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