Know. This is a very special subject. Quantum mechanics is is quite unique in in your undergraduate training in the sense that is the piece of physics, which is it is the great intellectual accomplishment of the last century. It is the piece of physics which is least understood, relativity and so on. But is is perfectly clear and tied up. Of course there is the theory of everything or whatever on the frontier of the subject.
But here we have a piece of undergraduate physics which is, it's universally agreed, is not properly understood in its deepest underpinnings. It's still fundamentally mysterious and it's also quite extraordinarily specific to physics, and yet it completely underpins it's absolutely fundamental. You cannot understand anything about your body, the table, the sun, anything. Everything is a manifestation of quantum mechanics in a very direct way. So so it's an extraordinarily exciting subject.
And I think it's what's particularly exciting for you should be that there's that there's are pieces of this still to be put in place. There are mysteries here still to be resolved. And I think there's no reason why the person who does that or people who contribute to doing that should not be here in the audience today. I don't think it can be done by people of my generation. It's looks as if it's it's something that still needs to be done by people coming along with a fresh look at it.
So it's extraordinarily worth working out. Physics isn't easy, and quantum mechanics is one of the hard bits of physics. This course isn't easy, but it's it's extremely well worth working at. And I would stress the importance of working on the problems. There are masses of problems, right? Too many problems. There may be you will think when you look at the problem sets, which you just some abstract to the problems.
There are already too many problems but it is it is the way to learn and understand about physics is to is to work with with the apparatus and think about the meaning of the solutions that you get and so on and so forth. And 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, is because you would even if you can solve the problem yourself, you might find it interesting to see how I solved the problem, you know, 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. And it's by its it's by constant practice and and experience that you'll deepen that understanding.
Okay. So 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. Then maybe 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. Quantum mechanics. Sorry. Physics is about predicting the future. It's about saying what's going to happen if you lean a ladder up against a wall.
You would like to know if you try in a letter whether it it's going to slip and fall down, that kind of thing.
And if the data on which we work are always uncertain and the systems with which we work and never isolated, and our theory crudely always applies to something which for which we physics is an apparatus where if you put in certain statements about what the system is, for example, with a ladder, what the roughness of the ground is, what the roughness of the wall is, what the weight of the latter 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 in the data. You can't say exactly how rough the the floor is because the roughness on the floor varies from place to place. You're not quite sure where you put down the ladder and so on. So the data that you're working with are uncertain.
So what you should really do 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 that of slipping from that position is such and such. In simple cases, you have a very sharp you have 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 and 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 to the to the extreme or to as hard as you can, you will have to calculate the probability distribution and calculating the probable distribution is hard. In classical physics, it's hard, and we will find that in quantum mechanics is actually rather easier to calculate, probably solutions.
With a quantum mechanical apparatus than it is with the classical physical operations, which is just as well because in quantum mechanics, where we're we're working on the 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 in contact with it's always interacting with the electromagnetic field. But the electromagnetic field is, it turns out, always, always quivering.
It's so we never know what the electromagnetic field, even 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 Sandown 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. It's not an isolated system. It may have eaten something. It didn't approve of a breakfast that morning, etc. So 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 a shell shot out of a out of a howitzer, you 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 count. Well, three numbers, I suppose the X, Y and Z coordinates of the shell to calculate it each time. Uh, you don't have to calculate a probability distribution. Well, in simple. In the simple case, you don't.
What if you're considering what will happen when electron leads 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 if you're calculating a probability distribution for the electron is going to go, and calculating a probability distribution for every possible value of X is clearly
going to be a [INAUDIBLE] of a lot more work than calculating one particular value of X, right? So that's 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 Blondell this course, he's already talked about the laws of probability.
Yeah. Good. So the things we need to we need to just remind ourselves, obviously, if we've got two independent events, the probability that we get, the probability that we get the event A and the event B is going to be the product of what's A and P. So we multiply the probabilities of independent events. Such as that. If you throw to dice the probability that one day comes up with number one and the other one comes up with number six.
