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Wavepackets

Mark Sheard Child

Published Aug 21, 2014• 0 citations
Publisher

Abstract

Abstract A wavepacket spreading under quadratic potentials is shown to have an exact classical analogue in the spreading of a classical ensemble taken from the Wigner distribution. The wavepacket motion under anharmonic potentials is complicated by greater spreading combined with additional interferences. Both are shown to be well handled by Heller’s frozen Gaussian swarm technique. Applications are outlined to spectral quantization and Franck–Condon transitions by Fourier analysis of the time correlation function. The chapter concludes with an introduction to the popular unitary Herman–Kluk coherent state propagation scheme. Advantages over semiclassical Van Vleck propagation are discussed, and a Filinov–based scheme for handling weakly chaotic systems is described.

DOI
10.1093/acprof:oso/9780199672981.003.0008

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Abstract A wavepacket spreading under quadratic potentials is shown to have an exact classical analogue in the spreading of a classical ensemble taken from the Wigner distribution. The wavepacket motion under anharmonic potentials is complicated by greater spreading combined with additional interferences. Both are shown to be well handled by Heller’s frozen Gaussian swarm technique. Applications are outlined to spectral quantization and Franck–Condon transitions by Fourier analysis of the time correlation function. The chapter concludes with an introduction to the popular unitary Herman–Kluk coherent state propagation scheme. Advantages over semiclassical Van Vleck propagation are discussed, and a Filinov–based scheme for handling weakly chaotic systems is described.
An exact analysis of coupled coherent state (CCS) theory in the moving locally quadratic Hamiltonian approximation is shown to reproduce both the linearized coherent state matrix element of the Herman–Kluk propagator and the coherent state overlap with Heller’s thawed Gaussian wave function. The derivation is applicable to anharmonic as well as harmonic systems, because the quadratic approximation is taken to apply only in the vicinity of a particular classical trajectory. New compact expressions for the linearized Herman–Kluk coherent state matrix element are given, and improvements for the practical application of CCS theory are discussed.
The semiclassical Herman–Kluk initial value approximation of the quantum-mechanical propagator, which is heavily used in dynamical studies of atomic and molecular systems, takes into account nonlinearity of the underlying classical dynamics by using multiple classical initial value solutions. In applications to the propagation of Gaussian wavepackets non-Gaussian distortions in the course of time and therefore a realistic description of the quantum dynamics are possible. This is in contrast to the single-trajectory approximations investigated in a recent paper by Baranger et al , one of which was erroneously termed Herman–Kluk approximation.
We present a comprehensive study of semiclassical phase-space propagation in the Wigner representation, emphasizing numerical applications, in particular as an initial-value representation. Two semiclassical approximation schemes are discussed. The propagator of the Wigner function based on van Vleck's approximation replaces the Liouville propagator by a quantum spot with an oscillatory pattern reflecting the interference between pairs of classical trajectories. Employing phase-space path integration instead, caustics in the quantum spot are resolved in terms of Airy functions. We apply both to two benchmark models of nonlinear molecular potentials, the Morse oscillator and the quartic double well, to test them in standard tasks such as computing autocorrelation functions and propagating coherent states. The performance of semiclassical Wigner propagation is very good even in the presence of marked quantum effects, e.g., in coherent tunneling and in propagating Schrodinger cat states, and of classical chaos in four-dimensional phase space. We suggest options for an effective numerical implementation of our method and for integrating it in Monte-Carlo-Metropolis algorithms suitable for high-dimensional systems.
Abstract Semiclassical mechanics, which stems from the old quantum theory, has seen a remarkable revival in recent years as a physically intuitive and computationally accurate scheme for the interpretation of modern experiments. The main text concentrates less on the mathematical foundations than on the global influence of the classical phase space structures on the quantum mechanical observables. Further mathematical detail is contained in the appendices. Worked problem sets are included as an aid to the student. One important thread concerns the sharp separation between quantum interference in the classically accessible phase space and exponential decay into the inaccessible shadow regions. Another is the use of catastrophe theory to eliminate singularities at the classical boundaries between the two types of region. The theory goes on to show how different parts of the wavefunction may be stitched together to obtain accurate quantization formulae for a wide variety of tunnelling and curve-crossing problems. A careful account of classical angle–action transformations is extended to include the evaluation of matrix elements. A variety of extensions of the familiar spectroscopic RKR inversion procedure are given. It is also shown how bifurcations in the phase space structure signal changes in the quantum eigenvalue spectrum from, for example, normal mode to local mode behaviour. Other quantization techniques are shown to extend even to weakly chaotic systems. A chapter on Gaussian wavepacket propagation leads up to the currently popular Herman–Kluk technique. The book concludes with chapters on semiclassical aspects of elastic, inelastic and reactive scattering.
A simple and elegant semiclassical theory for propagating wavepackets is presented. The development of the theory avoids the explicit use of the Green-function formalism. It involves expanding the potential to a quadratic time-dependent form and the use of multiple reference trajectories. The classical analogue in phase space predicts the accuracy of the results. The theory is applied to the calculation of the autocorrelation function in a Coulomb field in one dimension. Remarkable agreement with an exact quantum calculation is obtained in many circumstances in spite of the Coulomb singularity. All the detailed quantum behaviour, including the long-time recurrences and the spectrum, is reproduced.
Exact eigenfunctions for a particle in a box are obtained using Gaussian wavepacket dynamics. The eigenfunctions are obtained by propagating, without approximation, an infinite set of Gaussian wavepackets that collectively satisfy the boundary conditions of the problem, being coherent states appropriate to this problem. The method of images is applied to enforce these boundary conditions. This technique may be applied to the quantum billiard problem whenever the particle is confined to any open or closed region that tessellates space, regardless of the dimension of the region. Also, it is shown that the use of frozen Gaussians along with the De Leon-Heller spectral quantisation method gives the exact solution for the one-dimensional problem as well as for the above multi-dimensional problems, provided the components of the momentum of the wavepackets are chosen at random.