 Mae'r cyfeirioneddau yma yn eistedd o bwysig yma yn ei ddiweddio'r cyfodol. Mae'r cyfrannu gynhyrch yn ymddangos a ydy'n pethau trofyniaeth yma yn eistedd i ddweud i gael yma, ac mae'n modd rifnodd companhau ymdwysoedd yr oeddech chi'n greu gwirio'n cyfrannu fydd yn iesaf, ac mae'n cyfrannu ddysgu, mae'n cefwyr ar mwyloedd mewnabb fwyloedd hwnnw, is the piece of physics which is least understood, relativity and so on is perfectly clear and tied up, of course there are, there's a theory of everything or whatever on the frontier of the subject, but here we have a piece of undergraduate physics which is universally agreed is not properly understood. Its deepest un Allanings, it's still fundamentally mysterious and it's also quite extraordinarily specific to physics. ond mae'n unig yw'r rhaid i'r llyfr yn ddylam. Yn gyfnodol yn ffuntimentol, gallw'r bod yn ei gael arall gael y bod yn ymgrifftol, y llyfr, y Llyfr, y bydd. Mae anodol yn y llyfr sydd gan gweithio'r cyfnodol yn y fawr iawn. Ym oedd ymwysig yn fwyroio. Mae'n amlwg yn ymwysig. Mae'n amlwg yn fwyroio. Felly weithio yn glywed o'r rhai yr bwysfyn ystafell. Roedd y mystery yw ystafell dros yma. Felly weithio yn cael ei wneud i'r wneud yr ysgol cyffredinol o'r ffordd yng Nghymru a hynny... ...ad o'r rhai rôl yw chi nesaf y tu yn ei fawr i ni ddefnyddol yn y gwir. Rydyn ni'n gallu fod yn gyffredin расскo Brackie oherwydd ymlaen y gynchderach fydd yr ysgol sy'n du'r beth hynny. Efallai'r dyfodol iawn ysgol ar y gyf newlypus, o'i angen i'r gael. Mynd yw'r wahanol yn ffysiwch, nid yw'r wahanol yn srifyllfa'i unig arfer y fphysics. Mae'n rhan fyddi yn ffysiwch. Maen nhw'n hynny'n ymddiweddol yn ffysiwch. Rwy'n meddwl i'n blwyddyn hwbodaeth yr hyn yn y rhydw i'n blwyddyn. Mae'r ffordd o profhed, o'n profhed. Mae'n ddweud am ei wneud mwy o bobl yn y trogedd, ond hefyd yn oed mwy o bobl yn bros. Ond yw'n deithas i'r wahanol, is to work with the apparatus and think about the meaning of the solutions that you get and so on and so forth. So I really would urge you to work as hard as you can at those problems and the reason for providing solutions to many of the problems is because even if you can solve the problem yourself, you might find it interesting to see how I solve the problem and develop your technique in that way. Of course, quantum mechanics has a very funny way of looking at the world, that's part of the problem. It's by constant practice and experience that you'll deepen that understanding. Einstein, as everybody knows, didn't like quantum mechanics, but I think the reason why he didn't like quantum mechanics, which he expressed as God doesn't play dice, was not a good reason. It may be that one shouldn't like quantum mechanics, but that's not a good reason. Let's just think about that for a moment. Physics is about predicting the future, it's about saying what's going to happen. If you lean the ladder up against a wall, you would like to know if you tread on the ladder whether it's going to slip and fall down, that kind of thing. If the data on which we work are always uncertain and the systems with which we work are never isolated and our theory crudely always applies to physics and apparatus where if you put in certain statements about what the system is, for example with the ladder, what the roughness of the ground is, what the roughness of the wall is, what the weight of the ladder is, and so on and so forth, if you describe the system accurately, then you will get a precise prediction out. But in the real world, not only can you not, there's always uncertainty in the data, you can't say exactly how rough the floor is because the roughness of the floor varies from place to place, you're not quite sure what you put down the ladder and so on, so the data that you're working with are uncertain. So what you should really do, what the best that you can actually do, if you really want to push yourself to the most precise results, is derive probability distributions. You can say that the probability of the ladder slipping from this position is such-and-such, the probability of the ladder slipping from that position is such-and-such. In simple cases, you have a very sharp, you have a very narrow range of probabilities, the probability in certain positions is almost one that it won't slip in other places, it's almost one that it will slip and so we can give a simple answer when we say, well, the critical angle for it slipping is 43 degrees and 34 minutes or whatever else. But if you really, really, really want to know something accurately, if you really want to push your predictions to the extreme or as hard as you can, you will have to calculate a probability distribution. Calculating a probability distribution is hard. In classical physics, it's hard. We will find that in quantum mechanics, it's actually rather easier to calculate probability distributions with the quantum mechanical apparatus than it is with the classical physical apparatus, which is just as well, because in quantum mechanics, we're working on a theory, it arose out of attempts to understand things that are so small that they are always seriously not isolated. So an electron carries a charge, consequently it is always in contact with, it's always interacting with the electromagnetic field. But the electromagnetic