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.
Similar content being viewed by others
References
Tomamichel, M., Hanggi, E.: The link between entropic uncertainty and nonlocality. J. Phys. A Math. Theor. 46, 055301 (2013)
Gisin, N.: Quantum nonlocality: how does nature do it? Science 326, 1357–1358 (2009)
Barret, J., Gisin, N.: How much measurement independence is needed to demonstrate nonlocality? Phys. Rev. Lett. 106, 100406 (2011)
Banik, M., Gazi, M.R., Ghosh, S., Kar, G.: Degree of complementarity determines the nonlocality in quantum mechanics. Phys. Rev. A 87, 052125 (2013)
Kar, G., Ghosh, S., Choudhary, S., Banik, M.: Role of measurement incompatibility and uncertainty in determining nonlocality. Mathematics 4, 52 (2016)
Oppenheim, J., Wehner, S.: The uncertainty principle determines the nonlocality of quantum mechanics. Science 330, 1072–1074 (2007)
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)
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)
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)
Ringbauer, M., Giarmatzi, C., Chaves, R., Costa, F., White, A.G., Fedrizzi, A.: Experimental test of nonlocal causality. Sci. Adv. 2–6, e1600162 (2016)
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)
Ya Slepyan, G.: Heisenberg uncertainty principle and light squeezing in quantum nanonantennas and electric circuits. J. Nanophotonics 10, 046005 (2016)
Agrawal, G.S.: Heisenberg’s uncertainty relations and quantum optics. Fortr. der Phys. 50, 575–582 (2002)
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)
Laplante, P.: Heisenberg uncertainty. Softw. Eng. Notes 15, 21–22 (1990)
Majumdar, A.S., Pramanik, T.: Some applications of uncertainty relations in quantum information. Int. J. Quantum Inf. 14, 1650022 (2016)
Hofer, W.A.: Heisenberg, uncertainty, and the scanning tunneling microscope. Front. Phys. 7, 218–222 (2012)
Johri, S., Steiger, D.S., Troyer, M.: Entanglement spectroscopy on a quantum computer. Phys. Rev. B 96, 195136 (2017)
Brennen, G.K., Song, D., Williams, C.J.: A quantum computer architecture using nonlocal interactions. Phys. Rev. A 67, 050302 (2003)
Ratanje, N., Virmani, S.: Generalised state spaces and non-locality in fault tolerant quantum computing schemes. Phys. Rev. A 83, 032309 (2011)
Boyer, M., Brodutch, A., Mor, T.: Entanglement and deterministic quantum computing with one qubit. Phys. Rev. A 95, 022330 (2017)
Jozsa, R., Linden, N.: On the role of entanglement in quantum computational speed-up. Proc. R. Soc. A 459, 2011–2032 (2003)
Simon, J.Z.: Higher derivative Lagrangians, non-locality, problems, and solutions. Phys. Rev. D 41, 3720 (1990)
Simon, J.Z.: Higher Derivative Expansions and Non-Locality. PhD thesis, University of California, Santa Barbara, August (1990)
Suykens, J.A.K.: Extending Newton’s law from nonlocal-in-time kinetic energy. Phys. Lett. A 373, 1201–1211 (2009)
Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 367 (1948)
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)
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)
Kamalov, T.F.: Classical and quantum-mechanical axioms with the higher time derivative formalism. J. Phys. Conf. Ser. 442, 012051 (2013)
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)
El-Nabulsi, R.A.: Massive photons in magnetic materials from nonlocal quantization. Magn. Magn. Mater. 458, 213–216 (2018)
El-Nabulsi, R.A.: Dynamics of pulsatile flows through microtube from nonlocality. Mech. Res. Commun. 86, 18–26 (2017)
El-Nabulsi, R.A.: On nonlocal complex Maxwell equations and wave motion in electrodynamics and dielectric media. Opt. Quantum Electron. 50, 170 (2018)
Kamalov, T.F.: Physics of non-inertial reference frames. AIP Conf. Proc. 1316, 455–458 (2010). arXiv:0708.1584
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)
Kamalov, T.F.: Model of extended mechanics and non-local hidden variables for quantum theory. J. Russ. Laser Res. 30(5), 466–471 (2009)
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
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
Caianiello, E.R.: Is there a maximal acceleration. Lett. Nuovo Cimento 32, 65–70 (1981)
Caianiello, E.R.: Geometry from quantum mechanics. Nuovo Cimento 59, 350–366 (1980). (13)
Caianiello, E.R.: Quantum and other physics as systems theory. Riv. Nuovo Cimento 15, 1–65 (1992)
Caianiello, E.R.: Maximal acceleration as a consequence of Heisenberg’s uncertainty relations. Lett. Nuovo Cimento 41, 370–372 (1984)
Pati, A.K.: A note on maximal acceleration. Europhys. Lett. 18(4), 285–289 (1992)
Pati, A.K.: On the maximal acceleration and the maximal energy loss. Nuovo Cimento B 107, 895–901 (1992)
Papini, G.: Revisiting Caianiello’s maximal acceleration. Nuovo Cimento B 117, 1325–1331 (2003)
Wood, W.R., Papini, G., Cai, Y.Q.: Maximal acceleration and the time–energy uncertainty relation. Nuovo Cimento B 104, 361–369 (1989)
Caianiello, E.R., de Filippo, S., Marmo, G., Vilasi, G.: Remarks on the maximal-acceleration hypothesis. Lett. Nuovo Cimento 34, 112–114 (1982)
Caianiello, E.R., Landi, G.: Maximal acceleration and Sakharov’s limiting temperature. Lett. Nuovo Cimento 42, 70–72 (1985)
Sharma, C.S., Srirankanatham, S.: On Caianiello’s maximal acceleration. Lett. Nuovo Cimento 44, 275–276 (1985)
Papini, G.: Spin and maximal acceleration. Galaxies 5, 103 (2017)
Tawfik, A.N., Diab, A.M.: A review of the generalized uncertainty relation. Rep. Prog. Phys. 78, 126001 (2015)
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)
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)
Griffiths, D.: Introduction to Quantum Mechanics, 2nd edn. Addison-Wesley, Boston (2005)
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)
Alloul, H.: Electro transport in solids. In: Introduction to the Physics of Electrons in Solids. Graduate Texts in Physics. Springer, Berlin (2011)
Strickland, M., Noronha, J., Denicol, G.: The anisotropic non-equilibrium hydrodynamic attractor. Phys. Rev. D 97, 036020 (2018)
Kempf, A., Mangano, G., Mann, R.B.: Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 52, 1108–1118 (1995)
Nozari, K., Azizi, T.: Some aspects of minimal length quantum mechanics. Gen. Relat. Gravity 38, 735–742 (2006)
Zotchev, S.B.: Polyene macrolide antibiotics and their applications in human therapy. Curr. Med. Chem. 10, 211–223 (2003)
Wimpfheimer, T.: A particle in a box laboratory experiment using everyday compounds. J. Lab. Chem. Educ. 3, 19–21 (2015)
Yabuzaki, J.: Carotenoids database: structures, chemical fingerprints and distribution among organisms. Database 2017, 1–11 (2017). (Article ID bax004)
Khalil, M.K.: Some implications of two forms of the generalized uncertainty principle. Adv. High Energy Phys. 2014, 619498 (2014)
Herrmann, R.: Infrared spectroscopy of diatomic molecules—a fractional calculus approach. Int. J. Mod. Phys. B 27, 1350019 (2013)
Herrmann, R.: Fractional phase transition in medium size metal clusters. Phys. A 389, 3307–3315 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40509-018-0170-1