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

@article{Farina2014SolvingSE,
  title={Solving Schr{\"o}dinger equation via Tartaglia/Pascal triangle: a possible link between stochastic processing and quantum mechanics},
  author={Alfonso Farina and Marco Frasca and M. Sedehi},
  journal={Signal, Image and Video Processing},
  year={2014},
  volume={8},
  pages={27-37}
}
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… 
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References

SHOWING 1-10 OF 25 REFERENCES
Tartaglia-Pascal’s triangle: a historical perspective with applications
The aim of this paper is to provide a historical perspective of Tartaglia-Pascal’s triangle with its relations to physics, finance, and statistical signal processing. We start by introducing
Quantum groups, coherent states, squeezing and lattice quantum mechanics
By resorting to the Fock--Bargmann representation, we incorporate the quantum Weyl--Heisenberg algebra, $q$-WH, into the theory of entire analytic functions. The $q$--WH algebra operators are
Properties of Bethe-Salpeter Wave Functions
A boundary condition at t = ± ∞ (t being the “relative” time variable) is obtained for the four-dimensional wave function of a two-body system in a bound state. It is shown that this condition
Quantum Groups
This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions
Quantum mechanics is the square root of a stochastic process
We prove a theorem showing that quantum mechanics is not directly a stochastic process characterizing Brownian motion but rather its square root. This implies that a complex-valued stochastic process
Is quantum mechanics equivalent to a classical stochastic process
The authors analyze the connection between the theory of stochastic processes and quantum mechanics. It is shown that quantum mechanics is not equivalent to a Markovian diffusion process as claimed
Coherent states: Theory and some Applications
In this review, a general algorithm for constructing coherent states of dynamical groups for a given quantum physical system is presented. The result is that, for a given dynamical group, the
Quantum Groups
Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups
Dynamical Theories of Brownian Motion
These notes are based on a course of lectures given by Professor Nelson at Princeton during the spring term of 1966. The subject of Brownian motion has long been of interest in mathematical
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