Part 1:

This wavefunction, at , has width (in units of ) and is localized at the center of the well.

In order to calculate the dynamics, we need to write this wave function as a superposition of eigenstates of the well. The coefficients are given by

Here are the magnitudes of the first 25 coefficients--as we can see from the plot, 25 is all we need.

Question: Why is every other coefficient zero?

Thus, we can write

Since we know the time dependencies of the , we know the time dependence of .
Can you predict what will happen?

The wave packet has no net momentum , thus, the center of the wave packet does not move. However, since we know the approximate position of the packet at , by the uncertainty principle, there must be a spread ) in the momentum. Thus the packet spreads out, although it eventually reforms. (This is known in the lingo as "collapse and revival.")

Question: What is the period of oscillation in this case?

For the wave packet to go somewhere, it needs to have intial momentum. Let's add some...