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Chinese Physics C

Spatial-Temporal Evolution of the Wave Packet and Avoided Level Crossing

Fang Liu

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

In the two cases that the deformed nuclear system is regular and chaotic, the spatial temporal evolution of the coherent states of a two dimensional harmonic oscillator system as an initial state was analyzed with the Fourier transformation, especially by comparing the Fourier transformation of the width of the wave packet in the whole time stage with that the wave packet approaching to the saturation in the chaotic system. It was found that since a large number of avoided level crossings destroy the regular structure of the state space of the system, the components in the wave packet are no longer coherent and the motion of the wave packet becomes chaotic. Afterwards, the contribution coming from the different components tends to uniform, the influence of the avoided level crossings on the motion of the wave packet is not obvious also

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In the two cases that the deformed nuclear system is regular and chaotic, the spatial temporal evolution of the coherent states of a two dimensional harmonic oscillator system as an initial state was analyzed with the Fourier transformation, especially by comparing the Fourier transformation of the width of the wave packet in the whole time stage with that the wave packet approaching to the saturation in the chaotic system. It was found that since a large number of avoided level crossings destroy the regular structure of the state space of the system, the components in the wave packet are no longer coherent and the motion of the wave packet becomes chaotic. Afterwards, the contribution coming from the different components tends to uniform, the influence of the avoided level crossings on the motion of the wave packet is not obvious also
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.
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.
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.
Coherent states of two dimensional asymmetrical harmonic oscillator,which prop- agate in additional octupole deformation potentials, are used to investigate the complexity of the density matrix and the sensitivity against the strength of disturbance. It was found that according to the regular, partly chaotic and overall chaotic characteristics of the phase space of the classical system,its quantum analogy bears out corresponding characters, which were associated with the existence of the negative curvature at the potential surface and the magni- tude of the negative curvature.
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.