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Pilot-Wave Quantum Theory with a Single Bohm’s Trajectory

Abstract

The representation of a quantum system as the spatial configuration of its constituents evolving in time as a trajectory under the action of the wave-function, is the main objective of the de Broglie–Bohm theory (or pilot wave theory). However, its standard formulation is referred to the statistical ensemble of its possible trajectories. The statistical ensemble is introduced in order to establish the exact correspondence (the Born’s rule) between the probability density on the spatial configurations and the quantum distribution, that is the squared modulus of the wave-function. In this work we explore the possibility of using the pilot wave theory at the level of a single Bohm’s trajectory, that is a single realization of the time dependent configuration which should be representative of a single realization of the quantum system. The pilot wave theory allows a formally self-consistent representation of quantum systems as a single Bohm’s trajectory, but in this case there is no room for the Born’s rule at least in its standard form. We will show that a correspondence exists between the statistical distribution of configurations along the single Bohm’s trajectory and the quantum distribution for a subsystem interacting with the environment in a multicomponent system. To this aim, we present the numerical results of the single Bohm’s trajectory description of the model system of six confined planar rotors with random interactions. We find a rather close correspondence between the coordinate distribution of one rotor, the others representing the environment, along its trajectory and the time averaged marginal quantum distribution for the same rotor. This might be considered as the counterpart of the standard Born’s rule when the pilot wave theory is applied at the level of single Bohm’s trajectory. Furthermore a strongly fluctuating behavior with a fast loss of correlation is found for the evolution of each rotor coordinate. This suggests that a Markov process might well approximate the evolution of the Bohm’s coordinate of a single rotor (the subsystem) and, under this condition, it is shown that the correspondence between coordinate distribution and quantum distribution of the rotor is exactly verified.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Publications, New York (1972)

    MATH  Google Scholar 

  2. Bartsch, C., Gemmer, J.: Dynamical typicality of quantum expectation values. Phys. Rev. Lett. 102, 110403 (2009)

    ADS  Article  Google Scholar 

  3. Berezovsky, J., Mikkelsen, M.H., Stoltz, N.G., Coldren, L.A., Awschalom, D.D.: Picosecond coherent optical manipulation of a single electron spin in a quantum dot. Science 320, 349 (2008)

    ADS  Article  Google Scholar 

  4. Biercuk, M.J., Uys, H., VanDevender, A.P., Shiga, N., Itano, W.M., Bollinger, J.J.: Optimized dynamical decoupling in a model quantum memory. Nature 458, 07951 (2009)

    Article  Google Scholar 

  5. Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev. 85, 166 (1952)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. II. Phys. Rev. 85, 180 (1952)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. Bohm, D.: Proof that probability density approaches \(|\psi |^2\) in causal interpretation of the quantum theory. Phys. Rev. 89(2), 15 (1953)

    MathSciNet  Article  Google Scholar 

  8. Braverman, B., Simon, C.: Proposal to observe the nonlocality of bohmian trajectories with entangled photons. Phys. Rev. Lett. 110, 060406 (2013)

    ADS  Article  Google Scholar 

  9. Brody, T.A., Flores, J., French, J.B., Mello, P.A., Pandey, A., Wong, S.S.M.: Random-matrix physics: spectrum and strength fluctuation. Rev. Mod. Phys. 53, 385 (1981)

    ADS  MathSciNet  Article  Google Scholar 

  10. de Broglie, L.: Electrons et Photons, Rapport au Ve Conseil Physique Solvay. Gauhier-Villiars, Paris (1928)

    Google Scholar 

  11. de Broglie, L.: An Introduction to the Study of Wave Mechanics. E.P. Dutton and Company, New York (1930)

    MATH  Google Scholar 

  12. Christov, I.P.: Time-dependent quantum monte carlo: preparation of the ground state. N. J. Phys. 9, 70 (2007)

    Article  Google Scholar 

  13. Cohen-Tannoudji, C., Diu, B., Laloë, F.: Quantum Mechanics. Wiley, New York (1977)

    MATH  Google Scholar 

  14. Colin, S., Struyve, W.: Quantum non-equilibrium and relaxation to equilibrium for a class of de broglie-bohm-type theories. N. J. Phys. 12, 043008 (2010)

