Abstract
By probability theory the probability space to underlie the set of statistical data described by the squared modulus of a coherent superposition of microscopically distinct (sub)states (CSMDS) is non-Kolmogorovian and, thus, such data are mutually incompatible. For us this fact means that the squared modulus of a CSMDS cannot be unambiguously interpreted as the probability density and quantum mechanics itself, with its current approach to CSMDSs, does not allow a correct statistical interpretation. By the example of a 1D completed scattering and double slit diffraction we develop a new quantum-mechanical approach to CSMDSs, which requires the decomposition of the non-Kolmogorovian probability space associated with the squared modulus of a CSMDS into the sum of Kolmogorovian ones. We adapt to CSMDSs the presented by Khrennikov (Found. Phys. 35(10):1655, 2005) concept of real contexts (complexes of physical conditions) to determine uniquely the properties of quantum ensembles. Namely we treat the context to create a time-dependent CSMDS as a complex one consisting of elementary (sub)contexts to create alternative subprocesses. For example, in the two-slit experiment each slit generates its own elementary context and corresponding subprocess. We show that quantum mechanics, with a new approach to CSMDSs, allows a correct statistical interpretation and becomes compatible with classical physics.
Similar content being viewed by others
References
Ballentine, L.E.: The statistical interpretation of quantum mechanics. Rev. Mod. Phys. 42(4), 358–381 (1970)
Home, D., Whitakker, M.A.B.: Ensemble interpretation of quantum mechanics. A modern perspective. Phys. Rep. 210(4), 223–317 (1992)
Leggett, A.J.: Testing the limits of quantum mechanics: motivation, state of play, prospects. J. Phys., Condens. Matter 14, R415–R451 (2002)
Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge (2004)
’t Hooft, G.: On the free-will postulate in quantum mechanics (2007). quant-ph/0701097
Bohr, B.: Maxwell and modern theoretical physics. Nature 128, 691–692 (1931)
Khrennikov, A.Yu.: EPR-Bohm experiment and Bell’s inequality: Quantum physics meets probability theory. Theor. Math. Phys. 157, 99–115 (2008)
Accardi, L.: Snapshots on quantum probability. Vestn. Samara State Univ., Nat. Sci. Ser. 67(8/1), 277–294 (2008)
Khrennikov, A.Yu.: The principle of supplementarity: a contextual probabilistic viewpoint to complementarity, the interference of probabilities and incompatibility of variables in quantum mechanics Found. Phys. 35(10), 1655–1693 (2005)
Chuprikov, N.L.: New approach to the quantum tunnelling process: wave functions for transmission and reflection. Russ. Phys. J. 49, 119–126 (2006)
Chuprikov, N.L.: New approach to the quantum tunnelling process: characteristic times for transmission and reflection. Russ. Phys. J. 49, 314–325 (2006)
Chuprikov, N.L.: On a new mathematical model of tunnelling. Vestn. Samara State Univ., Nat. Sci. Ser. 67(8/1), 625–633 (2008)
Chuprikov, N.L.: On the generalized Hartman effect: scattering a particle on two identical rectangular potential barriers. arXiv:1005.1323v2
Hartman, T.E.: Tunneling of a wave packet. J. Appl. Phys. 33, 3427–3433 (1962)
Olkhovsky, V.S., Recami, E., Salesi, G.: Superluminal tunnelling through two successive barriers. Europhys. Lett. 57, 879–884 (2002)
Olkhovsky, V.S., Recami, E., Jakiel, J.: Unified time analysis of photon and particle tunnelling. Phys. Rep. 398, 133–178 (2004)
Winful, H.G.: Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox. Phys. Rep. 436, 1–69 (2006)
Lunardi, J.T., Manzoni, L.A.: Relativistic tunnelling through two successive barriers. Phys. Rev. A 76, 042111 (2007)
Nimtz, G.: On virtual phonons, photons, and electrons. Found. Phys. (2009). doi:10.1007/s10701-009-9356-z
