The evolution of oscillator wave functions

TL;DR

This paper reviews methods to analyze how wave functions evolve over time in the harmonic oscillator, focusing on transformations that simplify their analysis and reveal properties like uncertainty and squeezing.

Contribution

It introduces transformations that fix the wave function's centroid and spreads, simplifying the study of their time evolution in harmonic oscillators.

Findings

Wave functions can be transformed to have a stationary centroid.

The evolution of spreads in position and momentum relates to energy and uncertainty.

All wave functions' evolution can be derived from those at rest with fixed spreads.

Abstract

We consider some of the methods that can be used to reveal the general features of 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 period of oscillation. 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 $\Delta_x$ and momentum $\Delta_p$ throws light on energy and uncertainty and on squeezed and coherent states. Finally we show that any wave function can be transformed so that $\Delta_x$ and $\Delta_p$ 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 $\Delta_x$ and $\Delta_p$.