field is, it turns out, always, always quivering. So we never know what the electromagnetic field, even under the most precise control of the electromagnetic field, you put your electron inside some resonant cavity, you call the resonant cavity as close to absolute zero as you can and so on. No matter how hard you work, it turns out that electromagnetic field is in an unknown configuration, consequently your electron is subject to uncertain disturbances. Consequently, what the electron is going to do, the best you can do, is predict probabilistically, in the same sense, that when the horses are racing at Sandan Park or whatever, the results are going to be probabilistic. You don't know what a particular horse is going to do on a particular day because of all the, it's not an isolated system, it may have eaten something it didn't approve of at breakfast that morning, etc. So it is natural that we should be working with probabilities. It is natural that the calculation, so whereas in classical physics, when you're talking about a cricket ball or a shell shot out of a howitzer, you operate under the fiction, which is a very good fiction, that at every point in the trajectory, at every time, every precisely measured time, the shell, central mass of the shell has a very precise coordinates and these coordinates progress in a very accurately calculable way. You have only one number to calculate, well three numbers I suppose, the x, y and z coordinates of the shell to calculate it each time. You don't have to calculate a probability distribution, well in the simple case you don't. If you're considering what will happen when electron leaves an electron gun, because of the quivering electromagnetic field, whatever uncertainty there was in the configuration of electron before it, shot out of the gun and so on, it's inevitable that you're calculating a probability distribution for where the electron is going to go and calculating a probability distribution for every possible value of x is clearly going to be a hell of a lot more work than calculating one particular value of x. So that's why it's going to be mathematically complex and why it's going to involve probabilities and let's just remind ourselves of some basic facts about probabilities which I think is this correct that in Professor Blundell's course he's already talked about the laws of probability? Yeah, good. So the things we need to just remind ourselves of is that if we've got two independent events, the probability that we get, the probability that we get, that A and the event B is going to be the product of P A, whoops, P A and P B. So we multiply the probabilities of independent events such as that if you throw two dice, the probability that one die comes up with number one and the other one comes up with number six. So P A might be the probability that the first die, the red one comes up with number one and P B might be the probability that the black die comes up with a number six. Then this is the probability that the red one comes up with one, what did I say, six, whatever. This is the probability of that particular configuration and you get a product and the other rule that's important for us is that the probability of A or B is equal to P A plus P B if they're exclusive events. So that's the probability that if I throw a single die that I get either a one or a six because I can't get both a one and a six simultaneously, I either get a one or I get a six, so these are exclusive events and the probability that I get either a one or a six is just some of these two probabilities. So those are the... and following on from that, if we have a X is a random variable so that's something like what happens when we... like the number we get when we throw a die, then we define a thing called the expectation of X to be the sum of the probability of the Ith outcome times the value that X takes on the Ith outcome and it's sort of roughly speaking, it's often called the average of X, but that is to say if you make a number N of trials and work out the average value that you get of X, you're hoping to get a value you should get a value which is close to this, it will never really agree with this, but the idea is that as you do more and more experiments, the average that you have all those experiments will converge in a... will rattle in a narrower and narrower range around this expectation value. Then we have a few simple rules that if we have two random variables and add their results and then take the expectation value, then that is the expectation value of X plus the expectation value of Y. That's always the case, whether the variables are independent or not. So, and zillions of branches of science use probabilities, right? It's a major feature in medicine, it's a major feature in the financial markets, and they use probabilities in just the same way that physicists do, but physicists have a unique way of calculating probabilities which nobody else uses, and I think this is a central mystery, and that's because in quantum mechanics we calculate these probabilities through amplitudes. That's to say every probability that we're interested in, probability is the mod square of some complex number, its amplitude. It's probability amplitude. So we never calculate this directly, we always calculate