Hagedorn Wavepackets in Time-Frequency and Phase Space

Lasser, Caroline; Troppmann, Stephanie

doi:10.1007/s00041-014-9330-9

Cited by 15 publications

(26 citation statements)

References 20 publications

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“…This link has been generalised in [18], showing that the Wigner functions of a Hagedorn wave packet factorises regardless of the structure of the wave packet itself. Another approach using ladder operators in the two-dimensional setting can be found in [4].…”

Section: Introductionmentioning

confidence: 87%

An invariant class of wave packets for the Wigner transform

Dietert

Keller

Troppmann

2017

Journal of Mathematical Analysis and Applications

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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.

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Section: Introductionmentioning

confidence: 87%

An invariant class of wave packets for the Wigner transform

Dietert

Keller

Troppmann

2017

Journal of Mathematical Analysis and Applications

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“…The initial inspiration for the work on the ladder operators for the Hagedorn wave packets came from Stephanie Troppmann's talk at the Summer School on Mathematical and Computational Methods in Quantum Dynamics at University of Wisconsin-Madison during the Summer 2013. I also thank Caroline Lasser for the helpful discussion during the workshop "Mathematical and Numerical Methods for Complex Quantum Systems" in March 2014 at the University of Illinois at Chicago regarding their work [15]. I would also like to take this opportunity to thank Shi Jin and Christof Sparber, the organizers of the summer school and workshop, for the generous travel support through the NSF Research Network in Mathematical Sciences "KI-Net."…”

Section: The Generating Function For the Hagedorn Wave Packetsmentioning

confidence: 99%

The Hagedorn–Hermite Correspondence

Ohsawa

2018

J Fourier Anal Appl

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We investigate the relationship between the semiclassical wave packets of Hagedorn and the Hermite functions by establishing a relationship between their ladder operators. This Hagedorn-Hermite correspondence provides a unified view as well as simple proofs of some essential results on the Hagedorn wave packets. Particularly, we show that Hagedorn's ladder operators are a natural set of ladder operators obtained from the position and momentum operators using the symplectic group. This construction reveals an algebraic structure of the Hagedorn wave packets, and explains the relative simplicity of Hagedorn's parametrization compared to the rather intricate construction of the generalized squeezed states. We apply our formulation to show the existence of minimal uncertainty products for the Hagedorn wave packets, generalizing Hagedorn's one-dimensional result to multi-dimensions. The Hagedorn-Hermite correspondence also leads to an alternative derivation of the generating function for the Hagedorn wave packets based on 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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“…as a multiplicative decomposition thereof. For the corresponding explicit formulae for Hermite and Hagedorn functions see [LT14].…”

Section: Phase Space Discretisationmentioning

confidence: 99%

Discretising the Herman–Kluk propagator

Lasser

Sattlegger

2017

Numer. Math.

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The Herman-Kluk propagator is a popular semi-classical approximation of the unitary evolution operator in quantum molecular dynamics. In this paper we formulate the Herman-Kluk propagator as a phase space integral and discretise it by Monte Carlo and quasi-Monte Carlo quadrature. Then, we investigate the accuracy of a symplectic time discretisation by combining backward error analysis with Fourier integral operator calculus. Numerical experiments for two-and six-dimensional model systems support our theoretical results. arXiv:1605.00588v1 [math.NA] 2 May 2016It is parametrised by a phase space point z = (q, p) ∈ R 2d . Gaussian wave packets enjoy the striking property that any square integrable function ψ ∈ L 2 (R d ) can be decomposed according toThe precise meaning of the integral is given by the inversion formula of the Fourier-Bros-Iagolnitzer (FBI) transform. From this we get the formal equationso that lim M →∞ sup z∈R 2d |D ε M (z 1 , . . . , z M ; z)| = 0 implies (34). and for the integralwhere the last equation also uses Fubini's theorem.

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Hagedorn Wavepackets in Time-Frequency and Phase Space

Cited by 15 publications

References 20 publications

An invariant class of wave packets for the Wigner transform

An invariant class of wave packets for the Wigner transform

The Hagedorn–Hermite Correspondence

Discretising the Herman–Kluk propagator

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