Heisenberg's Uncertainty Principle - Part 1 of 2

hello today we're going to look at Heisenberg's uncertainty principle which generally relates two things at the atomic and subatomic level Eisenberg's uncertainty principle is usually written in this form Delta x times Delta P is greater or equal to H bar over 2 what does that mean well Delta X talks about the uncertainty in position if you want to know for example where an electron is how clear can you be how precise can you be about where it is Delta P talks about the uncertainty of that electrons momentum in other words its velocity and its direction and the idea is that the uncertainty in position multiplied by uncertainty of momentum is greater than h-bar over 2 so you cannot precisely define position and momentum at the same time here is let's say an electron that electron has a certain position X and it has a certain momentum P but we cannot measure both of them at the same time the reason for this is that electrons even atoms are very small compared with anything that you're going to use to measure them for example let's say we used light to look at an electron well here's a light wave which goes right past the electron without even noticing it in fact I haven't called this to scale at all because a light wave has a wavelength of approximately 5 times 10 to the minus 7 meters whereas even if this were an atom it would be approximately 10 to the minus 10 meters if it were a proton it would be 10 to the minus 15 meters and if it real Ektron it would be something like 10 to the minus 18 meters in other words even an at is a thousand times smaller than the wavelength of this light a proton would be about a hundred million times smaller than the wavelength of this light so any visible light will just go straight past the electron or the proton or the atom without noticing it if you want to actually see something of the size of an atom or a proton or electron you have to use some form of radiation whose wavelength is broadly comparable to the size of the thing you're trying to see and therein lies the problem because the smaller the wavelength the larger the momentum it's given by a formula that says that the momentum of a wave equals H Planck's constant divided by lambda and so you can see that as lambda gets smaller P gets larger where does this formula come from well we can derive it in a simplistic way we take the famous formula Reinstein e equals mc-squared and we say that momentum is classically given by mass times velocity now when we're talking about electromagnetic radiation like light photons which are the constituent parts of that light don't have a menace but we can say that the mass of a photon is kind of equivalent to e over C squared from this formula here mass is a over C squared and so you can say that the mentum is the mass e over C squared times the velocity which is C and that gives you II over C but you'll know that energy of an electromagnetic wave is HF Planck's constant times the frequency of the radiation and that is known as the packet of energy the photon packet of energy the quantized energy and so now we can say that P the momentum is e which is HF divided by C but C over F is lambda and so we derived the formula we start with P equals H over lambda and now we can see the problem but if we have an electron or a proton or whatever and we send in very low wavelength electromagnetic radiation in order to be able to see this proton it's going to have such a high momentum that although the wave may well reflect and we can detect it it will give a huge kick to this electron or proton and send it scurrying away is rather like hitting a beard ball with a super-fast billiard ball that knocks it for six and so you can't tell what its position and its momentum are we can perhaps explain this by reference to the so-called single slit experiment this is where you take a single slit very narrow and you pass ordinary light through it and what happens is that the light beam comes through the slit but as it goes through the slit it spreads out onto a screen and what you find is that the intensity of the light on the screen goes something like that in other words at these points it's virtually zero up here it's very high and it doesn't get very much bigger thereafter so it's a kind of a spread out across the screen and the spread is at an angle theta why does it do that well if we magnify the slit we can see what's actually happening here here is the slit of dimension D and here is the wave that is going up to reach this point here in other words the point where there's no light at all now if we drop a perpendicular here this angle as we've said is Theta that's that angle there and by geometry this angle here is also theta now what will cause these two waves when viewed by my eye here to cancel out by cancel out I mean that one wave will look like this and the other way will look like this and when they are superimposed they will simply produce a flat line which is the no light that you get here well the answer is that they have to be completely out of phase and how can that happen well that happens if this distance here from here to here is lambda because that's the difference between the phase of this wave and this wave why is that the case well let's blow this up again here is that famous triangle this is D this is this gap here this is lambda that's there and take a point half way now this is lambda over two and those this wave and this wave are going to be a half a wavelength apart and so they are going to be in precisely this position of superimposing so the light there will cancel the light there and the light there will cancel the light there because they're all lambda over two apart and similarly every position here will cancel a position here every point here every point there will cancel there every point there will cancel there so all the points in the upper half cancel all the points in the lower half and produce this minimum here and you can see that from this diagram here that lambda equals D sine theta so if you know the wavelength of the light you know what the angle is where you're going to get a minimum you can do the same thing instead of using light you can do the same thing with electrons

Channel: DrPhysicsA

Views: 170,817

Rating: 4.8652425 out of 5

Keywords: Heisenberg's, Uncertainty, Principle

Id: yrVi24pp_6I

Length: 8min 50sec (530 seconds)

Published: Sat Feb 18 2012