Classical limit of quantum mechanics induced by continuous measurements

Abstract

We investigate the quantum–classical transition problem. The main issue addressed is how quantum mechanics can reproduce results provided by Newton’s laws of motion. We show that the measurement process is critical to resolve this issue. In the limit of continuous monitoring with minimal intervention the classical limit is reached. The Classical Limit of Quantum Mechanic, in Newtonian sense, is determined by two parameters: the semiclassical time (τsc) and the time interval between measurements (Δτu). If is Δτu small enough, comparing with the τsc, then the classical regime is achieved. The semiclassical time for Gaussian initial states coincides with the Ehrenfest time. We also show that the classical limit of an ensemble of Newtonian trajectories, the Liouville regime, is approximately obtained for the quartic oscillator model if the number of measurements in the time interval is large enough to destroy the revival and small enough to not reach the Newtonian regime. Namely, the Newtonian regime occurs when τsc≫Δτu and the Liouvillian regime is mimicked, for the position observable, if Δτu∈[τsc,TR], where TR is the revival time.