Jimster41 said:
Trying to picture what happens if you suddenly increase the size of the box.
This is actually a fairly common exercise for students studying the one-dimensional particle in a box. Basically you start by assuming that the wave function ##\Psi(x,t)## does not change at the instant the box increases in size. If it was originally in the ground state, it goes from this:
to something like this:
Which doesn't look very interesting, does it? But, just let some time elapse!
In the original box, ##\Psi## is an energy eigenstate ##\psi_k(x) e^{-iE_k t / \hbar}## with a fixed energy. The probability distribution ##|\Psi|^2## maintains the same shape, so we call it a "stationary state".
In the new box, ##\Psi## is a superposition (linear combination) of the new energy eigenstates, e.g. :
for the new ground state and first excited state. Each of these eigenstates oscillates at a different frequency, so the probability distribution of the superposition does not maintain the same shape, that is, it is not a "stationary state." The probability distribution "sloshes" around inside the new box as time passes, starting from the probability distribution of the original wave function.
To see what this actually looks like for a specific case, you have to work out the coefficients A
k of the linear combination that expresses the spatial part of the original ##\psi(x)## in terms of the spatial part of the new energy eigenstates: $$\psi(x) = \Sigma {A_k \psi_k^\prime(x)}$$ Then you can find the new time-dependent wave function $$\Psi(x,t) = \Sigma A_k \psi_k^\prime(x) e^{-iE_k t / \hbar}$$ and the new probability distribution $$P(x,t) = |\Psi(x,t)|^2$$
and get yourself squared up. Moreso IMHO than regular question threads which can be pretty chaotic and often you are just trying to figure out what the conversation is about. Or you are having to frame a question without such lucid context, and are not even sure if your terms are right.