a probability amplitude, and having got it, we take its mod, which is a complex number, and we interpret the mod square of that complex number as the probability. And so all of quantum, my purpose in the next few lectures is to persuade you that all of quantum mechanics and all its strangeness follows from this business here, which nobody else uses. There's no other branch of knowledge. There are people in the city who talk about the quantum mechanical or even people who name their hedge funds, quantum, etc. They like to have a connection with quantum mechanics, but it's completely bogus because they never calculate probabilities in this way. Now the consequence of this is that the probability of spoting something can happen by two routes. So let's be specific. Let's suppose that we have an electron gun and we have a double slit arrangement. My drawings are never very good, something like this, and we're firing electrons out of here, sort of in scatterplatin, and some of them go through holes and then hit our detector, our screen over here, scintillator, photographic plate, whatever you want to use, and others bounce. We'll call this S and we'll call this T. If we focus on a particular place X here on the screen, there are two ways in which an electron can arrive there. It can go through the top hole or the bottom hole. And we'll call the path through the top hole, the path S, and the path through the bottom hole, the path T. So what we're interested in calculating is the probability that we get an electron arriving at X. So the probability of arriving at X should be calculated from some amplitude, and the rule is that that amplitude is the amplitude to take the path S plus the amplitude to take the path T. And then, of course, that gives us the amplitude to arrive there regardless. So this is like the probability rule up there, P, A, or B. This is the probability that it got there by either root S or root T is the sum, well, up there it's the sum of two probabilities, but the rule here is the sum of the amplitude for it is the sum of the amplitudes, and the probability is the square of this. What does that give us? That gives us, because we know how to take the mod square of two complex, the sum of two complex numbers, this is AS mod squared plus AT mod squared plus AS AT complex conjugate plus AS complex conjugate AT. So that stuff follows just from the ordinary rules for taking the amplitude of the sum of two complex numbers. But this we know is the probability that it got there through S. So that's P that it took root S plus P that it took root T plus this stuff, which can be, this stuff here can be written as twice the real part of AS A star T. So the probability that something happens when it can happen in two mutually exclusive ways because it either goes through the top hole or it goes through the bottom hole is the sum of the probabilities that it took either root plus this funny stuff down here. That's a consequence of calculating probabilities using amplitudes and this fundamental principle that if something can happen by this way then you add the amplitudes, you don't add the probabilities. Nobody knows why that's the right rule. You should reasonably ask me so how do I know that's the right rule? And the answer I think the proper answer to that question is that this is the fundamental cornerstone of quantum mechanics and our civilisation quite simply depends on quantum mechanics because we're all busy communicating with each other using electronics that has been designed using quantum mechanics. So, of course there are particular specific experiments that one could talk about but really it's not as persuasive as the point that without this quantum mechanics would make no sense and without quantum mechanics our civilisation would fall apart. Let's think a little bit more about this. What do we think that these individual probability distributions look like? In other words, if you covered up one of these things and we're just firing your bullets through one hole what would you imagine? Well, of course, your electrons, your bullets, your particles through one hole what you'd imagine was that the probability would be largest on the place which was formed by a straight line from the centre of the muzzle of the gun through the hole to the screen. So you would expect that p s look like so I want to draw a plot of this is going to be x I guess I better put this is x is 0 I would expect that p s look something like this some kind of vaguely Gaussian so it's most likely to arrive this is the point which is the geometrical is the intersection of the straight lines through the middle of the muzzle and some width because the slit has some width the muzzle has some width and doesn't fire bullets exactly in one direction but in some spray of directions and we would expect that p of t correspondingly was the same thing on the other side of the origin right so if these Gaussians are very narrow we're expecting that p p of x at some location here say if we chose this place we'd find that p of t was about equal to 0 p of s p s was some number here so this vanished this amplitude would vanish because this is the mod square of whatever complex number it is that sits underneath and the term would disappear and we would find, guess what, surprise