So might be the probability that the first die, the red one comes up with number one. And PB might be the probability that a black guy comes up with the number six. Then this is the probability that the red one comes up with one, which they say six, whatever. This is the probability of that particular configuration. Okay. And you get you get a product. And the other rule that's important for us is that the probability of A or B is equal to 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. 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 the sum of these two probabilities. So those are the and following on from that, if we have an X is a random variable.
So that's something like what happens when we like the number we get when we throw a die and then we define the thing called the expectation of X. To be the sum of the probability of the if outcome times the value that X takes on the 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 end of trials, work out the average values 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 of all those experiments will converge in a rattle in a narrow or narrow range around this expectation value.
And we have a few simple rules that if we have to 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 events or the variables are independent or not. So. And zillions of branches of of science use probabilities. Right. It's a major feature in medicine, major feature in the in the financial markets.
And they they use probabilities in just the same way the 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. P is the mod square of some complex number. It's amplitude. It's probability amplitude.
So we never calculate this directly. We always calculate a probability amplitude. And having got it, we take it, which is a complex number. And we interpret the mod square of that complex number as the probability. And so, so all of quantum I've got the purpose. My purpose in the next few lectures is to persuade you that all of quantum mechanics and all its strangeness follows from this. From this business here, which nobody else uses this. There's no other branch of knowledge.
You know, 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, with, with quantum mechanics, but it's completely bogus because they never calculate probabilities in this way. Okay. Now the consequence of this is that the probability of supposing something could happen by two routes, right?
So let's, let's be specific. Let's suppose that we have an electron gun and we have a double slit arrangement. Perhaps I draw drawing drawings. Never. Very good. Something like this. And we're firing electrons out of here, sort of in scatter pattern. And some of them go through holes and then hit our detector off screen over here centilitre, photographic plate, whatever you want to use and others bounce. Oops. And then so we'll call this s and we'll call this T.
There are two. 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 the path through the top hole, the path s and the path through the bottom hole, the path T. So if you what we're interested in calculating is the probability that we get the 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, right? So this is like the probability rule up there, P, A or B, this is the probability that it got there by either route S or route t is the sum well up there. It's the sum of two probabilities.
But the rule here is this the sum of the amplitude for is just the sum of the amplitudes, and the probability is the square of this. What does that give us? That gives us because we now undertake the mod square of two complex, the sum of two complex numbers. This is a mod squared plus a t squared plus a s a t complex conjugate plus a s complex conjugate a t. So that stuff follows just from the ordinary rules for taking the complex, the amplitude of 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 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 or it goes through the bottom hole is the probability that it is the sum of the probabilities that it took,
i.e. the 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 or by that 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 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 it's of course, there are Kong there are particular specific experiments that one could one could talk about.
But really, it's not as persuasive as the as as the point that without this, quantum mechanics would make no sense. And without quantum mechanics, our civilisation would fall apart. Yeah. Okay, so. So let's think a little bit more about this. Um, what do we think that these individual probability distributions look like? In other words, if you, 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, you know, your electrons, your bullets, your particles through one hole. What we would imagine was that the majority that that that the that the probability will be largest on the place which was formed by a straight line from the centre of the the muzzle of the gun through the hole to the screen. So you would expect. That piece looked like. Um, so I wanted to draw a plot of this is going to be X, I guess I better put this is X is nought.
I would expect that look something like this, some kind of vaguely Gaussian, you know, 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, in the middle of the whole right. And there's some width because the slit has some width of the muzzle, has some wit some width and doesn't far it 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. Um, so if these Gaussians are very narrow, we're expecting that P but P of X at some location here, say if we chose this place, we'd find that T was about equal to zero P of p of s was some number here. So this vanished, this amplitude would vanish because because this is the mod square of whatever complex number it is that sits underneath.
And so this term would disappear and we would find, guess what, surprise, surprise that the probability of arriving in X was indeed, indeed equal to the probability of arriving through S. But suppose these these things, and now we're interested in the more interesting case where these are these are really broad distributions and this is a really broad distribution, but very broad distribution. Okay. 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 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 that this number is about equal to this number. And these two numbers have comparable magnitudes. So let's in fact write a of as is equal to model A, let me put a subscript on it. Model S E to the I phi s okay.
