Skip to main content
SpringerLink
Log in

Nonlocal uncertainty and its implications in quantum mechanics at ultramicroscopic scales

  • Regular Paper
  • Published:
Quantum Studies: Mathematics and Foundations Aims and scope Submit manuscript

Abstract

In this paper, we have discussed the implications of quantum acceleratum operator which is one of the main consequences of nonlocal-in-time kinetic energy approach in quantum mechanics and molecular physics. We have constructed the nonlocal Heisenberg’s uncertainty relation by introducing a new extended quantum state and we have discussed a number of its implications in atomic and molecular physics, in particular, the \(\upbeta \)-carotene and the hydrogen atom. A number of properties were derived and revealed which prove the importance of nonlocality in theoretical and applied quantum sciences at ultramicroscopic scales.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Tomamichel, M., Hanggi, E.: The link between entropic uncertainty and nonlocality. J. Phys. A Math. Theor. 46, 055301 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Gisin, N.: Quantum nonlocality: how does nature do it? Science 326, 1357–1358 (2009)

    Article  Google Scholar 

  3. Barret, J., Gisin, N.: How much measurement independence is needed to demonstrate nonlocality? Phys. Rev. Lett. 106, 100406 (2011)

    Article  Google Scholar 

  4. Banik, M., Gazi, M.R., Ghosh, S., Kar, G.: Degree of complementarity determines the nonlocality in quantum mechanics. Phys. Rev. A 87, 052125 (2013)

    Article  Google Scholar 

  5. Kar, G., Ghosh, S., Choudhary, S., Banik, M.: Role of measurement incompatibility and uncertainty in determining nonlocality. Mathematics 4, 52 (2016)

    Article  MATH  Google Scholar 

  6. Oppenheim, J., Wehner, S.: The uncertainty principle determines the nonlocality of quantum mechanics. Science 330, 1072–1074 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hensen, B., Bernien, H., Dréau, A.E., Reiserer, A., Kalb, N., Blok, M.S., Ruitenberg, J., Vermeulen, R.F.L., Schouten, R.N., Abellán, C., Amaya, W., Pruneri, V., Mitchell, M.W., Markham, M., Twitchen, D.J., Elkouss, D., Wehner, S., Taminiau, T.H., Hanson, R.: Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometers. Nature 526, 682 (2015)

    Article  Google Scholar 

  8. Giustina, M., Versteegh, M.A.M., Wengerowsky, S., Handsteiner, J., Hochrainer, A., Phelan, K., Steinlechner, F., Kofler, J., Larsson, J.-A., Abellán, C., Amaya, W., Pruneri, V., Mitchell, M.W., Beyer, J., Gerrits, T., Lita, A.E., Shalm, L.K., Nam, S., Scheidl, T., Ursin, R., Wittmann, B., Zeilinger, A.: Significant-loophole-free test of Bell theorem with entangled photons. Phys. Rev. Lett. 115, 250401 (2015)

    Article  Google Scholar 

  9. Shalm, L.K., Meyer-Scott, E., Christensen, B.G., Bierhorst, P., Wayne, M.A., Stevens, M.J., Gerrits, T., Glancy, Scott, Hamel, D.R., Allman, M.S., Coakley, K.J., Dyer, S.D., Hodge, C., Lita, A.E., Verma, V.B., Lambrocco, C., Tortorici, E., Migdall, A.L., Zhang, Y., Kumor, D.R., Farr, W.H., Marsili, F., Shaw, M.D., Stern, J.A., Abellán, C., Amaya, W., Pruneri, V., Jennewein, T., Mitchell, M.W., Kwiat, P.G., Bienfang, J.C., Mirin, R.P., Knill, E., Nam, S.W.: A strong loophole-free test of local realism. Phys. Rev. Lett. 115, 250402 (2015)

    Article  Google Scholar 

  10. Ringbauer, M., Giarmatzi, C., Chaves, R., Costa, F., White, A.G., Fedrizzi, A.: Experimental test of nonlocal causality. Sci. Adv. 2–6, e1600162 (2016)

    Article  Google Scholar 

  11. Beche, B., Gaviot, E.: About the Heisenberg’s uncertainty principle and the determination of effective optical indices in integrated photonics at high sub-wavelength regime. Optik 127, 3643–3645 (2016)

    Article  Google Scholar 

  12. Ya Slepyan, G.: Heisenberg uncertainty principle and light squeezing in quantum nanonantennas and electric circuits. J. Nanophotonics 10, 046005 (2016)

    Article  Google Scholar 

  13. Agrawal, G.S.: Heisenberg’s uncertainty relations and quantum optics. Fortr. der Phys. 50, 575–582 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hartcourt, R.D.: The Heisenberg uncertainty principle: an application to the shell structure of atoms and orbit descriptions of molecules. J. Chem. Educ. 64, 1070 (1987)