surprise that the probability of arriving at x was indeed equal to the probability of arriving through s but suppose these things are now interested in the more interesting case where these are really broad distributions and this is a really broad distribution whoops really broad distribution then there will be places where there's a non-negligible amplitude coming from both sides and in fact by symmetry it's evident that at the origin in the geometrical middle of the screen there will be equal amplitudes coming from the equal probabilities expected from both sides so in this neighbourhood we're expecting that that this number is about equal to this number and these two numbers have comparable magnitudes so let's in fact write a of s is equal to mod a let me put a subscript on it mod a s e to the i phi s so this is a complex number that's the funny quantum mechanical thing so it has an amplitude this is a quantum amplitude but it has an amplitude in the sense of complex numbers a modulus sitting in front here and then it must have some phase up there similarly we'll write that a t is equal to a t e to the i phi t and both and everything here will be a function of position down on the screen this amplitude depends on where you are on the screen this does and we expect that this does we expect all of these bits of the complex number depend on position but when we're in the middle here so near centre of screen we're expecting that the modulus of a s is about equal to the modulus of a t because this is the square root of the probability of getting there and this is the square root of the probability of getting there through s only and we can't see any difference between the two so what does the combined probability look like then p of x is on the order of it's about equal to 2 times the probability of getting through shall we through s because p t is about equal to p s but now we're going to have plus twice s we're going to have a s mod squared no no right because we're saying that up there we've got a of s times a star t but we're saying that the modulus of a s is about equal to the modulus of a t so I can just put in a that product just becomes this times e to the i sorry times the real part of e to the i phi s minus phi t but this we recognise as the probability p s so this is about equal to 2 p s 1 plus in the real part of this of course is the cosine phi s minus phi t so this is what we're expecting at relation and it's only valued near the middle but the conclusion of this the implication of this rule for adding for adding the amplitudes is that the probability is a function of position near the centre is going to be what you would naively expect so this is the classical result the classical result is the probability of arriving there is twice the probability of getting there through either one of the slits each slits contributing the same probability but this is now being multiplied by 1 plus cosine of this totally quantum mechanical bit and this bit is called a quantum interference term and the extraordinary so the prediction is since this cosine so this difference will calculate well this difference between the phases is later on we can't put a number on it at the moment but we do expect phi s and phi t to be functions of position and so by default we have to expect that this thing is varying with position and as the cosine as the argument of the cosine varies with position through you know goes through 0 and 2 pi and so on cosine is going to go from 1 to minus 1 and this probability of arrival is going to go from nothing to twice the classical probability 4 times p s so what we're expecting is that at the end of the day p of x is going to do some kind of oscillation this is only valid in a small region of x but it is an unexpected it is surely a surprising result so this is 2 times classical probability and this is 0 so that's phenomenon of quantum interference is an inevitable consequence of this extraordinary rule for adding amplitudes and calculating probabilities from the sum of the amplitudes rather than adding the probabilities that is what makes quantum mechanics special and that is a phenomenon which nobody else uses probability encounters the need to do this only physicists encounter this need that I think is the real mystery how are we doing okay of course we have to ask why is it that this if you farm machine gun bullets through slits and stuff we're not expecting to find that there's a safe place to stand every yard or every millimetre or any distance these places where no machine gun bullets are going to arrive we don't believe exist and you have to ask the question why not and the answer we will calculate the answer later on but the answer is going to be that as the mass of the particles your firing goes up from the mass of an electron up to the mass of a bullet this pattern stays the same but it gets more and more and more and more compressed and there's this distance between places where it's safe to stand get smaller and smaller and smaller and smaller until it becomes ludicrously small in the case of machine gun bullets and when you make any measurement when you make any measurement with machine gun bullets you inevitably average over the places where the bullets are extremely likely to arise twice as likely to