This wavefunction, at , has width
(in units of
) and is localized at the center of the well.
![[Graphics:../Images/index_gr_42.gif]](../Images/index_gr_42.gif)
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
![[Graphics:../Images/index_gr_45.gif]](../Images/index_gr_45.gif)
Here are the magnitudes of the first 25 coefficients--as we can see from the plot, 25 is all we need.
![[Graphics:../Images/index_gr_46.gif]](../Images/index_gr_46.gif)
Question: Why is every other coefficient zero?
Thus, we can write
![[Graphics:../Images/index_gr_48.gif]](../Images/index_gr_48.gif)
Since we know the time dependencies of the , we know the time dependence of
.
Can you predict what will happen?
![[Graphics:../Images/index_gr_51.gif]](../Images/index_gr_51.gif)
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...