Particle In A Half Box

A question was recently asked in Physics Forums about how A particle in a half box changes over time. I thought it would be interesting to see how a finite difference time dependent Schrödinger solver I am working on handles the example. It provides the community with something interesting to play with.

How do you think that this wave function will evolve over time? To see if you are right, click the ▸ button to start the simulation, and see what happens.

Particle in a box. ◼ stops the simulation, ▸ restarts it. Other buttons set initial conditions when the simulation is stopped. Green is

Ψ^{*} Ψ

, red is the real part of the wave function, and purple is the imaginary part.

We can also check out some standard wave functions.

Button	Effect
$Ψ =$ Particle in a half box	Sets up the wave function from the physics forum question.
$Ψ =$ Gaussian wave packet	Sets up a wave function roughly corresponding to a free particle.
$Ψ = Ψ_{n}$	Sets $Ψ = n^{th}$ energy eigenfunction.

Check them out and see what you can learn from their different behavior. For example, $Ψ = Ψ_{1}$ , is extremely low energy so the change over time is very slow.

This is a numerical approximation, and if you look closely you can see that sometimes a little bit of the wave function leaks out of the box. In part this is because I can not use a true infinite value for the potential, but for compatibility with mobile devices I am limited to approximately 30,000 eV. Of course I will be looking for ways to improve this in the future.