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PROFESSOR: Levinson's theorem, in terms of derivations, that we do in this course, this is probably the most subtle derivation of the semester. It's not difficult, but it's kind of interesting and a little subtle. And it's curious, because it relates to things that seem to be fairly unrelated. But the key thing that one has to do is you have to use something. How all of the sudden are you going to relate phase shifts to bound states? The one thing you have to imagine is that if you have a potential and you have states of a potential, if the potential changes, the states change. But here comes something very nice. They never appear or disappear. And this is something probably you haven't thought about this all that much. Because you had, for example, a square, finite square well. If you made this more deep, you've got more bound states. If you made it less deep, the bound states disappear. What does it mean the bound states disappear? Nothing, reall...

The central limit theorem is absolutely remarkable. It is a very deep result, and highly nontrivial and non intuitive. There's no apparent reason why this random variable here, a standardized version of the sum of random variables, should have an approximately normal distribution. Furthermore, it is very useful, and one key reason is that it is universal. It doesn't matter what the distribution of the X's is. No matter what the distribution is, still in the limit, this standardized version of the sum is going to behave like a normal random variable. And if we wish to apply it to particular examples or models, the only thing that we need to know about the distribution of the X's are the corresponding means and variances, as we're going to see in multiple examples. When we apply it, it turns out to be very accurate, and it is also a very nice computational shortcut. Even if we knew, in detail, the distribution of the X's, in order to calculate the di...

PROFESSOR: The binomial theorem extends to a thing called the multinomial theorem, whereas instead of taking a product of a sum of two things, you'd take the product of a sum of k things to get the multinomial theorem. And what underlies it is a rule that we're going to call the bookkeeper rule, and here's why. So, the bookkeeper rule is about the question of, look at the word bookkeeper and ask how many different ways are there to scramble the letters in this word that actually are distinguishable? The point being that the two o's are indistinguishable, so the order in which they appear doesn't matter. Likewise, the three e's and the two k's. Well, how do we answer this question? The simple way to do it to begin with is to label all of the indistinguishable letters with subscripts to make them distinguishable. So, I'm going to put subscripts 1 and 2 on the o's, 1 and 2 on the k's, and 1, 2, and 3 on the e's. Now, all the 10 let...