Behavior of the differential equation
PROFESSOR: Now, what should happen? Somehow, this equation probably has solutions for all values of the energy, but those solutions diverge and are not normalizeable. It's the kind of thing you will find with a shooting method that you're doing with your computer. Count the solution and it suddenly diverges up or diverges down and cannot be normalized. But for some specific values, it can be normalized. So what we need is an intuition why this differential equation has normalizeable solutions, only if the energies, curly E, take specific values. That's intuition that we need. For that, we'll look at the equation a little closer, and try to understand what happens at the place where it can get in trouble, which is large X. That's where you expect it to get in trouble. So what does this equation become as u goes to plus minus infinity? Well this equation, at that stage, becomes like this. The second phi, the u squared is roughly equal to u squared phi. B...