Krekora, P., Su, Q., Grobe, R.: Effects of relativity on the time-resolved tunnelling of electron wave packets. Phys. Rev. A 63, 032107 (2001)
Li, Z.J.Q., Nie, Y.H., Liang, J.J., Liang, J.Q.: Larmor precession and dwell time of a relativistic particle scattered by a rectangular quantum well. J. Phys. A, Math. Gen. 36, 6563–6570 (2003)
Buttiker, M., Landauer, R.: Traversal time for tunneling. Phys. Rev. Lett. 49, 1739–1742 (1982)
Landauer, R., Martin, Th.: Barrier interaction time in tunnelling. Rev. Mod. Phys. 66, 217–228 (1994)
Buttiker, M.: Larmor precession and the traversal time for tunnelling. Phys. Rev. B 27, 6178–6188 (1983)
Home, D., Kaloyerou, P.N.: New twists to Einstein’s two-slit experiment: complementarity vis-à-vis the causal interpretation. J. Phys. A, Math. Gen. 22, 3253–3266 (1989)
Schlosshauer, M.: Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys. 76, 1268–1305 (2004)
Joos, E., Zeh, H.D.: The emergence of classical properties through interaction with the environment. Z. Phys. B 59, 223–243 (1985)
Ghirardi, G.C., Grassi, R., Rimini, A.: Continuous-spontaneous-reduction model involving gravity. Phys. Rev. A 42, 1057–1064 (1990)
Ballentine, L.E.: Failure of some theories of state reduction. Phys. Rev. A 43(1), 9–12 (1991)
Volovich, I.V.: Quantum cryptography in space and Bell’s theorem. In: Khrennikov, A.Y. (ed.) Proc. Conf. Foundations of Probability and Physics. Ser. Quantum Probability and While Noise Analysis, vol. 13, pp. 364–372. World Sci., River Edge (2001)
Volovich, I.V.: Towards quantum information theory in space and time. In: Khrennikov, A.Y. (ed.) Proc. Conf. Quantum Theory: Reconsideration of Foundations. Ser. Math. Modelling. Vaxjo Univ. Press., vol. 2, pp. 423–440. (2002)
Christian, J.: Disproofs of Bell, GHZ, and Hardy type theorems and the illusion of entanglement (2009). arXiv:0904.4259v4
Fine, A.: Hidden variables, joint probability, and the bell inequalities. Phys. Rev. Lett. 48, 291–295 (1982)
Accardi, L.: Topics in quantum probability. Phys. Rep. 77, 169–192 (1981)
Pitowsky, I.: Resolution of the Einstein-Podolsky-Rosen and Bell paradoxes. Phys. Rev. Lett. 48, 1299–1302 (1982)
Rastall, P.: The Bell inequalities. Found. Phys. 13(6), 555–570 (1983)
Hess, K., Michielsen, K., De Raedt, H.: Possible experience: from Boole to Bell. Europhys. Lett. 87(6), 60007 (2009)
Nieuwenhuizen, T.M.: Is the contextuality loophole fatal for the derivation of Bell inequalities? Found. Phys. (2010). doi:10.1007/s10701-010-9461-z
Andreev, V.A.: The correlation Bell inequalities. Teor. Mat. Fiz. 158(2), 234–249 (2009)
Accardi, L., Regoli, M.: Locality and Bell’s inequality (2000). arXiv:quant-ph/0007005v2
Khrennikov, A.: Non-Kolmogorov probability models and modified Bell’s inequality. J. Math. Phys. 41(4), 1768–1777 (2000)
Khrennikov, A.Y.: Contextual Approach to Quantum Formalism, p. 353. Springer, Berlin (2009)
Accardi, L.: Dialogs on Quantum Mechanics. Moscow, (2004) 447 p
Khrennikov, A.: Detection model based on representation of quantum particles by classical random fields: Born’s rule and beyond. Found. Phys. 39(9), 997–1022 (2009)
Hestenes, D.: Zitterbewegung in quantum mechanics. Found. Phys. 40(1), 1–54 (2010)
Hofer, W.A.: Unconventional approach to orbital-free density functional theory derived from a model of extended electrons. Found. Phys. (2010). doi:10.1007/s10701-010-9517-0
Slavnov, D.A.: The locality problem in quantum measurements. Phys. Part. Nucl. 41(1), 149–173 (2007). arXiv:1010.4412v1
Afshar, Sh.S., Flores, E., McDonald, K.F., Knoesel, E.: Paradox in wave-particle duality. Found. Phys. 37(2), 295–305 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chuprikov, N.L. From a 1D Completed Scattering and Double Slit Diffraction to the Quantum-Classical Problem for Isolated Systems. Found Phys 41, 1502–1520 (2011). https://doi.org/10.1007/s10701-011-9564-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-011-9564-1