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

Original Paper


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.


Stochastic processing Quantum mechanics Schrödinger equation Tartaglia-Pascal triangle 


  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)MATHGoogle 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)CrossRefMATHMathSciNetGoogle 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)CrossRefMATHMathSciNetGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHMathSciNetGoogle 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)MATHGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.Selex ESRomeItaly
  2. 2.MBDARomeItaly

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