Visualizing the collapse and revival of wavepackets in the infinite square well using expectation values

Abstract

We investigate the short-, medium-, and long-term time dependence of wave packets in the infinite square well. In addition to emphasizing the appearance of wave packet revivals, i.e., situations where a spreading wave packet reforms with close to its initial shape and width, we also examine in detail the approach to the collapsed phase where the position-space probability density is almost uniformly spread over the well. We focus on visualizing these phenomena in both position- and momentum-space as well as by following the time-dependent expectation values of and uncertainties in position and momentum. We discuss the time scales for wave packet collapse, using both an autocorrelation function analysis, as well as focusing on expectation values and find two relevant time scales which describe different aspects of the decay phase. In an Appendix, we briefly discuss wave packet revival and collapse in a more general, one-dimensional power-law potential given by $V_{(k)}(x)=V_0|x/a{|}^k$ which interpolates between the case of the harmonic oscillator ($k=2$) and the infinite well ($k=\infty$).

34 pages, 11 figures

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