PROFESSOR: Square well. So what is this problem? This is the problem of having a particle that can actually just move on a segment, like it can move on this er...

PROFESSOR: Last time we talked about particle on a circle. Today the whole lecture is going to be developed to solving Schrodinger's equation. This is very...

BARTON ZWIEBACH: --that has served, also, our first example of solving the Schrodinger equation. Last time, I showed you a particle in a circle. And we wrote th...

PROFESSOR: Would solving this equation for some potential, and since h is Hermitian, we found the results that we mentioned last time. That is the eigenfunction...

PROFESSOR: How about the expectation value of the Hamiltonian in a stationary state? You would imagine, somehow it has to do with energy ion states and energy....

PROFESSOR: We start with the stationary states. In fact, stationary states are going to keep us quite busy for probably a couple of weeks. Because it's a p...

PROFESSOR: This definition in which the uncertainty of the permission operator Q in the state psi. It's always important to have a state associated with me...

PROFESSOR: Uncertainty. When you talk about random variables, random variable Q, we've said that it has values Q1 up to, say, Qn, and probabilities P1 up ...

PROFESSOR: Let me do a little exercise using still this manipulation. And I'll confirm the way we think about expectations values. So, suppose exercise. ....

PROFESSOR: That brings us to claim number four, which is perhaps the most important one. I may have said it already.b The eigenfunctions of Q form a set of bas...

PROFESSOR: So here comes the point that this quite fabulous about Hermitian operators. Here is the thing that it really should impress you. It's the fact t...

PROFESSOR: It's a statement about the time dependence of the expectation values. It's a pretty fundamental theorem. So here it goes. You have d dt of ...

PROFESSOR: Expectation values of operators. So this is, in a sense, one of our first steps that we're going to take towards the interpretation of quantum ...

PROFESSOR: We got here finally in terms of position and in terms of momentum. So this was not an accident that it worked for position and wave number. It works...

PROFESSOR: What we want to understand now is really about momentum space. So we can ask the following question-- what happens to the normalization condition tha...

PROFESSOR: Time evolution of a free particle wave packet. So, suppose you know psi of x and 0. Suppose you know psi of x and 0. So what do you do next, if you...

PROFESSOR: We'll begin by discussing the wave packets and uncertainty. So it's our first look into this Heisenberg uncertainty relationships. And to be...

Three-dimensional case. Now, in the future homework, you will be doing the equivalent of this calculation here with the Laplacians-- it's not complicated--...

BARTON ZWIEBACH: After this long detour, you must think that one is just trying to avoid doing the real computation, so here comes, the real computation. The r...

PROFESSOR: Let's do a work check. So main check. If integral psi star x t0, psi x t0 dx is equal to 1 at t equal to t0, as we say there, .Then it must ho...

BARTON ZWIEBACH: We were faced last time with a question of interpretation of the Schrodinger wave function. And so to recap the main ideas that we were lookin...

PROFESSOR: interpretation of the wave function. --pretation-- the wave function. So you should look at what the inventor said. .So what did Schrodinger say? ...

PROFESSOR: This is very important. This is the beginning of the uncertainty principle, the matrix formulation of quantum mechanics, and all those things. I wan...

PROFESSOR: ih bar d psi dt equal E psi where E hat is equal to p squared over 2m, the operator. That is the Schrodinger equation. The free particle Schrodinger...

PROFESSOR: This is a wonderful differential equation, because it carries a lot of information. If you put this psi, it's certainly going to be a solution. ...

PROFESSOR: Last time, we talked about the Broglie wavelength. And our conclusion was, at the end of the day, that we could write the plane wave that correspond...

PROFESSOR: So what are we trying to do? We're going to try to write a matter wave. We have a particle with energy e and momentum p. e is equal to h bar om...

PROFESSOR: Let me demonstrate now with plain doing the integral that, really, the shape of this wave is moving with that velocity. So in order to do that, I bas...