    MathSciNet  Article  Google Scholar 

  15. Dürr, D., Goldstein, S., Zanghi, N.: Quantum equilibrium and the origin of absolute uncertainty. J. Stat. Phys. 67, 843 (1992)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  16. Figalli, A., Klein, C., Markowich, P., Sparber, C.: Wkb analysis of bohmian dynamics. Commun. Pure Appl. Math. 67, 0581–0620 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  17. Fresch, B., Moro, G.J.: Typicality in ensembles of quantum states: Monte carlo sampling versus analytical approximations. J. Phys. Chem. A 113, 14502 (2009)

    Article  Google Scholar 

  18. Fresch, B., Moro, G.J.: Emergence of equilibrium thermodynamic properties in quantum pure states. I. Theory. J. Chem. Phys. 133, 034509 (2010)

    ADS  Article  Google Scholar 

  19. Fresch, B., Moro, G.J.: Emergence of equilibrium thermodynamic properties in quantum pure states. II. Analysis of a spin model system. J. Chem. Phys. 133, 034510 (2010)

    ADS  Article  Google Scholar 

  20. Fresch, B., Moro, G.J.: Beyond quantum micocanonical statistics. J. Chem. Phys. 134, 054510 (2011)

    ADS  Article  Google Scholar 

  21. Fresch, B., Moro, G.J.: Typical response of quantum pure states. Eur. Phys. J. B 86, 233 (2013)

    ADS  MathSciNet  Article  Google Scholar 

  22. Garashchuk, S., Dell’Angelo, D., Rassolov, V.A.: Dynamics in the quantum/classical limit based on selective use of the quantum potential. J. Chem. Phys. 141, 234107 (2014)

    ADS  Article  Google Scholar 

  23. Garashchuk, S., Rassolov, V.A.: Semiclassical dynamics based on quantum trajectories. Chem. Phys. Lett. 364, 562 (2002)

    ADS  Article  Google Scholar 

  24. Garashchuk, S., Volkov, M.V.: Incorporation of quantum effects for selected degrees of freedom into the trajectory based dynamics using spatial domains. J. Chem. Phys. 137, 074115 (2012)

    ADS  Article  Google Scholar 

  25. Gardiner, C.W.: Handbook of Stochastic Methods for Physics. Chemistry and the Natural Sciences. Springer, New York (1986)

    MATH  Google Scholar 

  26. Goldstein, S., Lebowitz, J.L., Mastrodonato, C., Tumulka, R., Zanghi, N.: Approach to thermal equilibrium of macroscopic quantum systems. Phys. Rev. E 81, 011109 (2010)

    ADS  Article  Google Scholar 

  27. Goldstein, S., Lebowitz, J.L., Tumulka, R., Zanghi, N.: Canonical typicality. Phys. Rev. Lett. 96, 050403 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  28. Herman, M., Perry, D.S.: Molecular spectroscopy and dynamics: a polyad-based perspective. Phys. Chem. Chem. Phys. 15, 9970 (2013)

    Article  Google Scholar 

  29. Holland, P.R.: The Quantum Theory of Motion. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  30. Huang, K.: Statistical Mechanics. Wiley, New York (1987)

    MATH  Google Scholar 

  31. Khinchin, A.Y.: Mathematical Foundation of Statistical Mechanics. Dover, New York (1949)

    MATH  Google Scholar 

  32. Krasnoshchekov, S.V., Stepanov, N.F.: Polyad quantum numbers and multiple resonances in anharmonic vibrational studies of polyatomic molecules. J. Chem. Phys. 139, 184101 (2013)

    ADS  Article  Google Scholar 

  33. Linden, N., Popescu, S., Short, A.J., Winter, A.: Quantum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  34. Lopreore, C.L., Wyatt, R.E.: Quantum wave packet dynamics with trajectories. Phys. Rev. Lett. 82(26), 5190 (1999)

    ADS  Article  Google Scholar 

  35. Madelung, V.E.: Quantentheorie in hydrodynamischer form. Z. Phys. 40, 322 (1927)

    ADS  Article  MATH  Google Scholar 

  36. Neumann, P., Mizuochi, N., Rempp, F., Hemmer, P., Watanabe, H., Yamasaki, S., Jacques, V., Gaebel, T., Jelezko, F., Wrachtrup, J.: Multipartite entanglement among single spins in diamond. Science 320, 1326 (2008)

