Wave packet dynamics and chaos in the Hénon–Heiles system

Abstract

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)=ΔxΔyΔp_xΔp_y. 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.