The evolution of oscillator wave functions

We investigate how wave functions evolve with time in the harmonic oscillator. We first review the periodicity properties over each multiple of a quarter of the classical oscillation period. Then we show that any wave function can be simply transformed so that its centroid, defined by the expectation values of position and momentum, remains at rest at the center of the oscillator. This implies that we need only consider the evolution of this restricted class of wave functions; the evolution of all others can be reduced to these. The evolution of the spread in position Δ_x and momentum Δ_p throws light on energy and uncertainty and on squeezed and coherent states. Finally, we show that any wave function can be transformed so that Δ_x and Δ_p do not change with time and that the evolution of all wave functions can easily be found from the evolution of those at rest at the origin with unchanging Δ_x and Δ_p.