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

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 ($k=\infty$).Comment: 34 pages, 11 figure

Expectation value analysis of wave packet solutions for the quantum bouncer: short-term classical and long-term revival behavior

We discuss the time development of Gaussian wave packet solutions of the quantum bouncer' (a quantum mechanical particle subject to a uniform downward force, above an impermeable flat surface). We focus on the evaluation and visualization of the expectation values and uncertainties of position and momentum variables during a single quasi-classical period as well as during the long term collapsed phase and several revivals. This approach complements existing analytic and numerical analyses of this system, as well as being useful for comparison with similar results for the harmonic oscillator and infinite well cases.Comment: 20 pages, 7 separate .ps figure

Less than perfect quantum wavefunctions in momentum-space: How phi(p) senses disturbances in the force

We develop a systematic approach to determine the large |p| behavior of the momentum-space wavefunction, phi(p), of a one-dimensional quantum system for wich the position-space wavefunction, psi(x), has a discontinuous derivative at any order. We find that if the k-th derivative of the potential energy function has a discontinuity, there is a corresponding discontinuity in psi^{(k+2)}(x) at the same point. This discontinuity leads directly to a power-law tail in the momentum-space wavefunction proportional to 1/p^{k+3}. A number of familiar pedagogical examples are examined in this context, leading to a general derivation of the result.Comment: 22 pages, 2 figures. To appear in Am. J. Phy