Physical Review Letters

Semiclassical dynamics of chaotic motion: Unexpected long-time accuracy

Steven Tomsovic, Eric J. Heller

Published Aug 5, 1991• 227 citations

Abstract

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.

DOI: 10.1103/physrevlett.67.664

Related papers

Classical-quantum correspondence for chaotic motion

Eric J. Heller

We examine semiclassical propagation of nonstationary states. The fundamental object is the 1928 time dependent Van Vleck semiclassical Green`s function, as modified by Gutzwiller in 1967. This direct approach to semiclassical quantum mechanics has received very little attention over the years. The problems of evaluating the Green`s function (finding all trajectories which go from x{prime} to x in time t, for every x, x{prime}, and t sounds foreboding) and the development of caustics at an alarming rate in chaotic systems, may have suppressed its use. We have developed several strategies for evaluating it efficiently, including for chaotic dynamics. We found it can be used to accurately propagate smooth wave functions in spite of the caustics. The smooth wave function has the effect of strongly damping the caustics and leaving an accurate propagated state. Moreover many of the contributions to the semiclassical amplitude are legal, and allow a much longer time scale for breakdown of the semiclassical propagation than had previously been thought. This has allowed us to extract accurate spectra and eigenstates for chaotic dynamics in the stadium billiard and to calculate a complete molecular electronic absorption spectrum for two nonlinearly interacting vibrational modes, including line intensities and positions, usingmore » only classical trajectory input.« less

Bulletin of the American Physical SocietyMay 1, 19930 citations

Quantum versus chaos : questions emerging from mesoscopic cosmos

勝弘中村

Preface. 1. Genesis of Chaos and Breakdown of Quantization of Adiabatic Invariants. 2. Semiclassical Quantization of Chaos: Trace Formula. 3. Pseudo-Chaos Without Classical Counterpart in 1-Dimensional Quantum Transport. 4. Chaos and Quantum Transport in Open Magnetic Billiards: From Stadium to Sinai Billiards. 5. Chaotic Scattering on Hyperbolic Billiards: Success of Semiclassical Theory. 6. Nonadiabaticity-Induced Quantum Chaos. 7. Level Dynamics and Statistical Mechanics. 8. Towards Time-Discrete Quantum Mechanics. 9. Conclusions and Prospects. Index.

Kluwer Academic Publishers eBooksJan 1, 19971 citation

Quantum Chaos and Semiclassical Mechanics

Robert W. Batterman

This paper discusses the problem of finding and defining chaos in quantum mechanics. While chaotic time evolution appears to be ubiquitous in classical mechanics, it is apparently absent in quantum mechanics in part because for a bound, isolated quantum system, the evolution of its state is multiply periodic. This has led a number of investigators to search for semiclassical signatures of chaos. Here I am concerned with the status of semiclassical mechanics as a distinct third theory of the asymptotic domain between classical and quantum mechanics. I discuss in some detail the meaning of such crucial locutions as the "classical counterpart to a quantum system" and a quantum system's "underlying classical motion". A proper elucidation of these concepts requires a semiclassical association between phase space surfaces and wave-functions. This significance of this association is discussed in some detail.

PSA Proceedings of the Biennial Meeting of the Philosophy of Science AssociationJan 1, 199211 citations

Semiclassical approximation:quantization of chaotic motion

Zhu Hong

The Bohr-Sommerfield quantized rule of the old quantum theory has failed in chaotic motion. New semiclassical appoximation methods are developed to approach the quantization of chaotic motion in three domains: eigenvalue ,eigenstate and time evolution of the state.The term“post-modern quantum mechanics” is used to describe the new interest in semiclassical appoximation.

College PhysicsJan 1, 20000 citations

Exploring Quantum, Classical and Semi-Classical Chaos in the Stadium Billiard

Chris C. King

This paper explores quantum and classical chaos in the stadium billiard using Matlab simulations to investigate the behavior of wave functions in the stadium and the corresponding classical orbits believed to underlie wave function scarring. The simulations use three complementary methods. The quantum wave functions are modeled using a cellular automaton simulating a Hamiltonian wave function with discrete (square pixel) boundary conditions approaching the stadium in the classical limit. The classical orbits are computed by solving the reflection equations at the classical boundary thus giving direct insights into the wave functions and eigenstates of the quantum stadium. Finally, a simplified semi-classical algorithm is developed to show the comparison between this and the quantum wave function method. Quanta 2014; 3: 16–31.

QuantaJan 14, 20145 citations

Phase Space Structures in Chaotic Quantum Dynamics.

B. S. Helmkamp

I compare the quantum and classical dynamics of a particle moving in a cosine potential while subject to a time-dependent force. I concentrate here on the behavior of an initially well-localized wave packet at times before the classically chaotic motion is fully developed. I find that the quantum and classical dynamics are indistinguishable well beyond the Ehrenfest time where the wave packet delocalizes. The quantum and classical descriptions first differ precisely when the classical probability density is folded in the vicinity of a hyperbolic fixed point. At this point, the wave function acquires a nodal structure which I show to be the result of a simple beating phenomenon between paths in the semiclassical propagator. When the interaction of the classical manifold with the hyperbolic fixed point leads to escape from the remnant separatrix, representing the onset of classical chaos, the interference associated with the tendril leads to a long-lived accumulation of quantum amplitude on top of the cosine barrier. This effect also has a semiclassical interpretation, meaning that it arises from the interference between classical paths, but one must expand to second order in Planck's constant to describe the behavior correctly. Both the nodal structure and this pinning of amplitude on the barrier are dynamic mechanisms for the quantum inhibition of mixing. I then couple the system to a bath of harmonic oscillators in order to study the effect of the environment on these mechanisms. Although an oscillator bath environment brings dissipation as well as noise to the problem, the noise effect dominates for the high temperature, weak coupling regime I study. When there is sufficient noise to render the interfering classical paths indistinguishable, I find that the quantum interference gets erased. This dephasing occurs at very early times, long before there is appreciable dissipation of energy to the environment. Consequently, one can argue that the presence of an environment, even if its effect would be negligible in a nonchaotic setting, allows for the possibility of quantum mixing.

Civil War Book ReviewJan 1, 19950 citations