arise as in classical physics and the places where it's safe to stand so you inevitably average over these places and you end up with this average you're unable to measure anything but this average nobody has figured out a way to measure this anything but this average in the case of machine gun bullets so that's how we recover classical physics the quantum mechanics is asserting that there really are these places where it is safe to stand if you were small enough okay so now let's have a slightly let's talk about quantum states so my claim is that essentially everything follows from what we've already covered but it's all a consequence of this interference business through using probability amplitudes instead of probabilities but now we have to have some apparatus so we have got some we have in our lab some system some thing that we're trying to investigate so in the simplest case it would be a particle a spinless particle and let's fantasise about spinless particles so that's particles which do not have any that aren't gyros let's fantasise about them although it turns out that spinless particles are very rare things like electrons and neutrons and protons even are little gyros so if we had a spinless particle we could it's a system it's a dynamical system and you can ask yourself so how do I characterize the state of this particle well there are things you characterize its state of course by measuring something and what can you measure you can measure the x, y and z coordinates you can measure the px, py and pz momentum you could measure its energy you could measure its angular momentum these are all things that you could measure so there's a range of things that you could measure and in quantum mechanics these things you could measure are all called observables then you characterize the system by saying what results you would get if you made these measurements remember we've accepted that there's a probabilistic aspect so we don't expect to be able to say that if I measure x I will get the value 3.1415963 whatever right meters I expect to have to come clean and say well I don't know there's a probability distribution I think it's about round here that's just how life is going to be so what do you do what you do of course is you specify the quantum amplitudes to obtain certain results of measurements so we characterize the system the state of our system by measuring by giving quantum amplitudes to possible outcomes of measurements I think that's pretty reasonable and it turns out in quantum mechanics that the possible outcomes are sometimes but not always restricted so if you have an electron which is free to wander the universe then the possible outcomes of its x coordinate can be values from minus infinity to plus infinity all real numbers are on and the range of possible values are what are called the spectrum so the possible outcomes the numbers you can get they form the spectrum so the spectrum of x is generally minus infinity to infinity which is not a very interesting I mean so there's no interesting restriction there similarly the spectrum of px the momentum in the x direction is usually the same but the spectrum for example of the z component of angular momentum jz turns out to be only discreet values it turns out that we'll show that this is the case that you can have numbers like dot dot dot comma k minus one h bar k h bar k plus one h bar k plus two h bar and so on where k is equal to either for a particular particle it's either equal to naught or equal to a half so the spectrum can be discreet set of numbers or it can be a continuous set of numbers this is a property of the observable the spectrum of the energy is often a discreet set of numbers not always e0 e1 e2 that you have to calculate by hard grind and we'll spend a great deal of time calculating the spectrum of H it turns out to be a key to find out what that is for a particular system so all these observables have spectra and how you would characterise the state of the system if you were talking about its energy is you would give the amplitude so we could give we could possibly specify state of our system by giving the amplitude to get e1 sorry e0 the lowest energy the next energy above the amplitude etc so let's call this a0 a1 so the idea here is that for some systems if you know the complex number a0 who's a mod square gives you the probability if you would measure the energy that you've got and you also knew this number a1 whose mod square is this probability and if you knew this number whose mod square was the probability of getting the nth energy level and so on in general there will be an infinite number of these if you knew all of these amplitudes you would completely know you would have completely specified the dynamical state of that system what do I mean by that what I mean by that is if I knew all of those amplitudes I could calculate the amplitude to find any other amplitude that you might inquire about for example I could find the amplitude to find my system at the place x or I could calculate from those amplitudes I could calculate the amplitude to find that the momentum is the value p so we have the concept here of a set of amplitudes it's clear I hope it's clear you will need a set of amplitudes to define the state of a system in quantum mechanics