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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...

When you put a cold drink on the kitchen counter the counter surface temperature will decrease. But, if the cold drink is removed, the counter will eventually return to room temperature. If instead we place a cup of tea on the counter, the counter temperature rises; but if we remove the cup of tea, the counter top eventually returns to room temperature. We say that the counter at room temperature is a stable equilibrium. In this video, we discuss the world from the perspective of equilibrium and stability, and in particular linear stability. This video is part of the Linearity video series. Many complex systems are modeled or approximated as linear because of the mathematical advantages. All the world is an initial value problem, and the matter merely state variables. However, and less poetically, there are alternative interpretations of physical, and indeed social, systems that can prove very enlightening. The purpose of this video is to introduce you to the framework of...

Chemical properties: so let's put something up as a hypothesis. Let's say, well, maybe it has to do with the energy that it takes to remove electrons. And, one other thing that I failed to point out, if you take a look at the energies associated with the outermost electrons, in this case, lithium, you see it's 0.5 megajoules per mole, and then what's the 6.26? That's associated with one, n equals one, inner shell. There is a huge difference between the energies in the outermost shell and the inner shells, which tells you that it's unlikely that any electrons except those in the outermost shell are going to be active. The inner electrons are so tightly bound that they are for all intents and purposes, immobilized when it comes to reactivity. Again, boron is a good example. Look, 0.8, 1.36. This is 2p. This is 2s. Yeah, they're different but they are roughly on the order of about 1 MJ per mole. If you go down to 1s, it's 20 MJ per mole. So, t...

If you put water in the freezer, you'll end up with ice. If you leave ice on your countertop, you'll end up with liquid water. You've almost certainly seen these phase changes in your everyday experience. But there's more to freezing and melting than meets the eye, and we can use these seemingly simple phenomena to make buildings significantly more energy efficient. In this video, we'll explain the concept of "latent heat" and see how it can dramatically reduce heating and cooling costs in homes and skyscrapers alike. This video is part of the Conservation video series. In order to analyze or modify a system, it is important to understand how the laws of conservation place constraints on that system. Hi. My name is Stephen Ray and I am a graduate student in the Department of Mechanical Engineering at MIT. My research in the Building Technology Lab under the guidance of Professor Leon Glicksman focuses on energy efficient buildings. In order t...

Let us now put to use our understanding of the coin-tossing model and the associated binomial probabilities. We will solve the following problem. We have a coin, which is tossed 10 times. And we're told that exactly three out of the 10 tosses resulted in heads. Given this information, we would like to calculate the probability that the first two tosses were heads. This is a question of calculating a conditional probability of one event given another. The conditional probability of event A, namely that the first two tosses were heads, given that another event B has occurred, namely that we had exactly three heads out of the 10 tosses. However, before we can start working towards the solution to this problem, we need to specify a probability model that we will be working with. We need to be explicit about our assumptions. To this effect, let us introduce the following assumptions. We will assume that the different coin tosses are independent. In addition, we will assume...