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An invariant class of Hermite type multivariate polynomials for the Wigner transform

Helge Dietert, Johannes Keller, Stephanie Troppmann

Published May 22, 2015• 6 citations
OpenAlex

Abstract

Hagedorn wave packets appear as the eigenstates of multidimensional harmonic oscillators, and are given by a polynomial times a Gaussian. We recognise the arising polynomials for normalised and unnormalised Hagedorn wave packets as generalised multivariate Hermite polynomials, which are not tensor products in general. We provide formulas for generating functions and ladder operators, and prove a direct connection to the Laguerre polynomials. As our main result, we show that the class of Hagedorn wave packets is invariant under the Wigner transform, and present an explanation for the product structure of normalised Hagedorn wave packets in phase space, which has recently been discovered.

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Integral-free Wigner functions

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Wigner phase space quasi-probability distribution function is a Fourier transform related to a given quantum mechanical wave function. It is shown that for the wave functions of type $ψ(q)=e^{-aq^2}ϕ(q)$, the Wigner function can be defined in terms of differential operators acting on a given function, independently from the integral formula which appears in the standard definition. Gaussian wave packet, harmonic and singular oscillators are given as the examples.

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Hagedorn wave packets appear as the eigenstates of multidimensional harmonic oscillators, and are given by a polynomial times a Gaussian. We recognise the arising polynomials for normalised and unnormalised Hagedorn wave packets as generalised multivariate Hermite polynomials, which are not tensor products in general. We provide formulas for generating functions and ladder operators, and prove a direct connection to the Laguerre polynomials. As our main result, we show that the class of Hagedorn wave packets is invariant under the Wigner transform, and present an explanation for the product structure of normalised Hagedorn wave packets in phase space, which has recently been discovered.
We develop an alternative view of the semiclassical wave packets of Hagedorn---often called the Hagedorn wave packets---stressing the roles of the symplectic and metaplectic groups along with the Heisenberg--Weyl group. Our point of view clarifies the relationship between the Hagedorn wave packets and the Hermite functions by building a bridge between the ladders of wave functions in both theories. This Hagedorn--Hermite correspondence provides an elegant view as well as simple proofs of some essential results on the Hagedorn wave packets. We build the theory starting from fundamental properties of ladder operators. Particularly, we show that the ladder operators of Hagedorn are a natural set obtained from the position and momentum operators using the symplectic group. The idea that pervades our view of the Hagedorn wave packets is so-called symplectic covariance; it generalizes some of fundamental results concerning the Hagedorn wave packets as well as simplifies their proofs. We apply our formulation to show the existence of minimal uncertainty products for the Hagedorn wave packets; it generalizes the one-dimensional result by Hagedorn to multi-dimensions. The Hagedorn--Hermite correspondence also leads to an alternative derivation of the generating function for the Hagedorn wave packets and clarifies its relationship with the generating function for the Hermite functions. This result, in turn, reveals the relationship between the Hagedorn polynomials and the Hermite polynomials.
Hagedorn wave packets form an orthonormal basis of the L2-functions and are known solutions of the semiclassical Schrödinger equation with quadratic Hamiltonians. In this thesis we analyse their structure, in particular the polynomial part, their connection to the Hermite functions and their representation in phase space. Based on these findings we utilise them to investigate the time evolution generated by the Schrödinger equation with a non-Hermitian, quadratic Hamiltonian. Operators of this type appear for example in the context of diffusion models and absorbing potentials and are thus of interest in physics and chemistry. We provide explicit formulas for the propagated wave packets and thereby show that the non-unitary evolution activates lower excited states. At last we apply our results to the Lindblad master equation with quadratic internal Hamiltonian and linear Lindblad terms.
In this paper, we show that the sum rules for generalized Hermite polynomials derived by Daboul and Mizrahi (2005 J. Phys. A: Math. Gen. 38 427-48) and by Graczyk and Nowak (2004 C. R. Acad. Sci., Ser. 1 338 849) can be interpreted and easily recovered using a probabilistic moment representation of these polynomials. The covariance property of the raising operator of the harmonic oscillator, which is at the origin of the identities proved in Daboul and Mizrahi and the dimension reduction effect expressed in the main result of Graczyk and Nowak are both interpreted in terms of the rotational invariance of the Gaussian distributions. As an application of these results, we uncover a probabilistic moment interpretation of two classical integrals of the Wigner function that involve the associated Laguerre polynomials.
Recent developments in laser physics have called renewed attention to Laguerre–Gaussian (LG) beams of paraxial light. In this paper we consider the corresponding LG modes for the two-dimensional harmonic oscillator, which appear in the transversal plane at the laser beam's waist. We see how they arise as Wigner transforms of Hermite–Gaussian modes, and we proceed to find a closed form for their own Wigner transforms, providing an alternative to the methods of Simon and Agarwal. Our main observation is that the Wigner transform intertwines the creation and annihilation operators for the two classes of modes.
Wigner phase space quasi-probability distribution function is a Fourier transform related to a given quantum mechanical wave function. It is shown that for the wave functions of type ψ(q)=e−aq2ϕ(q)ψ(q)=e^{-aq^2}ϕ(q)ψ(q)=e−aq2ϕ(q), the Wigner function can be defined in terms of differential operators acting on a given function, independently from the integral formula which appears in the standard definition. Gaussian wave packet, harmonic and singular oscillators are given as the examples.