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American Journal of Physics

Decay of wave packet revivals in the asymmetric infinite square well

Todd Keene Timberlake, Seth Camp

Published May 13, 2011• 3 citations
Publisher

Abstract

In the infinite square well, any wave function will return to its initial state at integer multiples of the revival time. Most quantum systems do not exhibit perfect revivals, but some exhibit partial revivals in which the wave function returns close to its initial state. Subsequent partial revivals usually deteriorate in quality. We discuss the reasons for the perfect revivals in the infinite square well and how a small change in the potential disrupts the revivals. As an example, we examine partial revivals of a Gaussian wave packet in an infinite square well with a step. First-order and second-order perturbation theory show that the rate at which revivals decay depends on the location of the step.

DOI
10.1119/1.3549714

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In the infinite square well, any wave function will return to its initial state at integer multiples of the revival time. Most quantum systems do not exhibit perfect revivals, but some exhibit partial revivals in which the wave function returns close to its initial state. Subsequent partial revivals usually deteriorate in quality. We discuss the reasons for the perfect revivals in the infinite square well and how a small change in the potential disrupts the revivals. As an example, we examine partial revivals of a Gaussian wave packet in an infinite square well with a step. First-order and second-order perturbation theory show that the rate at which revivals decay depends on the location of the step.
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 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=∞).
The temporal evolution of a wave packet ψ( x,t ) confined to an infinite square well (occupying the region 0 ≤ x ≤ L ), as manifested by fractional revivals of the probability density P ( x,t ) ≡ | ψ( x,t ) | 2 and the time-dependence of the expectation values of position, momentum and force ( X ( t ) ≡ ⟨ x ⟩ t , Π( t ) ≡ ⟨ ⟩ t and Φ( t ) ≡ ⟨ ⟩ t , respectively) are studied, using wave packets for which Π(0) = 0. In the region accessible to the wave packet (0 < x < L ), and the Hamiltonian Ĥ commute; still, is, in general, not a constant of the motion, and the three expectation values are all time-dependent, except when the initial packet has a definite parity, that is ψ( x ,0) = ±ψ( L - x , 0), which implies X ( t ) = L /2 and Π( t ) = 0 = Φ( t ). For a packet with no definite parity, Y ( t ) ≡ X ( t ) - L /2 stays, if the packet is sufficiently narrow at t = 0 close to zero nearly all the time, but executes large excursions in the neighbourhood of instants at which the probability density experiences a mirror revival or a revival. This behaviour, with no classical counterpart, can be understood by paying attention to the presence of infinite potential barriers (at x = 0 and x = L ) and taking account of the net force felt by an initially off-centred wave packet. It is shown that though plots (against time) of expectation values catch a glimpse of the revival structure of a wave packet, such plots fail to reveal some fractional revivals.
We consider the revival properties of quantum systems with an eigenspectrum E_{n} ~ n^{2}, and compare them with the simplest member of this class - the infinite square well. In addition to having perfect revivals at integer multiples of the revival time t_{R}, these systems all enjoy perfect fractional revivals at quarterly intervals of t_{R}. A closer examination of the quantum evolution is performed for the Scarf potentials, and comparison is made with the infinite square well using quantum carpets.