The wavefunctions above are eigenfunctions, so their probability density plots are independent of time. This is not the case for a superposition of eigenstates.
Example 1: 
Here is the simplest case of a superposition of eigenstates--equal parts of 
and 
Question: Can you calculate the period of oscillation?
Example 2: Gaussian wave packet
It is intereresting to make a connection with the classical case.To do this, consider a Gaussian wave packet.
We choose a Gaussian wave function because
(1) It is localized in space
(2) In free space, a Gaussian wave function travels without changing shape
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...
Part 2:
We'll give the wave packet an initial "kick" of momentum 
.
We calculate the coefficients as before:
Question:Explain why the magnitudes of the coefficients 
peak where they do?
(Hint: using the eigenenergy 
, calculate a charactaristic momentum associated with this state)
Here's one with smaller momentum
