Exponentially Accurate Semiclassical Dynamics: Propagation, Localization, Ehrenfest Times, Scattering and More General States

TL;DR

This paper proves six theorems demonstrating that semiclassical quantum dynamics can be approximated exponentially accurately over finite, Ehrenfest, and infinite times, including scattering and localization properties.

Contribution

It provides new proofs for known results and extends semiclassical accuracy to more general initial states and infinite times with exponential precision.

Findings

Exact and approximate dynamics agree exponentially well for localized wave packets.

Wave packets remain localized near classical orbits with exponentially small errors.

Scattering results are accurate exponentially in time.

Abstract

We prove six theorems concerning exponentially accurate semiclassical quantum mechanics. Two of these theorems are known results, but have new proofs. Under appropriate hypotheses, they conclude that the exact and approximate dynamics of an initially localized wave packet agree up to exponentially small errors in $\hbar$ for finite times and for Ehrenfest times. Two other theorems state that for such times the wave packets are localized near a classical orbit up to exponentially small errors. The fifth theorem deals with infinite times and states an exponentially accurate scattering result. The sixth theorem provides extensions of the other five by allowing more general initial conditions.