in classical mechanics what do you need to know you need to know for a particle you need to know x and p x, y and z and px, py and pz because then you've pinned down where the thing is and how fast it's moving and when you know that you're all done six numbers done because from that you could calculate the energy but in quantum mechanics life isn't going to be so simple because we've agreed that you probably don't know what x is and you probably don't know what px is the best you can hope to know is what these probability distributions are and we've agreed that these probability distributions are for reasons that nobody understands going to be defined in terms of these complex numbers the quantum amplitudes whose mod squared give the probabilities so knowing specifying a complete set of information is is a matter of writing down a long list unfortunately of quantum amplitudes the good news is that you don't need to know all possible you don't need to write down all possible amplitudes quantum amplitudes because there are rules which we're going to develop for calculating from a complete set of quantum amplitudes all other quantum amplitudes that might be of interest and we'll do a concrete example probably next time maybe we are already maybe we do have time to just do this ok so let's have a look at this so I said that electrons and protons and neutrons and quarks a huge number of elementary particles have gyros so they have an intrinsic spin they are gyroscopes they have an intrinsic spin and they're called spin-a-half particles for reasons that will become apparent in a moment let's just use this we'll develop the theory of this properly next term but I want to use this as an example of a complete set of amplitudes and what it enables you to do ok so total angular momentum of these particles is always the same they spin at a certain rate so that they have an angular momentum that's root three quarters of h bar where h bar is Planck's constant over two pi so that's the amount of spin they have and they just have that spin and you can never change it it's always the same but what does happen is that the direction that this angular momentum points in changes so whereas the total angular momentum is this angular momentum in some particular direction for example the z direction if you measure it it turns out that you can only get two answers plus or minus a half of h bar and moreover there is an amplitude a plus is the so let this be the amplitude to measure that equals a plus a half h bar and obviously a minus will be the amplitude to measure that j z is minus a half h bar now in ordinary talk what we say is what everybody says and you'll find me saying this but it's immoral I shouldn't is that if j z is plus a half h bar its spin is pointing upwards and you imagine it to be a little particle it's going upwards and when j z is minus a half h bar you say it's pointing downwards now this is a fundamental mistake because if you square a half h bar and take the square root you don't get that this three indicates that actually this particle has a quarter of h bar it has a half h bar associated with the x and y directions as well so it's actually not a good idea to think of it as spin up as having a spin pointing upwards the most that we can say is that really it's pointing sort of not downwards it's pointing vaguely up and this one is pointing vaguely down I don't really know which way it is in the x y plane so that's just a little word of caution people get themselves into a real tangle by imagining this means the spin is up and that means the spin is down and you'll find me saying that but just when you find yourself saying that just have a little trip in the brain which says hang on a moment I mustn't take that too literally because it does have angular momentum in the x and y directions even though I've measured j z and found it up in j z or j z and I found it down ok so the good news is that the set a plus comma a minus is a complete set of amplitudes and what that means is if I know those two complex numbers if I know both of those two complex numbers I can calculate the amplitude and therefore the probability to find the particle with its spin in any direction that I want that you specify is either plus a half h bar in that direction or minus a half h bar in that direction and we'll work that out in some detail so so from these we will maybe I want to write the formula down I'm not sure moment with the notes I don't think we do yet we're not ready to write that down we just want to make that statement that it's a complete set of amplitudes in the sense that we will derive rules such that we can calculate b plus is a function of a plus and a minus and this is the amplitude to measure j in some direction theta to be and the theory consists so what the theory of quantum mechanics is about it's about finding the rules which enable you to calculate the amplitude for an event that you want to know what's the probability of something happening given your current state of information which is a complete set of amplitudes for something else to happen that's what the apparatus consists of and is there to do so I think that it probably is an appropriate moment to stop because the next section requires a bit of space which we shouldn't take now