Quantum-Classical Correspondence and Quantum Chaos in the Periodically Kicked Pendulum

The quantum-classical correspondence is studied analytically and numerically for the periodically kicked pendulum from the Ehrenfest-Hamilton perspective. Correspondence means that the mean-value trajectory shadows the classical trajectory and the rms deviations are negligible compared to the mean values. Initially, a minimum uncertainty gaussian wave packet with well-defined position and momentum is constructed for the quantum state. Under certain conditions, the wave packet propagates as a gaussian and the correspondence holds provided the angular deviation remains much less than one. Correspondence, however, does not require the wave packet to propagate as a gaussian. In general, shadowing occurs as long as the deviations remain negligible. With chaos, the wave packet broadens exponentially fast. By comparison, the spreading of the wave packet is slower without chaos. Therefore, correspondence is doomed to failure faster with chaos. Failure of the correspondence signifies that the classical description is no longer valid. After the breakdown of correspondence, the behavior of the quantum time evolution depends on the quasienergy spectrum. Therefore, although a mean-value trajectory is chaotic for a short time interval because its classical counterpart is, its long time behavior may not be.