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Decoherence and Determinism in a One-Dimensional Cloud-Chamber Model

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The hypothesis (Sparenberg et al. in EPJ Web Conf 58:01016, [1]. that the particular linear tracks appearing in the measurement of a spherically-emitting radioactive source in a cloud chamber are determined by the (random) positions of atoms or molecules inside the chamber is further explored in the framework of a recently established one-dimensional model (Carlone et al. Comm Comput Phys 18:247, [2]. In this model, meshes of localized spins 1/2 play the role of the cloud-chamber atoms and the spherical wave is replaced by a linear superposition of two wave packets moving from the origin to the left and to the right, evolving deterministically according to the Schrödinger equation. We first revisit these results using a time-dependent approach, where the wave packets impinge on a symmetric two-sided detector. We discuss the evolution of the wave function in the configuration space and stress the interest of a non-symmetric detector in a quantum-measurement perspective. Next we use a time-independent approach to study the scattering of a plane wave on a single-sided detector. Preliminary results are obtained, analytically for the single-spin case and numerically for up to 8 spins. They show that the spin-excitation probabilities are sometimes very sensitive to the parameters of the model, which corroborates the idea that the measurement result could be determined by the atom positions. The possible origin of decoherence and entropy increase in future models is finally discussed.

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  1. The present apparatus hidden variables are thus standard physical quantities. They are “hidden”, in the sense that they are not easily accessible experimentally, but they are not “hidden variables” of the kind postulated by Einstein, Podolsky and Rosen [3], which would have a different—and unknown—nature.


  1. Sparenberg, J.-M., Nour, R., Manço, A.: EPJ Web Conf. 58, 01016 (2013).

    Article  Google Scholar 

  2. Carlone, R., Figari, R., Negulescu, C.: Comm. Comput. Phys. 18, 247 (2015).

    Article  ADS  Google Scholar 

  3. Einstein, A., Podolsky, B., Rosen, N.: Phys. Rev. 47, 777 (1935)

    Article  ADS  Google Scholar 

  4. Mott, N.F.: The wave mechanics of alpha-ray tracks. Proc. R. Soc. A126, 79 (1929). Also available in [15]

    Article  ADS  MATH  Google Scholar 

  5. Mott, N.F., Alexandrov, A.S. (eds.): Sir Nevill Mott, 65 years in Physics. World Scientific, Singapore (1995)

    Google Scholar 

  6. Barra, F., Gaspard, P.: Scattering in periodic systems: from resonances to band structure. J. Phys. A 32, 3357 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Anderson, P.W.: Phys. Rev. 109, 1492 (1958)

    Article  ADS  Google Scholar 

  8. Dell’Antonio, G.F.: Found. Phys. 45, 11 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  9. Cacciapuoti, C., Carlone, R., Figari, R.: Spin-dependent point potentials in one and three dimensions. J. Phys. A 40, 249 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Figari, R., Teta, A.: Quantum Dynamics of a Particle in a Tracking Chamber. Springer, Berlin (2014).

    Book  Google Scholar 

  11. Figari, R., Teta, A.: From quantum to classical world: emergence of trajectories in a quantum system. Math. Mech. Compl. Syst. 4, 235 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  12. Bethe, H.: Ann. Phys. 397, 325 (1930).

    Article  Google Scholar 

  13. Beceiro-Novo, S., Ahn, T., Bazin, D., Mittig, W.: Prog. Part. Nucl. Phys. 84, 124 (2015).

    Article  ADS  Google Scholar 

  14. Kirkby, J., et al.: Ion-induced nucleation of pure biogenic particles. Nature 533, 521 (2016).

    Article  ADS  Google Scholar 

  15. Allahverdyan, A.E., Balian, R., Nieuwenhuizen, T.M.: Understanding quantum measurement from the solution of dynamical models. Phys. Rep. 525, 1 (2013).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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This text presents research results of the IAP program P7/12 initiated by the Belgian-state Federal Services for Scientific, Technical, and Cultural Affairs. We thank Ruben Ceulemans (KU Leuven) for useful discussions and for the check of our numerical results.

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Correspondence to Jean-Marc Sparenberg.

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Sparenberg, JM., Gaspard, D. Decoherence and Determinism in a One-Dimensional Cloud-Chamber Model. Found Phys 48, 429–439 (2018).

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