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PROFESSOR: So we go back to the integral. We think of k. We'll write it as k naught plus k tilde. And then we have psi of x0 equal 1 over square root of 2pi e to the ik naught x-- that part goes out-- integral dk tilde phi of k naught plus k tilde e to the ik tilde x dk. OK. So we're doing this integral. And now we're focusing on the integration near k naught, where the contribution is large. So we write k as k naught plus a little fluctuation. dk will be dk tilde. Wherever you see a k, you must put k naught plus k tilde. And that's it. And why do we have to worry? Well, we basically have now this peak over here, k naught. And we're going to be integrating k tilde, which is the fluctuation, all over the width of this profile. So the relevant region of integration for k tilde is the range from delta k over 2 to minus delta k over 2. So maybe I'll make this picture a little bigger. Here is k naught. And here we're going to be going and integrate ...

STEPHEN CARPENTER: We think about conventional activities to expand thematic and conceptual exploration. So the traditional ways or maybe more conventional ways you might think about art education or learning studio practices-- sculpture, working with clay, working with wood, working with paint, or drawing practices-- those are fine if we want to develop a set of skills. But what about thinking about concepts and themes and issues? Well, we might as well get started and do some of that now. So we're going to make a little transition. I'm going to set up a still life over here. And we're going to do some drawing. The range of ways in which people draw is perfectly acceptable, perfectly acceptable. So if you're concerned that it's tough for you to draw a conclusion or draw a straight line, don't be concerned by that. It's the process that we're interested in, not the visual representation. We have the three chairs in the center. With your mar...

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...

Hi! I'm Pritish Today, we will be talking about Trigonometry. This is the first video on Trigonometry. I would like to start with "Why do we want to study Trigonometry?" and "What is Trigonometry?" You would certainly know that Euclidean geometry is very fundamental to how we think about nature. The way we visualize things in Physics, for example, we use Euclidean geometry in the way we think. And often, Euclidean geometry requires some amount of innovative thinking to reason about problems. And often when you want to talk about Physics, you don't want geometry to be your bottleneck in thinking about Physics. So Trigonometry and Coordinate Geometry are two subjects which try to make a more systematic study of Geometry. And what is "Trigonometry"? Trigonometry is about a study of angles in an "algebraic" way. You would feel more comfortable doing Algebra Algebra is something you can do in a more mechanical way. Whereas, Geome...

In this problem, we will be helping Romeo and Juliet meet up for a date. And in the process, also we'll review some concepts in basic probability theory, including sample spaces and probability laws. This problem, the basic setup is that Romeo and Juliet are trying to meet up for a date. And let's say they're trying to meet up for lunch tomorrow at noon. But they're not necessarily punctual. So they may arrive on time with a delay of 0, or they may actually be up to 1 hour late and arrive at 1:00 PM. So the other thing that we assume in this problem is that all pairs of arrival times-- so the time that Romeo arrives paired with the time they Juliet arrives-- all of these pairs are equally likely. And I've put this in quotes, because we haven't really specify exactly what this means. And we'll come back to that in a little bit. The last important thing is that each person will wait for 15 minutes for the other person to arrive. If within that ...

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 omega. So you can get the omega of the wave. And p is equal to h bar k. You can get the k of the wave. So de Broglie has told you that's the way to do it. That's the p and the k. But what is the wave? Really need the phase to-- how does the wave look like? So the thing is that I'm going to do an argument based on superposition and very basic ideas of probability to get-- to find the shape of the wave. And look at this possibility. Suppose we have plane waves-- plane waves in the x plus direction. A particle that is moving in the plus x direction. No need to be more general yet. So what could the wave be? Well, the wave could be sine of kx minus omega t. Maybe that's the de Broglie wave. Or maybe the de Broglie wave is cosine of kx minus omega t. But maybe it's neither one of them. Maybe it is an e t...

PROFESSOR: Last time, we spoke about photons in the context of an interferometer. The Mach-Zehnder interferometer. And we saw the very unusual properties of photons and interference, and how relatively simple interference a effect can be used to produce a very surprising measurement. Today we're going to backtrack and go from the beginning, and think about photons as physicists did 100 years ago, and how, by thinking about photons, they pretty much came up with quantum mechanics. So we want to trace this back. And the best place to start, probably, is with a photoelectric effect. The photoelectric effect is an experiment done by Hertz in 1887, in which he irradiated plates. That means shine light, high energy beams of light, on metal plate, and he found that electrons were released. Those were called photo electrons. And therefore, you would get a photoelectric current from those electrons. So this is the effect we want to discuss now, is the photoelectric effect. And...

Hi. In this problem, we have an absent-minded professor who will inadvertently give us some practice with exponential random variables. So the professor has made two appointments with two students and inadvertently made them at the same time. And what we do is we model the duration of these appointments with an exponential random variable. So remember, an exponential random variable is a continuous random variable that takes on non-negative values, and it's parametrized by a rate parameter, lambda. And the exponential random variable is often used to model durations of time-- so time until something happens, so for example, in this case, time until the student leaves or the appointment is over. Or sometimes you will also use it to be as a model of time until something fails. And one thing that will be useful is the CDF of this exponential random variable. So the probability that it's less than or equal to some value, little t, is equal to 1 minus e to the minu...

STUDENT: Rainbows-- while we may not notice them, there are rainbows all around us as we go about our daily lives. They are hidden in the reflection of soap bubbles, the shine on a CD, and even in oily puddles on the street. In this video, we'll explore the phenomenon of light wave interference and how it creates the colors we see on the surfaces of thin films. Since soap and oil are usually colorless, why do they have iridescence? Let's start with the laws of light reflection and refraction, then peruse through some visual simulations of these fundamental principles. When waves travel through space and hit an interface or a surface, some of the wave reflects off the surface and the rest refracts, continuing through the new medium at a different angle. When the refracted wave hits another surface, part of it reflects back out of the medium and combines with the first wave or interferes with the first wave. Since the first and second reflected waves travel differen...

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 very grateful to V and the V studio for their inspiring and generous Presence at Cs and MIT and we are looking forward to the presentation this season V aoni is one of the most important figures in our architecture today from his days as the poet in the early 60s to his sem performance work in the 70s to the large scale architectural projects in recent years our country has pushed from one discipline to the next approaching each as a kind of challenging new vocation this is perhaps uh a unique occasion in which V will speak present uh uh the work together with the members of from the studio in his presentation uh I assume VTO they tell us among other things this story of his poetic philosophical Journey from art in art gallery through art in public space to art as architecture fashion and to other fields of art of design the three members of his seren Bast G ricardi and Lunes will hopefully speak talk about the project they have been involved with uh the Creative E...