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PROFESSOR: So, adiabatic results. So last time we just followed and tried to use an instantaneous eigenstate to construct the solution of the Schrodinger equation. Our result was that we couldn't quite construct the solution of the Schrodinger equation. What we wrote didn't exactly solve the Schrodinger equation. But we claimed that it was important and interesting. And therefore let me remind you of what we said. So we showed that the wave function, psi of t, could be written as a constant here, e to the i theta of t, e to the i gamma of t times this wave function. And that wave function here was what we called an instantaneous eigenstate of the Hamiltonian. So H of t acting on psi of t was, at any instant of time, an eigenstate. Nevertheless, we said that these eigenstates are not solutions of the Schrodinger equation in general. They solve this funny time-- inspired by the time-independent Schrodinger equation, this instantaneous eigenstate condition. But when ...

PROFESSOR: Let's do adiabatic evolution really now. Evolution. We're going to say lots of things, but the take away message is going to be the following. We're going to get maybe even confused as we do this, but the take away message is the following. You sort of begin in some quantum state, and you're going to remain in that quantum state as it changes. All the states are going to be changing. The quantum states are going to be changing in time. And you're going to remain on that quantum state with an extra phase that is going to have important information. That's basically all that's going to happen. There's a lot of subtleties in what I've said, and we have to unmask those subtleties. But you're going to remain in that state up to a phase. That phase is going to be called something, Berry's phase. And there's a dynamical phase as well that is simple and familiar, but Berry's phase is a little less familiar. So you...