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PROFESSOR: Here is your potential. It's going to be a smooth, nice potential like that. V of x. x, x. And now, suppose you don't know anything about the energy eigenstates. Now, this potential will be assumed to be symmetric. So here is one thing you can do. You can exploit some things that you know about this potential. And here's the wave function that we're going to try to plot. And we could say the following. Let's see. Whatever energy are here, for bound states, I'm going to eventually be in the forbidden region. So far on the right here, I will be in the forbidden region. And I must meet the wave function that looks like the forbidden region wave function. And the only possibility is something like that. You could say it's the [INAUDIBLE] of 1, but actually, if it's a [INAUDIBLE] then I could multiply by minus 1 and use this one conventionally. That wave function is always going to be like that over there. On the other hand, very impo...

PROFESSOR: Delta function potential. So it's still a one-dimensional potential-- potential is a function of x. We'll write it this way-- minus alpha delta of x, where alpha is positive. So this is a delta function in a negative direction. So if you want to draw the potential-- there's no way to draw really nicely a delta function. So you just do a thick arrow with it pointing down. It's a representation of a potential that, somehow, is rather infinite at x equals zero-- but infinite and negative. It can be thought of as the limit of a square well that is becoming deeper and deeper. And, in fact, that could be a way to analytically calculate the energy levels-- by taking carefully the limit of a potential. It is becoming thinner and thinner, but deeper and deeper, which is the way you define or regulate the delta function. You can imagine the delta function as a sequence of functions, in which it's becoming more and more narrow-- but deeper at the same ...