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Journal of Mathematical Physics

Wave packet revivals and quasirevivals in one-dimensional power law potentials

Richard W. Robinett

Published Apr 1, 2000• 42 citations
Publisher

Abstract

The harmonic oscillator and the infinite square well are two of the simplest one-dimensional quantum systems which exhibit wave packet revivals (trivially so in the case of the oscillator.) These two potentials can be thought of as special cases of the general one-dimensional power-law potential given by V(k)(x)=V0|x/a|k (with k=2 and k→∞ for the oscillator and square well, respectively.) Using an autocorrelation function approach and the WKB approximation for the quantized energy levels in such potentials, we exhibit numerical evidence for wave packet revivals in the case of arbitrary k>0 which agree well with more analytic results. We derive expressions for the revival and collapse time scales in terms of the physical quantities of the system as well as the wave packet parameters. We find that both times scale with the power-law exponent k as |(k+2)/(k−2)|. In this way, we can explicitly exhibit the approach to the two familiar limiting cases. We also briefly consider the case of a “half” well where an infinite wall in added at the origin.

DOI
10.1063/1.533213

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The harmonic oscillator and the infinite square well are two of the simplest one-dimensional quantum systems which exhibit wave packet revivals (trivially so in the case of the oscillator.) These two potentials can be thought of as special cases of the general one-dimensional power-law potential given by V(k)(x)=V0|x/a|k (with k=2 and k→∞ for the oscillator and square well, respectively.) Using an autocorrelation function approach and the WKB approximation for the quantized energy levels in such potentials, we exhibit numerical evidence for wave packet revivals in the case of arbitrary k>0 which agree well with more analytic results. We derive expressions for the revival and collapse time scales in terms of the physical quantities of the system as well as the wave packet parameters. We find that both times scale with the power-law exponent k as |(k+2)/(k−2)|. In this way, we can explicitly exhibit the approach to the two familiar limiting cases. We also briefly consider the case of a “half” well where an infinite wall in added at the origin.
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)=V0|x/a|k which interpolates between the case of the harmonic oscillator (k=2) and the infinite well (k=∞).
We have investigated quantum revivals of wave packets in a one-dimensional infinite square well potential containing a $\ensuremath{\delta}$ well in the middle. The time-dependent Schr\"odinger equation for this composite potential admits formally exact solutions. We present analytical results for revival properties in three physically motivated approximations: wave packets containing eigenstates with large numbers in the presence of an arbitrary $\ensuremath{\delta}$ well, ``shallow'' and ``deep'' $\ensuremath{\delta}$ wells. Analytical results in the case of a ``shallow'' $\ensuremath{\delta}$ well have been tested numerically.
We present an analytical investigation of revival phenomena in the finite square-well potential. The classical motion, revival, and super-revival time scales are derived exactly for wave packets excited in the finite well. These time scales exhibit a richer dependence on wave-packet energy and on potential-well depth than has been found in other quantum systems: They explain, for example, the difficulties in exciting wave packets with strong classical features at the bottom of a finite well, or with clearly resolved super-revivals in a shallow well. In the proper regions of validity, the time scales predict the instances of wave-packet reformation extremely accurately. Revivals at the bottom of the well are explored as a ``universal'' limit of the general theory, which offers the clearest connection with the series of fractional and full revivals seen in the dynamics of the infinite square-well potential.
We examine the long-term time dependence of Gaussian wave packets in a circular infinite well (billiard) system and find that there are approximate revivals. For the special case of purely m=0m=0m=0 states (central wave packets with no momentum) the revival time is ${T}_{\mathrm{rev}}^{(m=0)}=8\ensuremath{\mu}{R}^{2}/\ensuremath{\Elzxh}\ensuremath{\pi},$ where $\ensuremath{\mu}$ is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that ${T}_{\mathrm{rev}}^{(m\ensuremath{\ne}0)}=({\ensuremath{\pi}}^{2}{/2)T}_{\mathrm{rev}}^{(m=0)}\ensuremath{\approx}{5T}_{\mathrm{rev}}^{(m=0)}$ and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of $m\ensuremath{\ne}0$ states is increased. The dependence of the revival structure on the initial position, energy, and angular momentum of the wave packet and the connection to the energy spectrum is discussed in detail. The results are also compared to two other highly symmetrical two-dimensional infinite well geometries with exact revivals, namely, the square and equilateral triangle billiards. We also show explicitly how the classical periodicity for closed orbits in a circular billiard arises from the energy eigenvalue spectrum, using a WKB analysis.
We test the ability of semiclassical theory to describe quantitatively the revival of quantum wave packets-a long time phenomena-in the one dimensional quartic oscillator (a Kerr type Hamiltonian). Two semiclassical theories are considered: time-dependent WKB and Van Vleck propagation. We show that both approaches describe with impressive accuracy the autocorrelation function and wave function up to times longer than the revival time. Moreover, in the Van Vleck approach, we can show analytically that the range of agreement extends to arbitrary long times.
We develop a series solution for the bound-state energy levels of the quantum-mechanical one-dimensional finite square-well potential. We show that this general solution is useful for local approximations of the energy spectrum (which target a particular energy range of the potential well for high accuracy), for global approximations of the energy spectrum (which provide analytic expressions of reasonable accuracy for the entire range of bound states), and for numerical methods. This solution also provides an analytic description of dynamical phenomena; with it, we compute the time scales of classical motion, revivals, and super-revivals for wave-packet states excited in the well.