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Section 10.7: Wave Packet Dynamics
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In Section 10.6, we considered a two-state superposition within the infinite square well. We now consider a superposition of a large number of states so as to resemble an initial Gaussian wave packet at t = 0 in order to study the dynamics of such packets.6
We can examine the time dependence of such an initially localized state by choosing a Gaussian initially localized such that is is well within the well (0 < x < L) and has the form
ΨG(x,0) = (αħπ1/2)−1/2 exp[−(x − x0)2/(2α2ħ2)] exp[ip0(x − x0)/ħ] . (10.13)
One can show by direct calculation that <x>t = 0 = x0 , < p>t = 0 = p0 , and Δxt =0 = Δx0 =αħ/21/2 . The general expression for a time-dependent wave packet solution constructed from energy eigenfunctions of the well is
ΨG(x,t) = Σ cn exp[−iEnt/ħ] ψn(x) [sum from n = 1 to n = ∞] . (10.14)
where the expansion coefficients satisfy Σ |cn|2 = 1 . The expansion constants are determined by the integral
cn = ∫ ψ*(x)ΨG(x,0) dx [integral from 0 to L] . (10.15)
Once we determine these coefficients, we can use the equation for ψ(x,t) to reconstruct the wave packet and the corresponding packet dynamics. In the animation, we have two packets with x0 = L/2 , α = 1/(10 21/2), and one with p0 = 0 and one with p0 = 40π . We have set ħ = 2m = L = 1 and the time is given in terms of the time it takes the ground-state wave function to return to its original phase, i.e., Δt = 1 corresponds to an elapsed time of 2πħ/E1. You can also set the starting time for the animation to study the time evolution of the packet.
The time dependence of the wave packet in an infinite square well is determined by all of the exp(−iEnt/ħ) factors. There are two time scales for packets within an infinite square well, they are the classical period, Tcl, and the revival time, Trev. The classical period for the infinite square well is given by
Tcl = 2L/vn0 ,
where n0 is the mean quantum number of the packet's expansion in energy eigenfunctions. The analog of the classical speed, vn0 ≡ pn0/m, gives a result in agreement with classical expectations. For longer time scales, we require the revival time, Trev , which is given by7
Trev = 2πħ/E0 = 4mL2/(ħπ) = (2n0) Tcl . (10.16)
Note that in the animation we have set the revival time equal to 1. For the infinite square well, no longer time scales are present due to the purely quadratic dependence of the energy eigenvalues, En = n2π2ħ2/(2mL2). The revival time scale can clearly be much larger than the classical period. In the second animation for example, Trev/Tcl = 80.
The quantum revivals for this system are exact and any wave packet returns to its initial state after a time Trev. At half this time, t = Trev/2, the wave packet also reforms, but at a location mirrored about the center of the well and with an opposite momentum value.
At various fractional multiples of the revival time, pTrev/q, where p and q are integers, the wave packet can also reform as several small copies (sometimes called mini-packets or clones) of the original wave packet.8
6For a comprehensive review of this topic see: R. W. Robinett, "Quantum Wave Packet Revivals," Phys. Rep. 392, 1-119 (2004).
7In general these two time scales are calculated from the energy and they are
Tcl = 2πħ/|E'(n0)| and Trev= 2πħ/(|E''(n0)|/2),
where the primes represent d/dn. Longer time scales can be calculated from higher-order derivatives of the energy. See for example, R. Bluhm, V. A. Kostelecky, and J. Porter, "The Evolution and Revival Structure of Localized Quantum Wave Packets," Am. J. Phys. 64, 944-953 (1996).
8The original mathematical arguments showing how this behavior arises in general wave packet solutions were made by I. Sh. Averbukh and N. F. Perelman, "Fractional Revivals: Universality in the Long-term Evolution of Quantum Wave Packets Beyond the Correspondence Principle Dynamics," Phys. Lett A139, 449-453 (1989) and for the infinite square well by: D. L. Aronstein and C. R. Stroud, Jr., "Fractional Wave-function Revivals in the Infinite Square Well," Phys. Rev. A 55, 4526-4537 (1997).