Chaos and the correspondence limit in the periodically kicked pendulum

The correspondence limit is illustrated for the periodically kicked pendulum. The classical dynamics of this system can be represented by a discrete map. A corresponding quantum map is derived and then rendered in the Ehrenfest formulation for expectation values. The Ehrenfest representation is studied for a minimum uncertainty Gaussian wave packet. It is shown that as ħ gets very small, followed by a decrease in the variance of the wave packet, the quantum map shadows the classical map with an error approaching zero for a length of time approaching infinity. The Gaussian form of the wave packet is preserved by the time evolution provided ${\mathrm{\omega }}_0$T≪1. This constraint implies that the classical map is predominantly not chaotic with very small regions of very weak chaos. As ${\mathrm{\omega }}_0$T approaches 1, where the classical map becomes strongly chaotic, the propagation in time of the Gaussian wave packet completely breaks down. The possible significance of this breakdown is discussed.