Heisenberg algebra, umbral calculus and orthogonal polynomials

Journal Article · Thu May 15 00:00:00 EDT 2008 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.2909731· OSTI ID:21100290

Dattoli, G ^[1]; Levi, D ^[2]; Winternitz, P ^[3]

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1. ENEA, Dipartimento Fim, Cre Frascati, C.P. 65, Frascati, Rome 000044 (Italy)
2. Dipartimento di Ingegneria Elettronica, Universita Degli Studi Roma Tre and Sezione INFN Roma Tre, Via della Vasca Navale 84, 00146 Roma (Italy)
3. Centre de Recherches Mathematiques and Department de Mathematiques et de Statistiques, Universite de Montreal, C.P. 6128 Centre Ville, Montreal, Quebec H3C 3J7 (Canada)

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

• OSTI ID: 21100290
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 5 Vol. 49; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
COORDINATES
DIFFERENTIAL EQUATIONS
INTEGRALS
INTEGRO-DIFFERENTIAL EQUATIONS
LAGUERRE POLYNOMIALS
LASERS
QUANTUM MECHANICS
QUANTUM OPERATORS