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Symplectic Group, Ladder Operators, and the Hagedorn Wave Packets

Tomoki Ohsawa

Published Oct 4, 2015• 0 citations
OpenAlex

Abstract

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.

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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 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.
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.
We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian/symplectic point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostant’s coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics. The Hamiltonian formulation naturally gives rise to corrections to the potential terms in the dynamics of both the wave packet and the Wigner function, thereby resulting in slightly different sets of equations from the conventional classical ones. We numerically investigate the effect of the correction term and demonstrate that it improves the accuracy of the dynamics as an approximation to the dynamics of expectation values of observables.
We complete our study on the uncertainties in position and momentum associated with the semiclassical Hagedorn wave packets by first filling in a technical gap in the dynamics of bound states for isochronous potentials. We then consider scattered states and show that, if the packet is reflected from the potential or transmitted through a symmetric potential, then a minimal uncertainty “initial” state cannot in general lead to a “final” state with minimal uncertainty, and we give an explicit relationship for the difference in terms of a characteristic time associated with classical trajectories. We also characterize the behavior of the uncertainty product in the case where the underlying classical dynamics lead to capture by the potential.
We develop a systematic analysis of the metaplectic semigroup Mp+(d,C)\mathrm{Mp}_+(d,\mathbb{C})Mp+​(d,C) associated with positive complex symplectic matrices, a notion introduced almost simultaneously and independently by Hörmander, Brunet, Kramer, and Howe, thereby extending the classical metaplectic theory beyond the unitary setting. While the existing literature has largely focused on propagators of quadratic evolution equations, for which results are typically obtained via Mehler formulas, our approach is operator-theoretic and symplectic in spirit and adapts techniques from the standard metaplectic group Mp(d,R)\mathrm{Mp}(d,\mathbb{R})Mp(d,R) to a substantially broader framework that is not driven by differential problems or particular propagators. This point of view provides deeper insight into the structure of the metaplectic semigroup, and allows us to investigate its generators, polar decomposition, and intertwining relations with complex conjugation and with the Wigner distribution. We then exploit these structural results to characterize, from a metaplectic perspective, classes of time-frequency representations satisfying prescribed structural properties. Finally, we discuss further implications for parabolic equations with complex quadratic Hamiltonians, we study the boundedness of their propagators on modulation spaces, we obtain estimates in time of their operator norms. Finally, we apply our theory to the study of propagation of Wigner singularities.