    ADS  Article  Google Scholar 

  37. von Neunmann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955)

    Google Scholar 

  38. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, New York (2000)

    MATH  Google Scholar 

  39. Norsen, T.: The pilot-wave perspective on quantum scattering and tunneling. Am. J. Phys. 81, 258 (2013)

    ADS  Article  Google Scholar 

  40. Norsen, T.: The pilot-wave perspective on spin. Am. J. Phys. 82, 337 (2014)

    ADS  Article  Google Scholar 

  41. Philbin, T.G.: Derivation of quantum probabilities from deterministic evolution (2015)

  42. Popescu, S., Short, A.J., Winter, A.: Entanglement and the foundations of statistical mechanics. Nat. Phys. 2, 754 (2006)

    Article  Google Scholar 

  43. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  44. Reimann, P.: Foundation of statistical mechanics under experimentally realistic conditions. Phys. Rev. Lett. 101, 190403 (2008)

    ADS  Article  Google Scholar 

  45. Sanderson, C.: Armadillo: an open source c\(++\) linear algebra library for fast prototyping and computationally intensive experiments. Technical Report, NICTA (2010)

  46. Sawada, R., Sato, T., Ishikawa, K.L.: Analysis of strong-field enhanced ionization of molecules using bohmian trajectories. Phys. Rev. A 90, 023404 (2014)

    ADS  Article  Google Scholar 

  47. Shannon, C., Weaver, W.: The Mathematical Theory of Communication. University of Illinois, Urbana (1949)

    MATH  Google Scholar 

  48. Shtanov, Y.V.: Origin of quantum randomness in the pilot wave quantum mechanics (1997)

  49. Suter, D., Mahesh, T.S.: Spins as qubits: quantum information processing by nuclear magnetic resonance. J. Chem. Phys. 128, 052206 (2008)

    ADS  Article  Google Scholar 

  50. Towler, M.D., Russell, N.J., Valentini, A.: Time scales for dynamical relaxation to the born rule. Proc. R. Soc. A 468, 990 (2012)

    ADS  MathSciNet  Article  Google Scholar 

  51. Valentini, A.: Signal-locality, uncertainty, and the subquantum H-theorem. I. Phys. Lett. A 156, 5 (1991)

    ADS  MathSciNet  Article  Google Scholar 

  52. Valentini, A.: Signal-locality, uncertainty, and the subquantum H-theorem. II. Phys. Lett. A 158, 1 (1991)

    ADS  MathSciNet  Article  Google Scholar 

  53. Wigner, E.P.: Random matrices in physics. SIAM Rev. 9(1), 1 (1967)

    ADS  Article  MATH  Google Scholar 

  54. Wyatt, R.E.: Quantum wave packet dynamics with trajectoriers: application to reactive scattering. J. Chem. Phys. 111(10), 4406 (1999)

    ADS  Article  Google Scholar 

  55. Wyatt, R.E.: Quantum wave packet dynamics with trajectoriers: wavefunction synthesis along quantum paths. Chem. Phys. Lett. 313, 189 (1999)

    ADS  Article  Google Scholar 

  56. Zyczkowski, K.: Volume of the set of separable states. II. Phys. Rev. A 60, 3496 (1999)

    ADS  MathSciNet  Article  Google Scholar 

  57. Zyczkowski, K., Sommers, H.J.: Induced measures in the space of mixed quantum states. J. Phys. A 34, 7111 (2001)

    ADS  MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

The authors acknowledge the support by Univesità degli Studi di Padova through \(60\,\%\) grants. We thank the anonymous reviewers for critical comments and the suggestions.

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Correspondence to Giorgio J. Moro.

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Avanzini, F., Fresch, B. & Moro, G.J. Pilot-Wave Quantum Theory with a Single Bohm’s Trajectory. Found Phys 46, 575–605 (2016). https://doi.org/10.1007/s10701-015-9979-1

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  • DOI: https://doi.org/10.1007/s10701-015-9979-1

Keywords

  • Quantum statistical mechanics
  • de Broglie–Bohm theory
  • Pure quantum state statistics