    Article  Google Scholar 

  15. Laplante, P.: Heisenberg uncertainty. Softw. Eng. Notes 15, 21–22 (1990)

    Article  Google Scholar 

  16. Majumdar, A.S., Pramanik, T.: Some applications of uncertainty relations in quantum information. Int. J. Quantum Inf. 14, 1650022 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hofer, W.A.: Heisenberg, uncertainty, and the scanning tunneling microscope. Front. Phys. 7, 218–222 (2012)

    Article  Google Scholar 

  18. Johri, S., Steiger, D.S., Troyer, M.: Entanglement spectroscopy on a quantum computer. Phys. Rev. B 96, 195136 (2017)

    Article  Google Scholar 

  19. Brennen, G.K., Song, D., Williams, C.J.: A quantum computer architecture using nonlocal interactions. Phys. Rev. A 67, 050302 (2003)

    Article  Google Scholar 

  20. Ratanje, N., Virmani, S.: Generalised state spaces and non-locality in fault tolerant quantum computing schemes. Phys. Rev. A 83, 032309 (2011)

    Article  Google Scholar 

  21. Boyer, M., Brodutch, A., Mor, T.: Entanglement and deterministic quantum computing with one qubit. Phys. Rev. A 95, 022330 (2017)

    Article  MathSciNet  Google Scholar 

  22. Jozsa, R., Linden, N.: On the role of entanglement in quantum computational speed-up. Proc. R. Soc. A 459, 2011–2032 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Simon, J.Z.: Higher derivative Lagrangians, non-locality, problems, and solutions. Phys. Rev. D 41, 3720 (1990)

    Article  MathSciNet  Google Scholar 

  24. Simon, J.Z.: Higher Derivative Expansions and Non-Locality. PhD thesis, University of California, Santa Barbara, August (1990)

  25. Suykens, J.A.K.: Extending Newton’s law from nonlocal-in-time kinetic energy. Phys. Lett. A 373, 1201–1211 (2009)

    Article  MATH  Google Scholar 

  26. Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 367 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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)

    Article  MathSciNet  MATH  Google Scholar 

  28. El-Nabulsi, R.A.: 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)

    Article  Google Scholar 

  29. Kamalov, T.F.: Classical and quantum-mechanical axioms with the higher time derivative formalism. J. Phys. Conf. Ser. 442, 012051 (2013)

    Article  Google Scholar 

  30. El-Nabulsi, R.A.: Modeling of electrical and mesoscopic circuits at quantum nanoscale from heat momentum operator. Phys. E Low Dimens. Syst. Nanostruct. 98, 90–104 (2018)

    Article  Google Scholar 

  31. El-Nabulsi, R.A.: Massive photons in magnetic materials from nonlocal quantization. Magn. Magn. Mater. 458, 213–216 (2018)

    Article  Google Scholar 

  32. El-Nabulsi, R.A.: Dynamics of pulsatile flows through microtube from nonlocality. Mech. Res. Commun. 86, 18–26 (2017)

    Article  Google Scholar 

  33. El-Nabulsi, R.A.: On nonlocal complex Maxwell equations and wave motion in electrodynamics and dielectric media. Opt. Quantum Electron. 50, 170 (2018)

    Article  Google Scholar 

  34. Kamalov, T.F.: Physics of non-inertial reference frames. AIP Conf. Proc. 1316, 455–458 (2010). arXiv:0708.1584

    Article  Google Scholar 

  35. Kamalov, T.F.: The systematic measurement errors and uncertainty relation. New Technol MSOU 5, 10–12 (2006). (in Russian, English version: arXiv:quant-ph/0611053)

    Google Scholar 

  36. Kamalov, T.F.: Model of extended mechanics and non-local hidden variables for quantum theory. J. Russ. Laser Res. 30(5), 466–471 (2009)

    Article  Google Scholar 

  37. Kamalov, T.F.: Simulation the nuclear interaction. In: Studenikin, A.I. (ed.) Particle physics on the Eve of LHC, proceedings of thirteenth Lomonosov conference on elementary particle physics, Moscow, Russia, 23–29 August 2007, pp. 439–442. World Scientific, Singapore (2010). https://doi.org/10.1142/9789812837592_0076

    Google Scholar 

  38. El-Nabulsi, R.A.: On maximal acceleration and quantum acceleratum operator in quantum mechanics. Quantum Stud. Math. Found. (2017). https://doi.org/10.1007/s40509-017-0142-x

  39. Caianiello, E.R.: Is there a maximal acceleration. Lett. Nuovo Cimento 32, 65–70 (1981)

    Article  Google Scholar 

  40. Caianiello, E.R.: Geometry from quantum mechanics. Nuovo Cimento 59, 350–366 (1980). (13)

    Article  MathSciNet  Google Scholar 

  41. Caianiello, E.R.: Quantum and other physics as systems theory. Riv. Nuovo Cimento 15, 1–65 (1992)

    Article  MathSciNet  Google Scholar 

  42. Caianiello, E.R.: Maximal acceleration as a consequence of Heisenberg’s uncertainty relations. Lett. Nuovo Cimento 41, 370–372 (1984)

