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An invariant class of wave packets for the Wigner transform

Helge Dietert, Johannes Keller, Stephanie Troppmann

TL;DR

This work proves that the Wigner transform preserves the Hagedorn wave-packet structure under a phase-space lifting to doubled dimension: the Wigner function $\mathcal W^{\varepsilon}_{k,\ell}[Z,Y](z)$ is itself a Hagedorn packet $\Phi^{\varepsilon}_{(k,\ell)}[{\mathcal Z},{\mathcal Y}](z)$ in phase space. Central to the result is a phase-space lift of Lagrangian frames and a detailed analysis of the polynomial prefactors $q^M_k$, including a novel Laguerre connection that expresses these polynomials in terms of reduced matrices and Laguerre polynomials. The paper develops a generating function and ladder-operator framework for $q^M_k$, clarifying when tensor factorisation occurs (equivalently, when $M$ is block-diagonal) and revealing a Laguerre-type structure in the general case. The theory yields a precise phase-space interpretation of the tensor-product structure observed for Hagedorn wave packets, with explicit formulas for the phase-space Wigner functions and illustrative 2D examples. These results have potential applications in semiclassical approximations, quantum dynamics simulations, and phase-space methods by providing robust, structured representations of Wigner functions for a broad class of highly localised states.

Abstract

Generalised Hagedorn wave packets appear as exact solutions of Schrödinger equations with quadratic, possibly complex, potential, and are given by a polynomial times a Gaussian. We show that the Wigner transform of generalised Hagedorn wave packets is a wave packet of the same type in phase space. The proofs build on a parametrisation via Lagrangian frames and a detailed analysis of the polynomial prefactors, including a novel Laguerre connection. Our findings directly imply the recently found tensor product structure of the Wigner transform of Hagedorn wave packets.

An invariant class of wave packets for the Wigner transform

TL;DR

This work proves that the Wigner transform preserves the Hagedorn wave-packet structure under a phase-space lifting to doubled dimension: the Wigner function is itself a Hagedorn packet in phase space. Central to the result is a phase-space lift of Lagrangian frames and a detailed analysis of the polynomial prefactors , including a novel Laguerre connection that expresses these polynomials in terms of reduced matrices and Laguerre polynomials. The paper develops a generating function and ladder-operator framework for , clarifying when tensor factorisation occurs (equivalently, when is block-diagonal) and revealing a Laguerre-type structure in the general case. The theory yields a precise phase-space interpretation of the tensor-product structure observed for Hagedorn wave packets, with explicit formulas for the phase-space Wigner functions and illustrative 2D examples. These results have potential applications in semiclassical approximations, quantum dynamics simulations, and phase-space methods by providing robust, structured representations of Wigner functions for a broad class of highly localised states.

Abstract

Generalised Hagedorn wave packets appear as exact solutions of Schrödinger equations with quadratic, possibly complex, potential, and are given by a polynomial times a Gaussian. We show that the Wigner transform of generalised Hagedorn wave packets is a wave packet of the same type in phase space. The proofs build on a parametrisation via Lagrangian frames and a detailed analysis of the polynomial prefactors, including a novel Laguerre connection. Our findings directly imply the recently found tensor product structure of the Wigner transform of Hagedorn wave packets.

Paper Structure

This paper contains 16 sections, 9 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Generalised Hagedorn wave packets
  4. Lagrangian frames and ground states
  5. Excited states and spectral properties
  6. Analysis of the polynomial prefactor
  7. Generating function and raising operator
  8. Factorisation and eigenvectors
  9. Laguerre connection
  10. Hagedorn wave packets in phase space
  11. Examples
  12. Polynomial prefactor
  13. Hagedorn wave packets
  14. Acknowledgments
  15. Proof of Proposition \ref{['prop:polynomial_hagedorn_trafo']}
  16. ...and 1 more sections

Key Result

Lemma 2

Let $Z=(Q;P)$ be a normalised Lagrangian frame. Then, $Q$ and $P$ are invertible and In particular, for $\varepsilon>0$, is a square integrable function with $\|\varphi^{\varepsilon}_0[Z]\|_{L^2} = 1$.

Figures (3)

  • Figure 1: The nodal sets of three examplary two-dimensional polynomials, associated with the matrices $M^{(1)}$ (upper left), $M^{(2)}$ (upper right) and $M^{(3)}$ (lower). Regions with negative values are highlighted by grey coloring.
  • Figure 2: Intensity plot of the absolute value of the two examplary two-dimensional Hagedorn wave packets $\varphi_{4,6}^{0.1}[Z_1](x)$ (left) and $\varphi_{7,6}^{0.1}[Z_2](x)$ (right), $\varepsilon = 10^{-1}$. Darker colouring represents higher absolute values.
  • Figure 3: Intensity plot of the absolute value of the two-dimensional Hagedorn wave packets $\varphi_{6,5}^{\varepsilon}[Z_3](x)$ (left) and $\varphi_{3,7}^{\varepsilon}[Z_2,Z_3](x)$ (right), where $\varepsilon = 10^{-1}$.

Theorems & Definitions (19)

  • Definition 1
  • Lemma 2: Ground state
  • Definition 3: Generalised wave packets
  • Proposition 4
  • Lemma 5: Generating function
  • Lemma 6: Ladder operators
  • proof
  • Remark 7
  • Proposition 8: Laguerre connection
  • proof
  • ...and 9 more