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Heisenberg uncertainty principle

Video transcript

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 we know 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 as an 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 4pi so we have we have a constant divided by another constant so this just gives us a this is gives us a number on the right side and you might see something a little bit different in a different textbook doesn't really matter that much it just depends on how you define things and so the point is the 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 and this really this is just extremely simplified so let's just let's just see if we can understand that idea of inversely proportional so if you had an uncertainty of of two for the position all right and let's say you 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 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 moment and the momentum must increase to 4 because 1 times 4 is equal to 4 if I decrease the uncertainty in the position even Morris if I lower that to 0.5 I increase the uncertainty in the momentum that must go up to 8 so 0.5 times 8 gives us 4 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 right 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 there's a velocity going in that direction all right the reason why the Bohr model is useful is because we allowed us to understand things like quantized energy levels all right and we talked about the radius for the electron was running so if there's a circle here this radius for an electron in the ground state this would be the the radius of the first energy level is equal to 5 point 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 2 so 2 times that number would be equal to one point zero six times 10 to the negative 10 meters and this is just a a rough estimate of the size of the hydrogen atom using using the Bohr model so not with an electron in the ground state all right we we also did some calculations to 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 6 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 can hurt that to a decimal we just we just divide 10 by a hundred so we get ten percent 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 the momentum of that particle momentum is equal to mass times velocity all right if there's a whole if there's a ten percent uncertainty associate 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 right is nine point one one 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 percent 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 9.1 one 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 percent 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 unit's would be this was the mass and kilograms and the velocity was meters over seconds so kilograms times meters per second all right so listen this is the uncertainty associated with the momentum of our electron so let's plug it into our and our our uncertainty principle here so we had the uncertainty in the position of the electron alright times the uncertainty in the momentum of the electron must be greater than or equal to Planck's constant divided by 4 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 ten to the negative 25 this product must be greater than or equal to Planck's constant is six point six two six times ten to the negative 34 right and divide that by 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 two point six times ten to the negative ten and if you worked out your units you would get meters for this so the uncertainty in the position must be greater than equal to two point six times ten to the negative ten 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 a 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 and 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 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 if we had a particle let's make it a much bigger particle here so 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 could probably pretty accurately figure out the velocity and so we know the momentum and and that's true we do know these things fairly accurately but if you did a calculation using the uncertainty principle right so if you plugged in if you plug them some different numbers like if you increase the mass so instead of nine point one one times ten to the negative 31st let's say you're using nine kilograms right and you plugged in some some velocity here and you solve 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 the same idea with quantum mechanics quantum mechanics something makes absolutely no sense when you when you first encounter it that 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 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