, A.K.: On the maximal acceleration and the maximal energy loss. Nuovo Cimento B107, 895–901 (1992)CrossRefGoogle Scholar
  33. 33.
    Kiefer, C.: The semiclassical approximation to quantum gravity. In: Ehlers, J., Friedrich, H. (eds.), Canonical Gravity-From Classical to Quantum, pp. 170–212. Springer, Berlin (2005)Google Scholar
  34. 34.
    Sakho, I.: Relativistic theory of one-and two electron systems: valley of stability in the helium-like ions. J. Atom. Mol. Sci. 3, 23–40 (2012)CrossRefGoogle Scholar
  35. 35.
    Eager, D., Pendrill, A.-M., Reistad, N.: Beyond velocity and acceleration: jerk, snap and higher derivative. Eur. J. Phys. 37, 065008 (2016). (11pages)CrossRefGoogle Scholar
  36. 36.
    Garay, L.J.: Quantum gravity and minimum length. Int. J. Mod. Phys. A 10, 145–166 (1995)CrossRefGoogle Scholar
  37. 37.
    Behrooz, K., Behzad, K.: Time-evaluation of the modified position and momentum operators in harmonic oscillator based on the Kempf algebra. Mod. J. Lang. Teachnol. Methods. 6, 085–090 (2016)Google Scholar

Copyright information

© Chapman University 2017

Authors and Affiliations

  1. 1.Athens Institute for Education and Research, Mathematics and Physics DivisionsAthensGreece

Personalised recommendations