    Article  MathSciNet  Google Scholar 

  43. Pati, A.K.: A note on maximal acceleration. Europhys. Lett. 18(4), 285–289 (1992)

    Article  Google Scholar 

  44. Pati, A.K.: On the maximal acceleration and the maximal energy loss. Nuovo Cimento B 107, 895–901 (1992)

    Article  Google Scholar 

  45. Papini, G.: Revisiting Caianiello’s maximal acceleration. Nuovo Cimento B 117, 1325–1331 (2003)

    Google Scholar 

  46. Wood, W.R., Papini, G., Cai, Y.Q.: Maximal acceleration and the time–energy uncertainty relation. Nuovo Cimento B 104, 361–369 (1989)

    Article  MathSciNet  Google Scholar 

  47. Caianiello, E.R., de Filippo, S., Marmo, G., Vilasi, G.: Remarks on the maximal-acceleration hypothesis. Lett. Nuovo Cimento 34, 112–114 (1982)

    Article  MathSciNet  Google Scholar 

  48. Caianiello, E.R., Landi, G.: Maximal acceleration and Sakharov’s limiting temperature. Lett. Nuovo Cimento 42, 70–72 (1985)

    Article  Google Scholar 

  49. Sharma, C.S., Srirankanatham, S.: On Caianiello’s maximal acceleration. Lett. Nuovo Cimento 44, 275–276 (1985)

    Article  Google Scholar 

  50. Papini, G.: Spin and maximal acceleration. Galaxies 5, 103 (2017)

    Article  Google Scholar 

  51. Tawfik, A.N., Diab, A.M.: A review of the generalized uncertainty relation. Rep. Prog. Phys. 78, 126001 (2015)

    Article  Google Scholar 

  52. Li, Z.: The modification of generalized uncertainty principle applied in the detection technique of femtosecond laser. IOP Conf. Ser. Mater. Sci. Eng. 274, 012063 (2017)

    Article  Google Scholar 

  53. Li, H., Wang, J.-J., Yang, B., Wang, Y.-N., Shen, H.-J.: Low temperature thermodynamic properties of Fermi gas under generalized uncertainty. Acta Phys. Sin. 64, 80502–080502 (2015)

    Google Scholar 

  54. Griffiths, D.: Introduction to Quantum Mechanics, 2nd edn. Addison-Wesley, Boston (2005)

    Google Scholar 

  55. Moares, E.M.: Time varying heat conduction in solids. In: Vikhrenko, V. (ed.) Heat Conduction-Basic Research. INTECH, London (2011). (Chapter ISBN 978-953-307-404-7, 8)

    Google Scholar 

  56. Alloul, H.: Electro transport in solids. In: Introduction to the Physics of Electrons in Solids. Graduate Texts in Physics. Springer, Berlin (2011)

  57. Strickland, M., Noronha, J., Denicol, G.: The anisotropic non-equilibrium hydrodynamic attractor. Phys. Rev. D 97, 036020 (2018)

    Article  MathSciNet  Google Scholar 

  58. Kempf, A., Mangano, G., Mann, R.B.: Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 52, 1108–1118 (1995)

    Article  MathSciNet  Google Scholar 

  59. Nozari, K., Azizi, T.: Some aspects of minimal length quantum mechanics. Gen. Relat. Gravity 38, 735–742 (2006)

    Article  MATH  Google Scholar 

  60. Zotchev, S.B.: Polyene macrolide antibiotics and their applications in human therapy. Curr. Med. Chem. 10, 211–223 (2003)

    Article  Google Scholar 

  61. Wimpfheimer, T.: A particle in a box laboratory experiment using everyday compounds. J. Lab. Chem. Educ. 3, 19–21 (2015)

    Google Scholar 

  62. Yabuzaki, J.: Carotenoids database: structures, chemical fingerprints and distribution among organisms. Database 2017, 1–11 (2017). (Article ID bax004)

    Article  Google Scholar 

  63. Khalil, M.K.: Some implications of two forms of the generalized uncertainty principle. Adv. High Energy Phys. 2014, 619498 (2014)

    Article  MathSciNet  Google Scholar 

  64. Herrmann, R.: Infrared spectroscopy of diatomic molecules—a fractional calculus approach. Int. J. Mod. Phys. B 27, 1350019 (2013)

    Article  MathSciNet  Google Scholar 

  65. Herrmann, R.: Fractional phase transition in medium size metal clusters. Phys. A 389, 3307–3315 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics and Physics Divisions, Athens Institute for Education and Research, 8 Valaoritou Street, Kolonaki, 10671, Athens, Greece

    Rami Ahmad El-Nabulsi

Authors
  1. Rami Ahmad El-Nabulsi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rami Ahmad El-Nabulsi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El-Nabulsi, R.A. Nonlocal uncertainty and its implications in quantum mechanics at ultramicroscopic scales. Quantum Stud.: Math. Found. 6, 123–133 (2019). https://doi.org/10.1007/s40509-018-0170-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40509-018-0170-1

Keywords

Search

Navigation