Heisenberg Algebra, Umbral Calculus and Orthogonal Polynomials

Details Heisenberg Algebra, Umbral Calculus and Orthogonal Polynomials Heisenberg Algebra, Umbral Calculus and Orthogonal Polynomials

2007-12-18

arxiv.org/abs/0712.2957v1

Physics

Mathematical Physics

19 pages, 5 figures

Scientific paper

10.1063/1.2909731

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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