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Physica Scripta

The Force Exerted by the Walls of an Infinite Square Well on a Wave Packet: Ehrenfest Theorem, Revivals and Fractional Revivals

S. Waldenstrøm, Kalbe Razi Naqvi, Kjell J. Mork

Published Jan 1, 2003• 9 citations

Abstract

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.

DOI: 10.1238/physica.regular.068a00045

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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.

Fractional revivals of wave packets in an infinite square well are scrutinized with a viewpoint rooted in Fourier analysis, and a compact relation, expressing the wavefunction Ψ( x , t ) for certain values of t in terms of spatially displaced copies of Ψ( x ,0), is derived without appealing to a classical analogy ; conditions for the appearance of an interference pattern in a plot of the position probability density of a wave packet are deduced, along with related results obtained previously from different considerations, with minimal effort. The specific case of a packet of Gaussian shape is analysed in greater detail to provide concrete illustrations of fractional revivals with or without interference between overlapping copies of wave packets, depending on whether a finite value of the so-called classical time can or cannot be assigned to the wave packet.

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 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 calculate the Wigner quasi-probability distribution for position and momentum, P_W^(n)(x,p), for the energy eigenstates of the standard infinite well potential, using both x- and p-space stationary-state solutions, as well as visualizing the results. We then evaluate the time-dependent Wigner distribution, P_W(x,p;t), for Gaussian wave packet solutions of this system, illustrating both the short-term semi-classical time dependence, as well as longer-term revival and fractional revival behavior and the structure during the collapsed state. This tool provides an excellent way of demonstrating the patterns of highly correlated Schrodinger-cat-like `mini-packets' which appear at fractional multiples of the exact revival time.

If the probability density P ( x , t ) = |ψ ( x , t )| 2 of a non-stationary state happens to meet the condition P ( x , t 0 ) = P ( x , t 0 + Υ), the wave packet is said to experience exact revivals at intervals of Υ. The definition implies that a harmonic oscillator wave packet will experience revivals with a period Υ = 2π/ω, where ω is the angular frequency of the corresponding classical system. A harmonic oscillator wave packet of Gaussian form, centered initially ( t = 0) at x = x 0 , is shown to spread almost everywhere, if its initial width is sufficiently narrow, in a time very much shorter than the period of the corresponding classical oscillator. However, the packet reassembles itself twice during a single period: (i) it experiences a mirror revival, defined by the relation P ( x , ½Υ) = P (- x , 0), and (ii) an exact revival, P ( x , Υ) = P ( x , 0). These results are used to counter a recent claim (which apparently gives voice to a commonly held belief) that a harmonic oscillator wave packet ψ( x , t ) does not collapse and has no revivals. Plots of the recurrence probability |⟨ψ( x , 0)|ψ( x , t )⟩| 2 are presented and the need for formulating unambiguous definitions of the two concepts, revival and collapse, is stressed.

We calculate and visualize the Wigner quasi-probability distribution for the position and momentum, PW(n)(x,p), for the energy eigenstates of the infinite square well. We evaluate the time-dependent Wigner distribution, PW(x,p;t), for Gaussian wave packet solutions of this system, and illustrate the short-term semi-classical time dependence and the longer-term revival and fractional revival behavior. Our results indicate how the Wigner distribution can be used to examine the highly correlated dynamical position-momentum structure of quantum states. In particular, this tool provides an excellent way of demonstrating the patterns of highly correlated Schrödinger-cat-like “mini-packets” which appear at fractional multiples of the exact revival time.