Quantum numbers and orbitals
Heisenberg uncertainty principle
- 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 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 another textbook. It doesn't really matter that much, it just depends on how you define things. 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 have an uncertainty of two for the position, and let's say you had an uncertainty of two for the momentum. Alright, so two times two is equal to four, 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, because one times four is equal to four. If I decrease the uncertainty in the position even more, so if I lower that to point five, I increase the uncertainty in the momentum, that must go up to eight. So point five 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. So let's look at a picture of the Bohr model of the hydrogen atom. Alright, we know our negatively charged electron orbits the nucleus, like a planet around the sun. And, let's say the electron is going this direction, so there is a velocity associated with that electron, so there is velocity going in that direction. Alright, the reason why the Bohr model is useful, is because it allows us to understand things like quantized energy levels. And we talked about the radius for the electron, so if there's a circle here, there's a 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, with an electron in the ground state. Alright, we also did some calculations to figure out the velocity. 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 six meters per second. And since we know the mass of an electron, we can 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 point one. So we have point one 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. If there's a 10% uncertainty associated with the velocity, we need to multiply this by point one. So let's go ahead and do that. So we would have the mass of the electron 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 10% uncertainty, so we need to multiply all of that by point one. 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 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 is the mass in kilograms, and the velocity was in meters over seconds, so kilograms times meters per second. Alright, so this is the uncertainty associated with the momentum of our electrons. Let's plug it in to our uncertainty principle here: we had the uncertainty in the position of the electron, 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. Alright, 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, let's go ahead and do that math. So we have Planck's Constant, 6.626 times 10 to the negative 34, we divide that by 4, we need to divide that also by pi, and then we need to divide by the uncertainty in the momentum. So we also need to divide by the uncertainty in momentum, that's 2.0 times 10 to the negative 25, and that gives us 2.6 times 10 to the negative 10. So the uncertainty in the position must be greater than or equal to 2.6 times 10 to the negative 10 and if you worked our 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 it's 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 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 intuitions. So our every day 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 a calculation using the uncertainty principle, 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, 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. 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. 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.