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PROFESSOR: In order to learn more about this subject, we must do the wave packets. So this is the place where you really connect this need solution of Schrodinger's equation, the energy eigenstates, to a physical problem. So we'll do our wave packets. So we've been dealing with packets for a while, so I think it's not going to be that difficult. We've also been talking about stationary phase and you've practiced that, so you have the math ready. We should not have a great difficulty. So let's new wave packets with-- I'm going to use A equals 1 in the solution. Now, I've erased every solution, and we'll work with E greater than v naught to begin with. The reason I want to work with E greater than v naught is because there is a transmitted wave. So that's kind of nice. So what am I going to do? I'm going to write it this following way. So here is a solution with A equals to 1. e to the i kx plus k minus k bar over k plus k bar...

PROFESSOR: Welcome back to recitation. In this video we're going to do another solid of revolution problem. So what I'd like to do in this problem is to find the volume of the solid generated by rotating the region bounded by the following curves: y equals 0, x equal 4, and y equals square root of x, around the line x equals 6. And you can choose your favorite method to do this. What I would like first, when you're doing this kind of problem, is get a rough sketch of the region. So you have some picture of what's actually going to happen. You don't necessarily need a three-dimensional picture. But at least have the two-dimensional region and understand the where the rotation line is, with respect to that region. So I will give you a little bit of time to work on that problem. And when I come back I'll show you how I do it. OK, welcome back. So again, what we're doing in this video is we're going to be looking, finding the volume of a solid ...

DAVA NEWMAN: We want to make Earth as habitable and livable, and since we're accelerating this change, we want to live in balance with Spaceship Earth. SARAH HANSEN: In today's episode, we're taking another look at our planet from the outside in. DAVA NEWMAN: Simply we're trying to design a healthy relationship between people, technology, and our Earth systems for sustainability of humanity and all the living things on the planet. SARAH HANSEN: I'm your host, Sarah Hansen. Today on Chalk Radio, we're talking with Apollo Program Professor of Astronautics Dava Newman. Professor Newman is an expert in aerospace engineering, and she's used that knowledge to forge new perspectives on climate change. You can check out Professor Newman's course, 16.423J, Aerospace Biomedical and Life Support Engineering, on OCW as well as her climate-focused resources. In short, Professor Newman has worked on some very cool things, including better spacesuits for ...

We have to talk about composition of your images but it's very difficult to separate the idea of composition and point of view, background. And even lighting, they all go together. When you adjust your light -- you'll see later -- you're going to get a different kind of shadow and that shadow becomes part of the composition. So, we've attempted to separate them into three categories, but really they are very closely connected and it's something to keep in mind. So let's start with these leaves, these autumn leaves that I put on a light box, which you will see more of when we talk about light. In fact the light box itself is an important compositional element in this image because it's giving us a very strong negative space between the leaves. And, it could be an interesting thing to play around with, using the background as part of your composition. Something to think about. Taking a look at these crystals, which I just randomly placed on a bac...

Mujtaba: Today, we're going to learn about Ohm's Law. Mujtaba: I'm curious how many of you have heard of or know about Ohm's Law? Student: I know. Student: I do. Student: I've yeah, yeah. I've heard about it. Yeah. Mujtaba: So the simplest way it is represented is V equals IR. That is, the voltage drop, or the potential drop across a resistor in this case is just the current that passes through the resistor times its current. So if we were to draw a resistor right here with a value R, and this is the current, then the voltage drop across this resistor, this V is equal to the current times that resistor value that the resistor has so the resistance value of the resistor and the current path that passes through it. Current is the flow of electrons and V is the potential. So think of it as a force essentially, that allows or enables these electrons to move from an area of higher potential to low lower potential. So if you guys have drunk boba bubble t...

