Hi all, a couple of weeks ago, I was reading a book (Eisberg and Resnick) in which one of the questions asked was:

Basically, the book doesn't give the answer and I don't know it. I also can't work it out, despite the past two weeks of wracking my brain over this problem. Can anyone offer a resolution? Thank you!

First of all, in a stationary state, nothing "moves" in the sense of "changes".
Next, the notion of trajectory has a totally different meaning in quantum physics than it has in classical physics.

As such, you shouldn't picture a stationary state (described by the wavefunction you mention) as a kind of density of "presence" of a classical particle moving around on a certain trajectory, which is what is implicitly assumed in this question.

So, the purpose of the question is to make me realise what a stupid question it is..?
It seems very bizarre that there would be these locations in the potential where there is zero probability density. Is this merely because I have no intuition for quantum situations like this? Is it anything at all to do with the particle interfering with itself, thus in a way (I know this isn't quite the best way of looking at it) creating the standing waves found as solutions? In the same way that electrons, unobserved, passing through young's slits "interfere" with each other and create minima of the wave function...

I read somewhere on here that it is this same "interference" which causes probability densities for electons around nuclei to have high values here and low values there.

The electrons do not interfere with each other. You get an interference pattern even if you shoot one electron through the apparatus at a time, for a long enough period of time that you get a large number of total hits on the "screen" or whatever detector you're using.

An example: particle in the one dimensional well.
1. If we don't change the well in time, the situation is stationary and particle doesn't "move".
2. If we change the well or change the potential in time, then the "zero's" can move or even "zeros" could come to an end (psi function may by complex function in general case).

For example when we change the length of the well specifically in time we can get as a result 1 zero function (cooling) or 3 zero function (heating). "Zeros" will be borned or killed at the boundaries of the well.

Zero probability means nothing. We can have FLOW of probability in time from the left and from the right and they may kill each other.