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Journal of Physics A Mathematical and General

Generating function for the product of the associated Laguerre and Hermite polynomials

Istok P. Mendaš

Published Feb 7, 1993• 7 citations
Publisher

Abstract

The generating function for the product of the associated Laguerre and Hermite polynomials is formulated and used to determine the coordinate-space wave function of the displaced number states of the harmonic oscillator for the general case of a complex displacement parameter. The benefits, in this context, of an extended form of the associated Laguerre polynomials are advertised.

DOI
10.1088/0305-4470/26/3/005

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The generating function for the product of the associated Laguerre and Hermite polynomials is formulated and used to determine the coordinate-space wave function of the displaced number states of the harmonic oscillator for the general case of a complex displacement parameter. The benefits, in this context, of an extended form of the associated Laguerre polynomials are advertised.
One dimensional quantum harmonic oscillator is well studied in elementary textbooks of quantum mechanics. The wave function of one-dimensional oscillator harmonic can be written in term of Hermite polynomial. Due to the symmetry of the spring energy, the wave functions of two-dimensional and three-dimensional harmonic oscillators can be written as products of the one-dimensional case. Because of that, the wave functions of two- and three-dimensional cases are focused on cartesian coordinates. In this article, we utilize polar and spherical coordinates to describe the wave function of two- and three-dimensional harmonic oscillators, respectively. The radial part of the wave functions can be written in term of associated Laguerre polynomials.
In this paper, we derive two new generating functions of Laguerre-polynomials, which look like the negative binomial theorem for the Laguerre function Lnx, by adopting the bi-partite entangled state representation and the operator-Hermite-polynomial HnX method, where X is the coordinate operator. Their application in investigating the density operator of photon loss of chaotic light is presented, and a new partition of Fock space in terms of the density operator of the negative binomial field is obtained. We also point out that the power-series definition of the Laguerre-polynomial originates from the Hermite polynomial. As a by-product, we derive a new Laguerre-polynomials-operator generating formula by virtue of the method of integration within an ordered product of operators.
The angular momentum states may be realized by the wave functions of two dimensional harmonic oscillators in polar coordinates. Using the generating function of these wave functions, Schwinger's generating function of Clebsh-Gordan in Fock space and Gauss integral, we determine the generating function of the couplings function of two angular momentums in coordinates representations. We deduce from the last function the connection of the wave function of the hydrogen atom and the four dimensional harmonic oscillator. We also find the expression of the relation between the spherical and parabolic wave functions of hydrogen atom. In addition, we find a new bilinear generating function involving the products of Laguerre polynomials.
Hermite polynomials with many variables and many indices play a crucial role within the framework of phase-space formulation of classical or quantum mechanics. A generalized operational formalism useful to handle these polynomials is discussed and also a new set of creation–annihilation operators associated to the phase-space harmonic oscillator orthogonal functions is introduced. The Fourier transform of these orthogonal functions is studied by discussing the analogies with the ordinary case. Finally, the integral representation of the generalized Hermite polynomials and a simple technique to deal with the generalized heat equation are discussed.