Page 13 - Physlets and Open Source Physics for Quantum Mechanics:

Appendix

Wave Packet Revivals: Theory

Quantum-mechanical wave packet revivals have recently received considerable
theoretical and experimental attention [29]. Most often the theoretical research has
focused on initially localized wave packets in the infinite square well (ISW) because of
its well-known time scales: the classical time scale and the revival time scale. We
examine the time dependence of such an initially localized state by choosing a
standard Gaussian wave packet of the form

(A1)

where by direct calculation: <x> t = 0 = x 0, <p> t = 0 = p 0, and ∆x t = 0 = ∆x 0 = b/√2. The
wave packet can be constructed from a sum of energy eigenstates as:

(A2)

2
where the expansion coefficients satisfy Σ n |c n| = 1. The expansion coefficients are
determined by an ‘overlap’ integral of the individual energy eigenstates with the initial
Gaussian wave packet.

We find that over time the wave packet spreads in a characteristic way defined by the
2
spreading time, t 0 = mb /ħ. For longer times, we have the so-called classical period,
T cl, and the revival time, T rev, which are related to each other by:

2
T rev = 2πħ/|E''(n 0)|/2 = 2πħ/E 0 = 4mL /πħ = 2n 0 T cl. (A3)

The classical periodicity for the ISW agrees with the classically-expected result (T cl =
2L/v which corresponds to the time it takes a classical particle to traverse the well
twice). The so-called initial ‘bounce’ at T cl/4 with one wall is of interest because of the
similarities and differences between the classical and quantum-mechanical cases [21,
22]. Also of interest in the context of the ISW, but on a considerably longer time scale
than the classical time scale, are the well-known exact, mirror, and fractional revivals
[26-28] in which the wave packet or copies (sometimes called ‘mini-packets’ or ‘clones’)
of the original wave packet reform long after the original packet ‘collapses.’ The
analysis of these problems often requires specialized visualization techniques because
of the long times involved.

References

[1] E. Cataloglu and R. Robinett, “Testing the Development of Student Conceptual and
Visualization Understanding in Quantum Mechanics through the Undergraduate
Career,” American Journal of Physics, 70, 238-251 (2002).

[2] M. Belloni and W. Christian, “Physlets for Quantum Mechanics,” Computing in
Science and Engineering 5, 90 (2003). The Physlet-based quantum mechanics
exercises from this paper are available at: http://webphysics.davidson.edu/cise_qm.