Before I ask my question, I have to stress: I have absolutely no idea what the math is going on. I've read my textbook, several Wikipedia articles, scoured the internet, and don't feel anymore knowledgeable after hours of reading. I am frustrated beyond belief—I apologize if the quality of this post decreases because of that. Here's the question I'm working on (but any illumination of the surrounding concepts is much appreciated; the question itself doesn't have to be answered at all, I'd just like to have some idea of what I'm doing):
A finite square well 1.0 fm wide contains one neutron. How deep must the well be if there are only two allowed energy levels for the neutron?
I know that Schrödinger's Wave Equation is:
Which is all fine and dandy... but what is it? I understand what it is mathematically, but that isn't telling me anything of why it's useful. Did Schrödinger just plug De Broglie's relationships into the wave equation, call it a day, and get famous? What is this a wave of? What's it supposed to be? This equation is so out of all context for me that just working with it is made more difficult. My textbook says:
The reasoning followed by Schrödinger is somewhat difficult and not important for our purposes.
But then goes on to throw around some constants and make some assumptions about $\Psi(r,t)$ in order for it to... preform some undisclosed magical wave-matter-related things.
From what I've read, all I can gather is that—in solving this problem—I have to construct wave-equations for various potentials inside and outside of some "well"? What is the well? There's something about even and odd parity, which I think refers to there needing to be a whole-number of wavelengths within that 1 fm range (why?), but otherwise I'm pretty much dead in the water.
This isn't a Quantum Mechanics class in particular; it's a general "Modern Physics" class. So far, the whole of the class seems to be randomly tossing out equations, constants, and relationships without context, derivation, or motivation—this confusion has become a very unfortunate (and frustrating) standard for the class. (And I despise the textbook. Help me. :l)