Time evolution of a particle in an ISW after the well

[Ratpigeon]

Homework Statement

How does the state of a particle in an ISW evolve with time after the width of the well doubles - from a to 2a
If the particle starts in the ground state of the half width well, then immediately after the well doubles it will be undisturbed therefore the initial wave function is
|P(0)>=Integrate[Sqrt[2/a] Sin[3.1415... x/a], {x,0,a}] |x>
THe evolution operator is
|P(t)>=Exp[-i E t/hbar] |P(0)>

Homework Equations

The Attempt at a Solution

Does this mean that the solution is just stitching those two equations together? I plotted it and it came out that the solution just went up and down in the half well (or, if I changed the integration limits, it came out at just the second excited state of the full well).
I was under the impression that some sort of spreading had to occur, but I'm not sure how to get it...

 

[TSny]

What are the new energy eigenstate wavefunctions for the expanded well? These become the states that have a simple time evolution.

[EDIT: I don't understand this expression: |P(0)>=Integrate[Sqrt[2/a] Sin[3.1415... x/a], {x,0,a}] |x>]