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arXiv · 0712.2957

Heisenberg Algebra, Umbral Calculus and Orthogonal Polynomials

G. Dattoli·D. Levi·P. Winternitz
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Abstract

Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation $[\hat P,\hat M]=1$. In ordinary quantum mechanics $\hat P$ is the derivative and $\hat M$ the coordinate operator. Here we shall realize $\hat P$ as a second order differential operator and $\hat M$ as a first order integral one. We show that this makes it possible to solve large classes of differential and integro-differential equations and to introduce new classes of orthogonal polynomials, related to Laguerre polynomials. These polynomials are particularly well suited for describing so called flatenned beams in laser theory

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G. Dattoli, D. Levi, P. Winternitz. 2007-12-18. Heisenberg Algebra, Umbral Calculus and Orthogonal Polynomials. https://doi.org/10.1063/1.2909731

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