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Heisenberg uncertainty principle
- [Voiceover] The Heisenberg uncertainty principle is a principle of quantum mechanics. And so if we take a particle, let's say we have a particle here of mass m, moving with velocity v, the momentum of that particle, the linear momentum is equal to the mass times the velocity. And according to the uncertainty principle, you can't know the position and the momentum of that particle accurately at the same time. So if you know the position, if you know where that particle is in space, really well, you don't know the momentum, or you don't know the velocity of that particle, and vice versa. If you know the momentum really well, you don't know the position. So let's look at a mathematical description of the uncertainty principle. So the uncertainty in the position, so delta x is the uncertainty in the position, times the uncertainty in the momentum, so delta p is uncertainty in momentum, the product of these two must be greater than or equal to some constant. And that constant is Planck's constant, h, divided by four pi. So we have a constant divided by another constant. So this just gives us a number on the right side. And you might see something a little bit different in a different textbook, it doesn't really matter that much. It just depends on how you define things. And so the point is, the product of the two uncertainties must be greater than or equal to some number. So the uncertainties are inversely proportional to each other. If you increase one, you decrease the other. Let's go ahead and use some really simple numbers here, just so you can understand that point. So let's say, and this is just extremely simplified, so let's just see if we can understand that idea of inversely proportional. So if you had an uncertainty of two for the position, all right, and let's say you had an uncertainty of two for the momentum, all right. So two times two is equal to four. And so I won't even worry about greater than, I'll just put equal to here. So if two times two is equal to four, if I decrease the uncertainty of the position, so I decrease it to one, so the uncertainty in the momentum must increase to four, 'cause one times four is equal to four. If I decrease the uncertainty in the position even more, so if I lower that to 0.5, I increase the uncertainty in the momentum, that must go up to eight. So 0.5 times eight gives us four. And so, what I'm trying to show you, here, is as you decrease the uncertainty in the position, you increase the uncertainty in the momentum. So another way of saying that is, the more accurately you know the position of a particle, the less accurately you know the momentum of that particle, and that's the idea of the uncertainty principle. And so let's apply this uncertainty principle to the Bohr model of the hydrogen atom, and so let's look at a picture of the Bohr model of the hydrogen atom. Right, we know our negatively charged electron orbits the nucleus, like a planet around the sun. And let's say the electron is going in this direction, so there's a velocity associated with that electron, so that there's a velocity going in that direction. All right, the reason why the Bohr model is useful is because it allowed us to understand things like quantized energy levels. All right, and we talked about the radius for the electron, right, so there's a circle here, this radius for an electron in the ground state, this would be the radius of the first energy level, is equal to 5.3 times 10 to the negative 11 meters. So if we wanted to know the diameter of that circle, we could just multiply the radius by two. So two times that number would be equal to 1.06 times 10 to the negative 10 meters. And this is just a rough estimate of the size of the hydrogen atom, using the Bohr model. So with an electron in the ground state. All right, we also did some calculations to figure out the velocity, right. So the velocity of an electron, in the ground state, of a hydrogen atom using the Bohr model, we calculated that to be 2.2 times 10 to the sixth meters per second. And since we know the mass of an electron, we could actually calculate the linear momentum, so the linear momentum p is equal to the mass times the velocity. Let's say we knew the velocity with a 10% uncertainty associated with that number, so a 10% uncertainty, if we convert that to a decimal, we just divide 10 by 100, so we get 10% is equal to 0.1. So we have 0.1 here. If I want to know the uncertainty of the momentum of that electron, so the uncertainty in the momentum of that particle, momentum is equal to mass times velocity. All right, if there's a 10% uncertainty associated with the velocity, we need to multiply this by 0.1. So let's go ahead and do that. So we would have the mass of the electron, right, is 9.11 times 10 to the negative 31st. The velocity of the electron is 2.2 times 10 to the sixth, and we know that with a 10% uncertainty, so we need to multiply all of that by 0.1. So let's go ahead and do that. We're gonna multiply all those things together. So we take the mass of an electron, 9.11 times 10 to the negative 31st. And we multiply that by the velocity, 2.2 times 10 to the sixth, and we know there's a 10% uncertainty associated with the velocity, so we get an uncertainty in the momentum of 2.0 times 10 to the negative 25. So the uncertainty in the momentum is 2.0 times 10 to the negative 25. And the units would be, this was the mass in kilograms, and the velocity was in meters over seconds. So kilograms times meters per second. All right, so this is the uncertainty associated with the momentum of our electron, so let's plug it in to our uncertainty principle here. So we had the uncertainty in the position of the electron, all right, times the uncertainty in the momentum of the electron, must be greater than or equal to Planck's constant, divided by four pi. So we can take that uncertainty in the momentum, and we can plug it in here. So now we have the uncertainty in the position of the electron in the ground state of the hydrogen atom, times 2.0 times 10 to the negative 25. This product must be greater than or equal to Planck's constant is 6.626 times 10 to the negative 34. All right, and divide that by, four pi. So we could solve for the uncertainty in the position. So delta x must be greater than or equal to 2.6 times 10 to the negative 10, and if you worked out your units, you would get meters for this. So the uncertainty in the position must be greater than or equal to 2.6 times 10 to the negative 10 meters, let's go back up here to the picture of the hydrogen atom. 2.6 times 10 to the negative 10 meters, that's greater than the diameter of our hydrogen atom. So the uncertainty would be greater than this diameter, so the uncertainty in the position, would be greater than the diameter of the hydrogen atom, using the Bohr model. So the Bohr model is wrong. It's telling us we know the electron is orbiting the nucleus at a certain radius, and is moving at a certain velocity. The uncertainty principle says this isn't true. If we know the velocity fairly accurately, we don't know the position of the electron. The position of the electron is greater than the diameter according to the Bohr model. So this is just one reason why the Bohr model is wrong. But again, we keep the Bohr model around, because it is useful as a simple model, when you're just starting to get into chemistry. But this concept of the uncertainty principle goes against our natural intuition. So our everyday life doesn't really give us any experience with the uncertainty principle. For example, if we had a particle, let's make it a much bigger particle here, so a much bigger particle than an electron, so something that we can actually see in our real life. And so this has a much bigger mass, and moving with some velocity, logic tells us we can figure out pretty accurately where the position of that object is, and we can probably pretty accurately figure out the velocity, and so we know the momentum. And that's true, we do know these things fairly accurately. But if you did the calculation using the uncertainty principle, right, so if you plugged in some different numbers, like if you increased the mass, so instead of 9.11 times 10 to the negative 31st, let's say you're using nine kilograms. Right, and you plugged in some velocity here. And you solved for the uncertainty in the position, you're gonna get an uncertainty in the position that's extremely small, so you don't really notice those things on a macroscopic scale. So you only notice them when you think about the atomic scale. And so that's why this isn't really an intuitive concept. Same idea with quantum mechanics. Quantum mechanics is something that makes absolutely no sense, when you first encounter it. You have no experience with quantum mechanics, in your daily life. It just doesn't make any sense. You don't see these sorts of things. So this is just showing you an application at an atomic scale. And again, this is the uncertainty principle. We'll get more into quantum mechanics, and how quantum mechanics affects electrons and atoms in the next few videos.