Welcome to a new lecture on the binomial theorem. In this lecture, we will look at certain divisibility results which can be derived based on the binomial theorem. Once again, recall the binomial theorem is given by (x+y)^n = \sum{r=0}^{n}{C_r x^{n-r} y^r}, where we use C_r to denote nC_r. To give you a flavor of the kind of results we will be looking at today, let's look at the first example which asks you to show that 3^{2n+2} - 8n - 9 is divisible by 64 when n is a natural number. It asks you to show that this term is divisible by 64 for natural numbers n. At first sight, there doesn't seem to be a binomial expansion you can apply to generate this result so we would like to create a binomial expansion to derive this result. Notice that you have an exponent here, but you also have some terms here and these terms are, in general, not divisible by 64 for n being a natural number. You can plug in certain natural numbers and you can see that this term, which is bein...

This week we're going to take a look at exactly what you DO with the photographs after you've made them. And, once again, frankly we could devote a full six week course to just this idea alone--presenting your work. But we'll try to cover this topic as best we can in just a couple of tutorials for you for this week. So let's start with a few figures just to dissect them and get you to think more about how you create figures with your images. Let's first take a look at a figure that the researchers made for a grant submission. And here we see a host of the pieces of devices, and the full devices, all labeled. They wanted to show what they were capable of creating--of course which is important. But there's so much going on, the question is whether we want to bother looking anywhere. By the way, just a note, I didn't make any of these images of the research, but I knew that a few of the original images were in color. So first of all, why not use t...

We're going to take a look at how point of view and composition are very clearly connected. But, before we show the connections, we have to sort of define, visually define, what I mean by point of view, for example. So, take a look at these four images. Now, it doesn't look it, but these images are of the identical sample, just simply taken with four different points of view. There, it's true that I also changed the background on them, but, generally, what I did was I moved the sample around or the camera, and we get four really completely different-looking visual ideas of what this stuff was about. Which one is right, I'm not sure there is a right in any of this. That's one of the fun things about this kind of work: There are no hard, fast rules that we have to follow. But I wanted, to give you, this example to show you that just by simply changing the angle or-of your camera, or moving your material around, you see different things. Again, we're ...

PROFESSOR: Welcome to another help session on recombinant DNA. Today, we're going to be discussing about transformation and protein expression. As you can imagine, there are often many times we will need a large amount of protein. But it can be difficult to get it from the original source. For example, you need insulin to treat diabetes, but it's not exactly practical to get a lot of insulin from humans. In order to get a lot of the desired protein, often other organisms will be used to express this protein. But it's a multi-step process. For example, let's say we want to express our human insulin in bacteria. Well, the human gene has both introns and exons, as you remember from lecture. Exons are what are actually cut together in order to produce the final mature mRNA, which is later used to express the protein. Bacteria, on the other hand, don't have introns. They just have exons. So they are only capable of reading a gene that just has the exons and...

PROFESSOR: Welcome back to recitation. In this video what I'd like us to do is practice Taylor series. So I want us to write the Taylor series for the following function, f of x equals 3 x cubed plus 4 x squared minus 2x plus 1. So why don't you pause the video, take some time to work on that, and then I'll come back and show you what I get. All right, welcome back. Well, we want to find the Taylor series for this polynomial f of x equals 3 x cubed plus 4 x squared minus 2x plus 1. So what I'm going to do is I'm just going to write down Taylor's-- or the expression we have for the sum, for the Taylor series in general and then I'm going to start computing what I need and I'm going to see what I get. So what do I need to remember? Well let's remind ourselves what the formula is. We should get f of x is equal to the sum from n equals 0 to infinity of the nth derivative of f at 0 over n factorial times x to the n. So that's what we wan...

Hi. Welcome back to recitation. We've been talking about Taylor series for a number of functions and rules by which you can compute Taylor series. I have here an example that I don't think we did in lecture. So this is the function f of x equals secant of x. Now, unlike some of the other ones you've seen, there's not a really simple formula for the whole Taylor series of secant x. So what I'd like you to do is not to find, you know, a formula for the general term, but rather, just to use some of the tools that we've learned to compute the first few terms of the Taylor series for f of x equals secant x. Say, up through the x to the fourth term, if you wanted, or even a little further if you were feeling ambitious. So why don't you pause the video, have a go at that, come back, and we can do it together. So welcome back. I asked you to compute the first few terms of a Taylor series. One thing you can always do in this case, is you can go and you ...