Page 9 - Physlets and Open Source Physics for Quantum Mechanics:

In Figure 9 we show the short-term dynamics for a Gaussian wave packet (x 0 = 0.5, p 0
= 40π, and ∆x 0 = 0.05) in an infinite square well. As time goes on (from left to right in
the figure), the wave packet moves to the right, bounces, moves to the left, bounces
again, and moves to the right. At the same time the packet is spreading. Over one
classical period the packet has already spread so much that it covers the entire extent
of the well. Notice that at the times of T cl/4 and 3T cl/4 the wave packet is colliding with
one of the infinite walls. Unfortunately from this depiction, we cannot discern any of the
interesting (and perhaps unexpected) features of the collision with the wall.

Figure 10: The time evolution of the same quantum wave packet in shown in
Figure 9 between t = 0 and t = T cl (T rev is set to 1) using three different OSP
programs. Here we suppress the animation of the wave packet and show the
expectation value of position (QMSuperpositionExpectationXApp), expectation
value of momentum (QMSuperpositionExpectationPApp), and the quantum
carpet (QMSuperpositionCarpetApp) visualizations instead.

In Figure 10, we are again showing the short-term classical time scale, but now with
three different programs based on QMSuperposition (*ExpectationXApp,
*ExpectationPApp, *CarpetApp) that visualize the expectation value of position, the
expectation value of momentum, and the quantum carpet. All three of these
visualizations show the entire time evolution from t = 0 and t = T cl. Notice that the
expectation value of position appears to be decreasing in amplitude over time. This
effect is a direct result of packet spreading. In addition, there is a ‘softening’ of the
collision with the wall as compared to what one expects classically. This ‘softening’
also occurs for the expectation value of the momentum, since it obeys Ehrenfest’s
principle d<x>/dt = <p>/m. In addition, by zooming in on either the position or the
momentum expectation values at the classical collision time, T cl/4, one can see that the
packet has a negative momentum in contradiction to the classically-expected result of
zero. The position-space quantum carpet also shows the short-term time evolution of
the packet, but it shows a space-time diagram (time vs. position) for the wave function.
The colors represent the phase of the wave function and hence the lines of constant