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PROFESSOR: I've put on the blackboard here the things we were doing last time. We began our study of stationary states that are not normalizable. These are scattering states. Momentum eigenstates were not normalizable, but now we have more interesting states that represent the solutions of the Schrodinger equation, that are stationary states with some energy e. Because they are not normalizable, but we cannot directly interpret any of these solutions as the behavior of a particle. I kind of tell you a story, OK. So this is-- a particle is coming, colliding, doing something. These are not normalizable states. So part of what we're going to be trying to do today is connect to the picture of wave packets and see how this is used to really calculate what would happen if you send in a particle of potential. Nevertheless, we wrote a solution that has roughly that interpretation, at least morally speaking. We think of a wave that is coming from the left, that's ae to...

so I've got to complete some thoughts here about the refinement equation which dominated Lector 27 and bring in wavelets as the differences between one the coar scale and the fine scale V not is contained in V1 but there's some more stuff in there and that's the details because you know V1 has details in it that vot doesn't have and it'll be the wavelet space you not that shows us those details then to discuss some applications some of the many applications and good a very good question was asked in the before the lecture how did we get how did do ingred doesi who whose great paper was in 1988 how what question was she asking herself and what answers did she find okay the question she came out of physics and uh had friends in seismology so for example a question that was in the air was what kind of a basis could seismology just use to represent a a fast pulse that stops and 48 basis was not any good for that the signal was uh was not suited to function...

so um so I've written up on the board as usual the topics uh or some of the main topics uh for this lecture uh it it certainly will connect to the next one uh so this is uh least squares estimation and uh uh so I'm interested here in waiting in the waiting Matrix the the role of The covariance Matrix and in the recursive part uh but the number of parameters the parameters I'm looking for you hat well I'm looking for you and I settle for you hat is not changing I mean it's is improving but but uh we're we're at a fixed time so to speak so we've got these various steps Le squares hi I I think I started a moment early but I've written I haven't said anything yet that isn't already written um so this we know about this this this waiting Matrix I'll introduce early and then this recursive aspect so that's the those are the themes of this lecture and then uh on into the next lecture we'll come a dynamic problem in which th...

[MUSIC PLAYING] JOHN DOLHUN: I've done these demos in front of live audiences. I remember one time I did one demo in front of alumni. I forget what it was, but it was a failure. And this one little kid-- he was only about four years old-- he raised his hand. He said, you win some and you lose some. SARAH HANSEN: Today on Chalk Radio, we're talking about the course 5.310, Laboratory Chemistry. This is a course that not only teaches students about hands-on science, but also life. Because working in a lab is a lot like baking-- you can follow recipes, but your cake might still collapse. I'm your host Sarah Hansen. In this episode, we're joined by two MIT educators who have helped shape this standout course. They'll tell us more about what it looks like to support students with various academic interests as they develop lab skills, overcome setbacks, and apply their studies to real-world problems like water quality. JOHN DOLHUN: I'm John Dolhun, curren...