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Today, let's start our little look at history. We've actually been here before is sort of the theme I want to develop, because I want us to put us back as if you are entering the labor force around 1930. You've just come off what we call the roaring '20s. And while you should feel better because the economy was going so well in the 1920s and profits were up and the stock market was at record levels, you probably don't feel very optimistic. Because despite all this good news, family income was falling in the 1920s as it is today. Most of the income went to the top 10%, just as it is today in our society. And then of course 1929 comes along, and we get the Great Stock Market Crash, and everything starts to fall to pieces. At that point, over the next several years, if you were entering the labor force or an existing, more mature worker, you were expected to lose between 8% to 20% percent of your income. If you got unemployed, over 25% of the labor force,...

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 place where you get the intuition about solving Schrodinger's equation. So the stationary states are simple and useful solutions of the Schrodinger equation, very nice and simple. So what are they by definition? Well, we begin with a definition. And the intuition of a stationary state will follow. See the word stationary is not the same as static. Stationary is something that maybe it's kind of moving, but things don't change. Static is something that's just not moving. Stationary states have time dependence. It is very simple, as we will see. So, your definition of a stationary state has a factorized space and time dependencies. So this psi of x and t is a stationary state. If you can write it as a product of a function of time times a function of position. And now, I will try to be careful about thi...

We are going to start with the lecture notes from last Friday because I didn't finish them. What we were doing at the end of lecture last time was probably some of the hardest concepts that we've covered so far in this course, and maybe even in the whole course. And you will remember that we finally got around to talking about what does a wave function mean, and how does it actually represent the electron in our hydrogen atom. And we saw that the physically significant interpretation of the wave function was not actually the wave function itself, but the wave function squared. So psi here, which we learned, has to be labeled by three quantum numbers, n, l and m. And that is a function of three position variables, r, theta and phi. If you square that, the physical interpretation of that is a probability density. This means this has units. This has units of inverse volume. It's a probability per unit volume. That is very important. It's not a probability, bu...

I usually like to start a lecture with something new, but this time I'm going to make an exception, and start with the finishing up on Friday because it involves a little more practice with complex numbers. I think that's what a large number of you are still fairly weak in. So, to briefly remind you, it will be sort of self-contained, but still, it will use complex numbers. And, I think it's a good way to start today. So, remember, the basic problem was to solve something with, where the input was sinusoidal in particular. The k was on both sides, and the input looked like cosine omega t. And, the plan of the solution consisted of transporting the problem to the complex domain. So, you look for a complex solution, and you complexify the right hand side of the equation, as well. So, cosine omega t becomes the real part of this complex function. The reason for doing that, remember, was because it's easier to handle when you solve linear equations. It's m...

So, let's start with where we were. We were talking about exponential growth in populations. And, we said we could describe this as one over the dN/dt equals some growth rate, r. And, in this case, we're talking about, let me ask that is a question. As a model for population growth, what's wrong with this? What does this project? This is N. This is time. There's no stopping it. I mean, we'd be knee deep in everything if populations grew according to this model, OK, because it just goes off into infinity in terms of density. So, we know that this is inadequate. In fact, some people describe the entire field of population ecology as a field that tries to determine why real populations can't grow according to this model. In other words, the whole field is trying to understand what the mechanisms are in populations that limit their growth. So, they don't grow exponentially forever. So, in this case, this is really a maximum growth rate. We can call...