Coherent states for the bouncing pendulum and the paddle ball

Abstract

The coherent states of the simple harmonic oscillator with an impenetrable barrier at its center are studied. This half oscillator is the equivalent of a pendulum that bounces elastically off a vertical wall directly below the point of suspension with the angle of swing sufficiently small. The system can also be considered as a paddle ball, where the paddle is fixed and the ball is constrained by a spring attached to the paddle. The coherent states are almost the same as the familiar Gaussian coherent states of the full oscillator, except when they overlap the barrier. The solutions can be easily extended to two and three dimensions and gravity can be included if the impenetrable barrier is vertical. To better understand the form of the expectation values of the position and momentum, we investigate some general aspects of the effect of impenetrable barriers on the dynamics of wavepackets.