Heiles System Research Articles

Local ergodicity as a probe for chaos in quantum systems: Application to the Henon–Heiles system

Classical chaos is usually accompanied by ‘‘local’’ ergodicity—ergodic behavior in regions of phase space that are generally smaller than, but of the same dimensionality as, the energy surface. If there are strong bottlenecks in phase space impeding the relaxation to statistical equilibrium in the full region of ergodicity, it is often possible to define an approximate form of ergodic behavior in a smaller subregion where relaxation occurs temporarily but quickly. Such ‘‘pseudoergodicity’’ is also a symptom of chaos. We use the presence of quantum behavior that mimics local ergodicity and pseudoergodicity as a probe for the influence of classical chaos on quantum dynamics. We show that the quantum analog of a pseudoergodic region in phase space is generally formed by superpositions of energy eigenstates. The superposition states spanning a given pseudoergodic zone have similar expectation values and small off-diagonal elements to other states with similar energy for a certain class of operators. We perform calculations which identify the ergodic regions for two versions of the quantum Henon–Heiles system. For the ‘‘more classical’’ of these systems we find good agreement between the proportion of quantum states in pseudoergodic zones and the proportion of classical phase space occupied by chaotic trajectories. We also find that the quantum pseudoergodic regions can be identified with classical vague tori of the precessing and librating types. Different ergodic regions are separated from one another by what appear to be the quantum analogs of the precessor–librator separatrix and other classical bottlenecks. For the ‘‘less classical’’ of the systems, we find that classical chaos is reflected in the quantum dynamics to a smaller, but still noticeable, degree.

Lifetimes of local and hyperspherical vibrational resonances of ABA molecules

The complete spectrum of vibrationally excited ABA* molecular resonance lifetimes is evaluated using the simple Rosen–Thiele–Wilson model of coupled Morse oscillators. Two complementary methods are used: First, unimolecular dissociative resonance wave functions are propagated in time by the Fourier method, where the initial wave functions are obtained as an approximation by linear combinations of symmetry-adapted products of Morse functions. Second, bimolecular reaction S matrices are propagated along the hyperspherical radius of the system giving the diagonalized lifetime matrix, which is analyzed for resonance lifetimes and energies. The resulting uni- and bimolecular resonance energies agree within ±0.002 eV and the lifetimes within ±30%. Uni- and bimolecular assignments of gerade (+) and ungerade (−) ABA* symmetries agree perfectly. On the average, the unimolecular decay times decrease as the resonance energies increase from the ABA*→A+BA to about 3/4 of the A+B+A dissociation threshold; even more highly excited resonances tend to be slightly more stabilized. Superimposed on this overall nonmonotonous energy dependence is a strong, 1–2 orders of magnitude variation of lifetimes, indicating substantial mode selectivity for the decay of individual resonances, irrespective of the excitation energy. The mode selectivity is investigated for hyperspherical mode resonances with lobes extending across the potential valleys, in contrast with local mode resonances with frontier lobes pointing towards the valleys. On the average, hyperspherical mode resonances decay at a slower rate than local mode resonances. This conclusion agrees with our previous analysis of low energy ABA* resonances, and with Hose and Taylor’s analysis of the Hénon–Heiles system. However, these correlations are also violated by several important exceptions: the ABA* system has many slowly, but also a few rapidly, decaying hyperspherical resonances.

Occurrence of secular terms in the Carleman embedding

Secular terms occur in many perturbative solutions of nonlinear equation systems. In this work, an investigation is made of which cases they may occur in as the result of the application of the linear Carleman embedding to a system of nonlinear equations. The solution for the embedded system is written in a form that makes it convenient to see how these terms originate. Their occurrence for the general case is discussed and the results are exemplified by working out the Hénon–Heiles system.

Semiclassical ergodic properties of the Henon–Heiles system

This paper investigates the applicability of the concepts of classical ergodicity and the results of semiclassical ergodic theory to quantum mechanical systems that are away from the classical limit and that do not necessarily become ergodic as ℏ → 0. To carry out this study, a new quantity, called the ‘‘constancy’’ FA of classical property A, is introduced. This constancy is defined in terms of the autocorrelation function of A and its statistical equilibrium value. It measures, on an absolute scale, the extent to which the dynamics of A behave statistically. The FA for a set of quadratically integrable properties A can be used to define the ‘‘degree of ergodicity’’ of a classical system with respect to this set. This analysis motivates introduction of the quantum analog of FA as a means of judging the applicability of classical ergodicity concepts to quantum mechanical systems. To illustrate these ideas, calculations for the Henon–Heiles system are carried out which compare the quantum and classical analogs of the constancies and of the ‘‘almost microcanonical’’ autocorrelation functions from which they are formed. The results indicate that the quantum and classical systems exhibit similar forms of partial ergodicity at high energy. This conclusion supports the approach introduced here for identifying the quantum implications of classical ergodicity. As a consequence of the good quantum-classical agreement, the partially ergodic nature of the classical behavior is reflected in the distribution of certain quantum-mechanical matrix elements. At lower energy, the applicability of classical ergodicity concepts to this system is more limited due to differences in the quantum and classical dynamics. The kinds of quantum-classical discrepancies that limit the implications of classical ergodicity for quantum systems are identified.

