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PROFESSOR: Very good. So it's time to start. So today, I want to talk about general features of quantum mechanics. Quantum mechanics is something that takes some time to learn, and we're going to be doing some of that learning this semester. But I want to give you a perspective of where we're going, what are the basic features, how quantum mechanics looks, what's surprising about it, and introduce some ideas that will be relevant throughout this semester and some that will be relevant for later courses as well. So it's an overview of quantum mechanics. So quantum mechanics, at this moment, is almost 100 years old. Officially-- and we will hear-- this year, in 2016, we're celebrating the centenary of general relativity. And when will the centenary of quantum mechanics be? I'm pretty sure it will be in 2025. Because in 1925, Schrodinger and Heisenberg pretty much wrote down the equations of quantum mechanics. But quantum mechanics really begins e...

PROFESSOR: So far, so good. We've really calculated. And we could stop here. But there is a nice interpretation of this Darwin term that gives you intuition as to what's really happening. See, when you look at these terms overall-- these interactions with this class-- you say, well, the kinetic energy was not relativistic. That gives rise to this term. There is a story for the spin orbit as well. We think of the proton-- creates an electric field. You, the electron, are moving around. As you move inside the static-electric field, you see a magnetic field. The magnetic field interacts with your dipole moment of the electron. That's the story here. So what's the story for this term? Where does it come from? What is the physics behind it? That's what I want to discuss now. So the physics interpretation of the Darwin term is that the electron behaves not as if it would be a point particle but as if it would be, kind of, a little ball where the charge is sp...

Good. So let's do, then, our transitions. So we do the constant perturbation. Constant perturbation. So as we said, delta H is equal to V, and it's time independent. It just begins at time 0. And we'll examine what's going on by time t0. So what are we going to do? We're going to examine a transition to go from some initial state i, initial state, to a final state f. So we don't have to say much about what the Hamiltonian is or anything. For us, V is going to have a constant. It's going to have some matrix elements that once we do an example you can calculate, but for the time being we need not know too much. So I'm going to use the key formula that was derived already about perturbation theory and how you get the transition amplitude. So we know that the coefficient c associated to the m state to first order in perturbation theory at that time t0 can be computed as a sum over all n integral from 0 to t0 e to the i omega mn t prime delta Hm...

PROFESSOR: So far so good. So here is the kind of very entertaining thing that happens when you try to do some physics with this. And this was done by two physicists, Elitzur and Vaidman in Tel Aviv, they invented or fantasized about some sort of bombs-- things that explode. So they're called Elitzur-Vaidman bombs. And you could invent different things, but here is what an Elitzur-Vaidman bomb is-- some sort of bomb, and the way it works is with a photon detector. So there's a little tube in the bomb, and there's a photon detector. And you have your bomb, and you want to detonate it-- you send the photon in, you send the photon in through the tube. And the photon, it's detected by the detector. And the bomb explodes. On the other hand, if the bomb is defective, the photon goes in, and the detector doesn't work. The photon goes out. Just goes through. So that's an Elitzur-Vaidman bomb. And here is the puzzle for you-- suppose you have those bombs, a...

OK. Good. The final class in linear algebra at MIT this Fall is to review the whole course. And, you know the best way I know how to review is to take old exams and just think through the problems. So it will be a three-hour exam next Thursday. Nobody will be able to take an exam before Thursday, anybody who needs to take it in some different way after Thursday should see me next Monday. I'll be in my office Monday. OK. May I just read out some problems and, let me bring the board down, and let's start. OK. Here's a question. This is about a 3-by-n matrix. And we're given -- so we're given -- given -- A x equals 1 0 0 has no solution. And we're also given A x equals 0 1 0 has exactly one solution. OK. So you can probably anticipate my first question, what can you tell me about m? It's an m-by-n matrix of rank r, as always, what can you tell me about those three numbers? So what can you tell me about m, the number of rows, n, the number of colum...