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Signal, Image and Video Processing

, Volume 8, Issue 1, pp 27–37 | Cite as

Solving Schrödinger equation via Tartaglia/Pascal triangle: a possible link between stochastic processing and quantum mechanics

  • A. FarinaEmail author
  • M. Frasca
  • M. Sedehi
Original Paper

Abstract

In a recent paper (Farina et al. in Signal Image Video Process 1–16, 2011), it was shown a clean connection between the solution of the classical heat equation and the Tartaglia/Pascal/Yang-Hui triangle. When the time variable in the heat equation is substituted with the imaginary time, the heat equation becomes the Schrödinger equation of the quantum mechanics. So, a conjecture was put forward about a connection between the solution of the Schrödinger equation and a suitable generalization of the Tartaglia triangle. This paper proves that this conjecture is true and shows a new—as far as the authors are aware—result concerning the generalization of the classical Tartaglia triangle by introducing the “complex-valued Tartaglia triangle.” A “complex-valued Tartaglia triangle” is just the square root of an ordinary Tartaglia triangle, with a suitable phase factor calculated via a discretized version of the ordinary continuous case of the Schrödinger equation. So, taking the square of this complex-valued Tartaglia triangle gives back exactly the probability distribution of a discrete random walk. We also discuss about potential connections between the theories of stochastic processes and quantum mechanics: a connection debated since the inception of the theories and still lively hot today.

Keywords

Stochastic processing Quantum mechanics Schrödinger equation Tartaglia-Pascal triangle 

Notes

Acknowledgments

The authors wish to thank the following colleagues who kindly shared interest and gave comments to the paper topic: Professor F. Zirilli, Dr. Wu Biao, Dr. P. Natoli, Professor P. Teofilatto. We would also like to thank the Editor for the excellent way he managed the review process that was really difficult for this kind of paper and the referees for the insightful and very helpful comments that guided us to improve the paper in a significant way.

References

  1. 1.
    Farina, A., Giompapa, S., Graziano, A., Liburdi, A., Ravanelli, M., Zirilli, F.: Tartaglia and Pascal triangle: a historical perspective with applications; from probability to modern physics, signal processing, and finance. Signal Image Video Process. 1–16 (2011). doi: 10.1007/s11760-011-0228-6
  2. 2.
    Jazwinski, A.H.: Stochastic processes and filtering theory. Academic Press, London (1970)zbMATHGoogle Scholar
  3. 3.
    Fourier, J.: Théorie analytique de la chaleur. Firmin Didot Père et Fils, Paris (1822)Google Scholar
  4. 4.
    Einstein, A.: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik. 17(8), 549–560 (1905). doi: 10.1002/andp.19053220806. English translation: “Investigations on the theory of Brownian Movement”. Translated by Cowper A.D.
  5. 5.
    Wick, G.C.: Properties of Bethe-Salpeter wave functions. Phys. Rev. 96, 1124–1134 (1954)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Nelson, E.: Dynamical theories of Brownian motion, 2nd edn., pp. 105ff (2001). Princeton University Press, Princeton (1967)Google Scholar
  7. 7.
    Guerra, F.: Structural aspects of stochastic mechanics and stochastic field theory. Phys. Rep. 77, 263–312 (1981)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Grabert, H., Hänggi, P., Talkner, P.: Is quantum mechanics equivalent to a classical stochastic process? Phys. Rev. A 19, 2440–2445 (1979)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Štoviček, P., Tolar, J.: Quantum mechanics in a discrete space-time. Rep. Math. Phys. 20, 157–170 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Celeghini, E., De Martino, S., De Siena, S., Rasetti, M., Vitiello, G.: Quantum groups, coherent states, squeezing and lattice quantum mechanics. Ann. Phys, 241, 50–67 (1995)CrossRefzbMATHGoogle Scholar
  11. 11.
    Frasca, M.: Quantum mechanics is the square root of a stochastic process. arXiv:1201.5091v2 [math-ph]. (2012)Google Scholar
  12. 12.
    Papoulis, A., Pillai, S.U.: Probability, random variables and stochastic processes, 4th edn. McGraw Hill, NY (2002)Google Scholar
  13. 13.
    Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89 (1943)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Zhang, Wei-Min, Feng, Hsuan, Gilmore, R.: Coherent states: theory and some applications. Rev. Mod. Phys. 62, 867–927 (1990)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Schiff, L.I.: Quantum mechanics. McGraw-Hill, NY (1949)Google Scholar
  16. 16.
    Nahin, P.J.: Chases and escapes—the mathematics of pursuit and evasion, pp. 7–14 and 23–27. Princeton University Press, Princeton (2007)Google Scholar
  17. 17.
  18. 18.
    Feynman, R.P., Hibbs, A.R.: Quantum mechanics and path integrals. McGraw-Hill, NY (1965)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.Selex ESRomeItaly
  2. 2.MBDARomeItaly

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