The quantum normal form and its equivalents

A quantum analog, called the quantum normal form, of the classical Birkhoff–Gustavson normal form is presented. The algebraic relationship between the quantum and Birkhoff–Gustavson normal forms has been established by developing the latter using Lie transforms. It is shown that the Birkhoff–Gustavson normal form can be obtained from the quantum normal form. Using an anharmonic oscillator and a Henon–Heiles system as test cases, the equivalence between the quantum normal form and the Rayleigh–Schrödinger perturbation method is shown. This equivalence provides an algebraic connection between the Birkhoff–Gustavson normal form and the Rayleigh–Schrödinger perturbation approach. The question of Weyl and torus quantizations of the Birkhoff–Gustavson normal form is discussed in the light of the quantum normal form.

Dynamics of hyperspherical and local mode resonance decay studied by time dependent wave packet propagation

Time dependent wave packet propagation of resonance states of ABA molecules is used to demonstrate the correlation between the directionality of the lobes of the wave functions and mode selectivity of the unimolecular decay. This correlation was inferred by Hose and Taylor. The molecule is modeled by the Thiele–Wilson coupled Morse oscillators. A near-degenerate pair of resonances with extreme motions is studied in detail: The local ‘‘bond’’ mode with lobes pointing towards the exit valleys of the potential decays about 30 times faster than the hyperspherical ‘‘restricted precession’’ mode with dominant lobe on the potential ridge. This is in close analogy to mode selectivity in the Hénon–Heiles system. The wave function propagation technique also yields detailed insight into the dissociation mechanism. Out of several choices, only a single lobe penetrates into the exit valley. For the local mode resonance vibrational predissociation starts out primarily from extended vibrationally excited diatomic configurations, A↔B(ν*=1)⋅ ⋅ ⋅A→AB(ν′=0)+A. However, the hyperspherical mode resonance prefers compressed diatomic geometry just before dissociation, AB(ν*=3)⋅ ⋅ ⋅A→AB(ν′=0)+A. The results imply some general criteria for mode selective unimolecular chemical reactions, as well as a successful numerical test of the preparation of resonance wave functions and their propagation by the Fourier method.

Studying the nonintegrability of Hamiltonian systems using the Kolmogorov entropy: Application to the Hénon–Heiles system

The present work proposes to use the Kolmogorov entropy K as a criterion for the determination of the critical energy above which an Hamiltonian system becomes nonintegrable. Application to the Hénon–Heiles system shows that this system is totally nonintegrable, i.e., K>0 at all energies, though none of the other criteria used so far were sensitive enough to detect this fact at energies below 1/12.

Wave packet dynamics and chaos in the Hénon–Heiles system

The evolution of wave packets under the influence of a Hénon–Heiles potential has been investigated by direct numerical solution of the time-dependent Schrödinger equation. Coherent state Gaussians with a variety of mean positions and momenta were selected as initial wave functions. Three types of diagnostics were used to identify chaotic behavior, namely, phase space trajectories computed from the expected values of coordinates and momenta, the correlation function P(t)=‖〈ψ(0)‖ψ(t)〉‖2, and the uncertainty product or phase space volume V(t)=ΔxΔyΔpxΔpy. The three approaches lead to a consistent interpretation of the system’s behavior, which tends to become more chaotic as the energy expectation value of the wave packet increases. The behavior of the corresponding classical system, however, is not a reliable guide to regular or chaotic behavior in the quantum mechanical system.

Quantum dynamics of wave packets on phase space cells in nonlinearly coupled oscillators

AbstractThe time evolution of harmonic oscillator coherent states (minimum uncertainty wave packet) | Ψμ located on regular and random von Neumann lattice points in phase space is analyzed for nonlinearly coupled anharmonic vibrational systems. The Henon–Heiles system is studied as an example. A quasiprobability measure \documentclass{article}\pagestyle{empty}\begin{document}$ \bar P_{\mu \mu '} $\end{document} is introduced to investigate the relative probabilities for the transition between phase space cells within a narrow energy range. The probability \documentclass{article}\pagestyle{empty}\begin{document}$ \bar P_{\mu \mu '} $\end{document}, as well as the information entropy Sμ, is studied as a function of energy. A smooth transition from regular to chaotic motion is found indicated by a change of the fluctuations of these quantities as a function of energy.

