We consider a classically chaotic system that is described by a Hamiltonian H(Q,P;x), where x is a constant parameter. Specifically, we discuss a gas particle inside a cavity, where x controls a deformation of the boundary or the position of a "piston." The quantum eigenstates of the system are |n(x)>. We describe how the parametric kernel P(nmid R:m) = |<n(x)mid R:m(x(0))>|(2) evolves as a function of deltax = x-x(0). We explore both the perturbative and the nonperturbative regimes, and discuss the capabilities and the limitations of semiclassical as well as random waves and random-matrix-theory considerations.
Heller, E.J.Semiclassical wave packets. The Physics and Chemistry of Wave Packets; Yeazel, J.; Uzer, T. eds, Wiley: New York (1999).
Quantum and classical quasiresonant vibration-rotation energy transfer is investigated for ultracold He−H2 collisions. Classical trajectory computations show that extremely strong correlations between Δj and Δv persist at low energies, though the changes themselves are less than one quantum. Quantum computations show that quasiresonant transitions occur in the limit of zero collision energy but that threshold effects become important and that some quasiresonant channels close. The qualitative similarity between classical and quantum results suggests that they share a common mechanism.
A time domain approach employing the semiclassical approximation to the quantum mechanical propagator, as applied to Gaussian wavepackets, is used to study the barrier penetration problem. We have observed that qualitative agreement with the exact quantum calculations for the correlation function and the transmission probability can be achieved by considering only classically allowed trajectories. The results slowly tend to the classical step function at the barrier top as a function of the wavepacket center parameter, however. We suggest that a full semiclassical calculation, including nonclassical trajectories with complex energies, would improve the results.
The semiclassical approximation of Feynman’s path integral is used to calculate the S matrix for the positron-impact ionization of hydrogen. The formulation provides a full scattering amplitude, and more importantly does not require knowledge of the asymptotic three-body Coulomb state in the continuum. In the limit of vanishing excess energy, the results confirm Wannier’s classical model for fragmentation [Phys. Rev. 90, 817 (1953)]. The experimentally observable ratio of fragmentation versus total ionization (including positronium formation) is predicted.
The semiclassical techniques developed in the previous paper are applied to the understanding of the hierarchical structure underlying the spectra. This organization, as analyzed by Davis with statistical models, is revealed by continuously changing the energy resolution of the spectra and noting the branching pattern of the peaks. We argue that the greater part of this hierarchical organization can be understood with classical events in the time domain.
A semiclassical method for the propagation of arbitrary wave packets in a multidimensional Hamiltonian is presented. The method is shown to be valid for treating Hamiltonian systems whose classical phase space is a combination of chaotic and quasiperiodic motion (mixed dynamics). The propagation can be carried out long enough for the nonlinearities of the system to be important. The nonlinear dynamics is reflected in spectra and correlation functions. We suggest this new semiclassical method can be a tool for analyzing the nonlinear aspects of the vibrational spectra.
In a recent Letter [Phys. Rev. Lett. 67, 664 (1991)] we found semiclassical propagation to be remarkably accurate in the chaotic stadium billiard long after classical fine structure had developed on a scale much smaller than ħ. We give a complete account of that work and derive an approximate time scale for the validity of the semiclassical approximation as a function of ħ.
It has been a long-standing problem to understand the eigenfunctions of a system whose classical analog is strongly chaotic. We show that in some cases the eigenfunctions can be constructed by purely semiclassical calculations.
We review some of the issues facing semiclassical methods in classically chaotic systems, then demonstrate the long-time accuracy of semiclassical propagation of a nonstationary wave packet using the quantum baker's map of Balazs and Voros. We show why some of the standard arguments against the efficacy of semiclassical dynamics for long-time chaotic motion are incorrect.
We investigate the behavior of the quantum baker's transformation, a system whose classical analogue is completely chaotic, for time scales where the classical mechanics generates phase space structures on a scale smaller than Planck's constant (i.e., past the log time t∗ ≈ ln ħ-1). Surprisingly, we find that a semiclassical theory can accurately reproduce many features of the quantum evolution of a wave packet in this strongly mixing time regime.
Outlines two methods that have been developed for performing the evaluation and demonstrate the quantitative agreement between the quantum and semiclassical dynamics for systems with nonintegrable underlying dynamics. Method for calculating the semiclassical and semiclassical propagation of initial states and correlation functions for nonlinear and chaotic dynamics.
A new semiclassical approach that constructs the full semiclassical Green’s function propagation of any initial wave function directly from an ensemble of real trajectories, without root searching, is presented. Each trajectory controls a cell of initial conditions in phase space, but the cell area is not constrained by Planck’s constant. The method is shown to be accurate for rather long times in anharmonic oscillators, indicating the semiclassical time‐dependent Green’s function is clearly worthy of more study. The evolution of wave functions in anharmonic potentials is examined and a spectrum from the semiclassical correlation function is calculated, comparing with exact fast Fourier transform results.
Chaos introduces essential complications into semiclassical mechanics and the conventional wisdom maintains that the semiclassical time-dependent Green’s function fails to describe the quantum dynamics once the underlying chaos has had time to develop much finer structure than a quantum cell (h). We develop a method to evaluate the semiclassical approximation and test it for the first time under these circumstances. The comparison of the quantum and semiclassical dynamics of the stadium billiard shows remarkable agreement despite the very intricate underlying classical dynamics.
A new semiclassical propagator based on a local expansion of the potential up the wavepacket is proposed. Formulas for the propagator are derived and the implementation using grid and fast Fourier transform (FFT) methods is discussed. The semiclassical propagator can be improved up to the exact quantum mechanical limit by including anharmonic corrections using a split operator approach. Preliminary applications to the CH3I photodissociation problem show the applicability and accuracy of the proposed method.
The effect of Born–Oppenheimer potential energy surface crossings on energy transfer in polyatomic molecules is investigated, classically and quantum mechanically. The hopping from one energy surface to another is enough to cause classical chaos, and strong mixing of the levels quantum mechanically. The rate of classical mixing determines the extent of quantum mixing, even though classical mixing is complete at long times.
We recently published a new method for the calculation of the time evolution of a wave function. We used an accurate approximate method to calculate the time propagator for a finite time Δt. Numerical calculations showed that this scheme works quite accurately, but that it is not more efficient than conventional methods. In this paper we propose to use a very fast and simple, but less accurate semiclassical method for the calculation of the time propagator. The approximation consists in the replacement of the Hamiltonian by a quadratic approximation around the center of the evolving wave packet called thawed Gaussian dynamics. We show by numerical examples in one and two dimensions that, despite this crude approximation, we achieve nearly the same accuracy as in the foregoing paper, but with an efficiency that is typically more than an order of magnitude better. We further show that the method is able to describe tunneling and long time dynamics (e.g., 1000 vibrational periods).
We present an approach to quantum dynamics, based entirely on Cartesian coordinates, which covers vibrational as well as rotational motion. The initial state is represented in terms of multidimensional Gaussian wave packets. Rotational adaptation to angular momentum eigenstates is done by using angular momentum projection operators. This gives an initial state represented as a weighted superposition of Gaussians with different average orientation in space. It is shown that the subsequent dynamics can be determined from the dynamics of Gaussians corresponding to just one of these orientations. An application to the 3Dphotodissociationdynamics of ICN is presented. All six degrees of freedom which describe the internal motion of the triatomic are included, the only approximation introduced in the present calculation being the thawed Gaussian wave packet approximation for the dynamics. The total absorptionspectrum out of vibrational–rotational eigenstates of ICN as well as fully resolved final product distributions are calculated.