Time-dependent approach to semiclassical dynamics

TLDR

The authors introduce a new semiclassical dynamics method that decomposes heavy‑particle wavefunctions into minimally spreading, classically evolving wave packets. They assume Gaussian wave packets, expand the potential to quadratic order, derive Hamiltonian equations for the packet’s classical parameters and first‑order quantum equations for its spread and phase, and use superposition and projection to extract state‑specific quantum information. The approach reproduces both classically allowed and forbidden transitions, with the packet center acquiring the correct action‑integral phase, and yields accurate, flux‑conserving results in a numerical test on the collinear He+H₂ system.

Abstract

In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.