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理论物理通讯：英文版

The Spatiotemporal Evolution of Wave Packets under Chaotic Condition

LiuF, LiJQ, LuoYX, XuGO, ZuoW

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

Using the minimum uncertainty state of quantum integrable system Ho as initial state, the spatiotemporal evolution of the wave packet under the action of perturbed Hamiltonian is studied causally as in classical mechanics. Due to the existence of the avoided energy level crossing in the spectrum there exist nonlinear resonances between some pairs of neighboring components of the wave packet, the deterministic dynamical evolution becomes very complicated and appears to be chaotic. It is proposed to use expectation values for the whole set of basic dynamical variables and the corresponding spreading widths to describe the topological features concisely such that the quantum chaotic motion can be studied in contrast with the quantum regular motion and well characterized with the asymptotic behaviors. It has been demonstrated with numerical results that such a wave packet has indeed quantum behaviors of ergodicity as in corresponding classical case.

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Using the minimum uncertainty state of quantum integrable system Ho as initial state, the spatiotemporal evolution of the wave packet under the action of perturbed Hamiltonian is studied causally as in classical mechanics. Due to the existence of the avoided energy level crossing in the spectrum there exist nonlinear resonances between some pairs of neighboring components of the wave packet, the deterministic dynamical evolution becomes very complicated and appears to be chaotic. It is proposed to use expectation values for the whole set of basic dynamical variables and the corresponding spreading widths to describe the topological features concisely such that the quantum chaotic motion can be studied in contrast with the quantum regular motion and well characterized with the asymptotic behaviors. It has been demonstrated with numerical results that such a wave packet has indeed quantum behaviors of ergodicity as in corresponding classical case.
Using the minimum uncertainty state of quantum integrable system as initial state, the spatiotemporal evolution of the wave packet under the action of perturbed Hamiltonian is studied causally as in classical mechanics. Due to the existence of the avoided energy level crossing in the spectrum there exist nonlinear resonances between some pairs of neighboring components of the wave packet, the deterministic dynamical evolution becomes very complicated and appears to be chaotic. It is proposed to use expectation values for the whole set of basic dynamical variables and the corresponding spreading widths to describe the topological features concisely such that the quantum chaotic motion can be studied in contrast with the quantum regular motion and well characterized with the asymptotic behaviors. It has been demonstrated with numerical results that such a wave packet has indeed quantum behaviors of ergodicity as in corresponding classical case.
Taking coherent states as initial quantum states, the spatio-temporal evolution of the wave packet under the action of perturbed Hamiltonian showed one-to-one correspondence to the behavior of classical phase points. The character of the coherent state propagation in classically soft chaotic system is found to be associated with the statistical property of the spectrum contained in the coherent state wave packet. The spreading width of wave packet increases with time exponentially if the statistics of the spectrum contained in the wave packet show GOE distribution due to avoided level crossings.
The quantum correspondence of one particular signature of classical chaos--the exponential instability of motion--can be characterized by the initial exponential growth rate of the spreading of the propagating quantum wave packet. The growth rate is approximately twice the classical maximum Lyapunov exponent of the system. In the regular case, the dispersion of the wave packet is only due to the usual quantum effect that should vanish in the classical limit. In contrast, in the chaotic case, the evolution behavior of the wave packet is due to the dynamical effect associated with the nonlinearity and persists as long as the spatial extension of the initial wave packet is kept finite.
Studying quantum properties of a system has been quite popular in quantum mechanics. One of the most important systems that are very crucial to the framework of quantum mechanics is the system of harmonic oscillator a system whose classical evolution is known to exhibit peculiar chaotic dynamics. We are motivated to investigate the behavior of quantum properties for a system with position and time dependent perturbed. Starting with Hamiltonian, we determined the equation of motion and obtained the wave function. The energy of the whole system using the operator ordering method was found. We show that the quantum mechanical picture alludes to a chaotic dynamics as expected. This is evidenced through the appearance of energy level crossings. An additional signature to this chaotic dynamics is observed in the transition of Eigen values from real to imaginary. We also show numerically that one can give the behavior of the system is Poincare section. By so doing we confirmed that increasing and decreasing the perturbation amplitude of the system becomes chaotic.
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.
The notion of quantum-classical correspondence is carefully investigated in order to prepare firm grounds for studying the spatiotemporal evolution of quantum states in the same spirit as for corresponding classical cases. Three relevant problems, (1) the integrability of dynamical equations of quantum systems, (2) the initial minimum uncertainty states one-to-one correspondent to classical phase points, and (3) the effective Planck constants for systems having analogous dynamical properties but exhibiting different quantum effects, have been successfully resolved. Then the solution ${\ensuremath{\rho}}_{\ensuremath{\gamma}}(t)$ of the dynamical equation of a quantum integrable system is shown to be expressed as an analytical functional of the initial minimum uncertainty state ${\ensuremath{\rho}}_{\ensuremath{\gamma}}^{0}$ varying smoothly with $\ensuremath{\gamma}$ and t. Such a general character of the quantum regular motion serves as a reference for the study of quantum irregular motion under the action of perturbed Hamiltonian.