Ehrenfest time and chaos

Abstract

The Ehrenfest time gives the scale of time on which the Bohr correspondence principle (Bohr, 1920) remains valid for a quantum evolution of an initial state at high characteristic quantum numbers (or small effective Planck constant ) closely following the corresponding classical distribution. For a narrow initial wave packet the Ehrenfest theorem (Ehrenfest, 1927) guaranties that the average values of quantum operators are close to the corresponding classical averages. For systems with integrable classical dynamics the Ehrenfest time is rather long being generally inversely proportional to the Planck constant (or another power of it). The new nontrivial situation appears for classically chaotic dynamics when nearby trajectories diverge exponentially with time due to exponential instability of motion characterized by the positive Kolmogorov-Sinai entropy . Thus in such semiclassical systems the Ehrenfest time is logarithmically short . The properties of the Ehrenfest time of quantum dynamics of such chaotic systems, with related examples, are discussed below.