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arXiv:1505.06192 · Journal of Mathematical Analysis and Applications

An invariant class of wave packets for the Wigner transform

Helge Dietert, Johannes Keller, Stephanie Troppmann

Published Dec 28, 2016• 15 citations
arXivHTMLPDF
DOI
10.1016/j.jmaa.2016.12.041

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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.
This chapter presents some of the features of the phase space picture. For quantum mechanical systems Eugene Paul Wigner introduced a quantum analogue of a phase space probability distribution called Wigner distribution function or phase space picture. Using the Wigner distribution function it is possible to perform a canonical transformation in quantum mechanics. The phase space picture is a useful candidate to illustrate the transition from classical to quantum mechanics. The Poisson brackets and canonical transformations are part of the effective formulation of classical mechanics. In the infinite height potential video sequences of the evolution of Wigner functions for a wave function of a wave packet and its mirror wave packet is constructed. The chapter also presents the application of the canonical transformations and phase space distribution function in the study of wave packet spread and coherent and squeezed states.