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PROFESSOR: OK. Let's turn then to the Dirac equation and motivated, basically. Through the Dirac equation is the simplest way to do perturbation theory for the hydrogen atom. It helps us derive what we should think of perturbation. So I'll discuss that quickly. And there may be some of that done in recitation. So we're going to discuss now the Dirac equation. So the Dirac equation begins with the observation that we have E squared minus p squared c squared for a free particle is m squared c to the fourth. This dispersion relation that rates E and p for any particle. So if you wanted to describe the dynamics of a relativistic particle, you could say, look, the energy is the square root of p squared c squared plus m squared c to the fourth. Therefore, I should take the Hamiltonian to be that thing. I should take H' to be that. And work with a Schrodinger equation that has this H with the square root. Nobody does that, of course. But you can do a little bit w...

PROFESSOR: Let's then try to understand intuitively some issues with phase shift. And that, I think, is pretty valuable. One shouldn't rush through these things. I'm going to erase this formula already. It's a little messy. It's not like you're going to use it ever. You could use that formula if you were to compute higher phase shifts. But at least this homework, you don't have that. So let's try to understand a little what phase shifts do for you. Phase shifts are useful when the first few phase shifts dominate the cross section. If you have to do this sum for all values of l, it can get very difficult. So if the answer really necessitates the tail of l going to infinity, it's hard. So in general, we find that phase shifts are useful when the first few dominate the cross section. So phase shifts are useful if the first few dominate sigma. And that will tend to happen when k a is much less than 1. And this will happen in that case. That...

PROFESSOR: OK. Let me then say a little more about that extra term. Let's rewrite it in a slightly different way so that we get the feeling of what it has to do with Hamiltonians. So this is a coupling term. So we try to understand, what is that coupling psi k, psi n dot? How could we write it? So what is psi k. Psi n dot? To figure this out, we have no option, probably, except looking at the Schrodinger equation-- not really the Schrodinger equation-- the time-instantaneous state conditions-- En of t, psi n of t. So you look at this instantaneous state condition for n and say, OK, I'm going to differentiate, and then I'm going to make the overlap. What else could you do? This is the equation that tells you how the instantaneous energy eigenstate is supposed to change in time. So it must have the information of how to compute this, or how to rewrite it in terms of other things that are interesting. So let's differentiate with respect to time. Differentiate...

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: Our subject today then is Fermi's golden rule. So that's what we're going to develop. Fermi's golden rule. Fermi's golden rule. And this has to do with the study of transitions. And typically, the interesting and sophisticated thing about this subject is that you have a transition from some initial state to a state that is part of a continuum of states. That's what makes it complicated. The transition from one discrete state to another discrete state with a perturbation is kind of a simple matter to do. But when you can go into a continuum, you have to integrate over the set of final states, and that makes it a lot more interesting. So we go from a discrete state into a continuum. And that makes it somewhat challenging. So we will consider this in two forms. It's worth considering the case of what we call constant perturbations. And you might say, well, aren't we doing time dependent perturbation theory? Yes, we are. But this kin...

JUDY HOYT: And then on Thursday, we'll have these. I also want to mention, towards the end of lecture today, we'll have the course evaluations for you to fill out. OK, before we start the formal lecture for today, for those of you who were here last time, I wanted to go over something that we had talked about. We didn't have time to go through how you would do these calculations from the last handout, which was handout 36. On slide 27, there was a couple of questions that we were asking. And I thought we'd just spent about five minutes before we start today's formal lecture going through this, at least on the board, so people have an idea of how to make these back-of-the-envelope calculations. So what you're being asked to do here is you're looking at a silicon MOSFET with-- a these are the source drain extensions, these shallow extensions that have a certain depth XJ. These are the deep source and drain that have been silicided. Remember, the ...