Quantum dynamics of the Henon–Heiles system

In this paper, we investigate the quantum dynamics of wave packets of bound states in the nonlinear Henon–Heiles (HH) system. The time evolution of a wave packet was separated into two processes involving (a) the motion of its first moments and (b) its spreading. On the time scale when wave packet spreading can be neglected, we were able to establish relations between wave packet quantum dynamics expressed in terms of the initial-state population probability P(t) and classical trajectory dynamics. We have demonstrated that the initial states, which are characterized by a periodic time evolution of P(t) over many periods, are related to the five elliptic periodic orbits in the HH system. The decay of P(t) over a few vibrational periods is related to the classical features of quasiperiodic trajectories and can be accounted for in terms of overlap reduction effects. When the center of the initial wave packet is moved further away from the classical periodic orbits, P(t) is characterized by a single initial peak followed by a ’’noisy’’ background. There is no qualitative distinction between the ’’short-time’’ behavior of a wave packet initiated in distant integrable regions and in classical stochastic regions. For long times, wave packet spreading effects set in. Quantum mechanical estimates for the spreading time τs, which depends on the initial conditions, were provided. For wave packets initiated on periodic trajectories, τs is very long, becoming shorter as one moves away from the fixed points. The quantum mechanical long time (t≫τs) behavior of P(t) exhibits small-amplitude noisy time dependence for all initial conditions. This behavior may be described as ’’quantum mechanical stochasticity’’ and is related to the universal feature of irregularity of the spectral resolution of all wave packets in a nonlinear system. From this point of view, all wave packets exhibit ’’quantum stochastic behavior,’’ but there are vast quantitative differences for different initial conditions that are manifested in terms of the values of τs and of the average noisy amplitude. It should be emphasized that this ’’quantum stochasticity’’ is not related to classical stochasticity phenomena.

Quantum manifestations of classical stochasticity. II. Dynamics of wave packets of bound states

In this paper we consider some of the consequences of classical stochasticity phenomena in the quantum mechanical Henon–Heiles and Barbanis systems. To explore classical–quantum analogies we have introduced a quantum mechanical phase space based on the coherent state representation. Quantum Poincaré maps (QPMs) were constructed from contour plots of the stationary phase-space densities. We have observed a correlation between the topological features of a QPM at an eigenenergy E and the sensitivity ‖dE/dε‖ of that energy to the strength ε of the nonlinear coupling. States with extreme values of ‖dE/dε‖, corresponding to both high and to low values of ‖dE/dε‖, show regular regions in the quantum phase space, while states with intermediate values of ‖dE/dε‖ span large parts of the quantum phase-space regions. These general features of the QPMs apply for almost each energy multiplet, and no qualitative change in the character of the QPMs was observed above the classical critical energy Ec for the stochastic transition. To investigate the quantum analog of classical trajectories, we have studied the dynamics of initially coherent Gaussian wave packets. We have discovered two limiting types of time evolution of intially coherent wave packets, which exhibit quasiperiodic time evolution or rapid dephasing. Quasiperiodic quantum dynamics is analogous to quasiperiodic classical trajectory dynamics, while rapid dephasing of the wave packet reflects the effects of spreading of the wave packet or/and stochastic classical dynamics. We were able to establish a correspondence between the topology of the quantum phase-space density and the wave packet dynamics. Quasiperiodic time evolution and rapid dephasing are exhibited by wave packets initially located in the regular and in the irregular regions of the quantum phase space, respectively.

On correlation functions and the onset of chaotic motion

A variety of correlation functions computed over a microcanonical ensemble for the Hénon–Heiles system are investigated. We find the general trend is that of a gradual change to some form of decaying behavior as the motion becomes predominantly chaotic. The decay of a mode energy correlation function indicates a time scale for intramolecular energy redistribution.

Perturbative examination of avoided crossings

Quantum perturbation theory is used to examine the eigenvalues of a nonseparable Hamiltonian system in the classically regular and irregular regimes. As a function of the perturbation parameter, the eigenvalues obtained by exact (matrix diagonalization) methods undergo an avoided crossing. In the present paper perturbation theory is used as an approximate method to predict the locations of such avoided crossings in energy-parameter space. The sparsity of such avoided crossings in the Hénon–Heiles system is seen to produce regular sequences in the eigenvalues even when the classical motion is predominantly chaotic.

Quantum dynamics in classically integrable and non-integrable regions

The time evolution of wavefunctionsΦ( t) composed of superpositions of energy eigenstates are compared when the wavefunctions are initiated in (1) the non-integrable and (2) the integrable regimes in the phase space of the Henon - Heiles system. The behavior of ▪ is found to be very different in the two regions

An SCF-state interaction method for coupled oscillator systems

A computational method is presented for the treatment of coupled systems of oscillators by a SCF-like formalism. Extensions to state interaction (analogous to configuration interaction) are presented and explored numerically for the Henon—Heiles system. The formalism is presented for an MCSCF-like approach to such problems.

A spectral entropy method for distinguishing regular and irregular motion of Hamiltonian systems

Regular and irregular motions of bounded conservative Hamiltonian systems of N degrees of freedom can be distinguished by the structure of the frequency spectrum of a single trajectory. The spectral entropy S is introduced which provides a measure of the distribution of the frequency components. Numerical calculations on the model Henon and Heiles system and a realistic molecular model are performed. Power spectra are obtained from numerical solutions to Hamilton's equations using fast Fourier transforms and the Hanning method. For regular trajectories S is found to stabilise after a finite time of integration, while for irregular cases S increases erratically. Estimates of the relative volume of regular regions of phase space as a function of energy are given for the two systems.