So this is a complex number. That's, that's a funny quantum mechanical thing. So it has a it has an amplitude in the technical sense. So 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, i t and both. And everything here will be a function of position down on the screen.
Right? This 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 are expecting that the modulus of ACE is about equal to the modulus of 80 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 only this route 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 two times the probability of getting through shall we. Through through s because it's about equal to yes. But now we're going to have plus twice we're going to have a s mod squared. But you know what we. Right. Because. Because we're saying that that up there, we've got a of's times a start.
But we're saying that the modulus of ace is about equal to the modulus of 80. So I can just put in a, uh, that price just becomes this times e to the I try to three times the real part of e to the I phi is minus five t. But this we recognise is the probability piece. So this is about equal to two piece, one plus. And the real part of this of course is the cosine bias minus five t. So this is what we're expecting only it's only an approximate relation and it's only valid near the middle.
But, 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. Right. The classical result is the probability of arriving. There is twice the probability of getting there through either one of the slits, because each slits contribute in the same probability.
But this is now being multiplied by one plus cosine of this totally quantum mechanical bit. And this bit is called the quantum interference term. And the extra. So so the prediction is since this cosine so this difference will calculate well this difference between the phases is later on. We can't we can't put a number on it at the moment but. But we do expect fire and fight to be functions of position. And so by default, we have to expect that that this thing is varying with position.
And as the cosine is the argument, the cosine varies with position through, you know, go through not to pi and so on. The cosine is going to go from from 1 to -1. And this probability of arrival is going to go from nothing to four times the classical sorry to twice the classical probability four times. So what we're expecting is that, 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 it is an unexpected.
It is surely a surprising result. So this is two times classical. Probability and this is zero. So that's that's the phenomenon of quantum interference is a is an inevitable consequence of this ordinary 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 something is a phenomenon which doesn't see and 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 every yard or every millimetre or any or any distance in 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 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 the bullet, the pattern, this pattern stays the same, but it gets more and more and more and more compressed.
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 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 found that you're not able to measure anything. But this average, nobody has figured out a way to measure this anything but this average in the case of things like machine gun bullets. So that's how we recover classical physics. But 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, that it's all a consequence of this interference business through using probability amplitudes instead of probabilities. So but now we have to have some apparatus. So we have got some we we have in a lab, some system, something that we're trying to investigate. So in this case, it would be a particle spin this particle. Let's fantasise about spin this particles.
That's particles which do not have any they don't that aren't gyros. Let's fantasise about them. Although it turns out that spineless particles are very red things like electrons and neutrons and protons, even our little gyros. So if we had to spin this particle, we could. It's a it's a system. It's a dynamic system. And you can ask yourself, So how do I characterise the state of this particle?
Well, there are things used to characterise it state of course by measuring something and what, what can you measure? You can measure the X, Y and Z coordinates, you can measure the X, Y and z momenta. 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 measure, these things, you could measure a rule called observables.
Then you characterise the system by saying what results you would get if you made these measurements now in quantum mechanics, remember? Or you could you 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 metres. 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, 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 characterise the system. The state of our system. By measuring, by giving quantum amplitudes. Two possible outcomes of measurements. Comes. 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 or what are called the spectrum so that the possible outcomes. The numbers you can get. They formed the spectrum. The spectrum. Observable. So the spectrum of X. Just generally minus infinity to infinity, which is not a very interesting.
I mean, so there's no interesting restriction there. Similarly, the spectrum of X, the momentum in the X direction is usually the same, but the spectrum, for example, of the Z component of angular momentum. Jay Z Uh, turns out to be only discrete values. It turns out that will show that that 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 for a particular particle, it's either equal to nought or it's equal to a half. So the spectrum can be discrete 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 is often a discrete set of numbers, not always e0e what e to, but you have to calculate by hard grind and we'll spend a great deal of time calculating the spectrum of h. It's a very 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 so we could give, we could possibly specify the state of our system. By giving. The amplitude to get e one. Sorry, e zero. The lowest energy. The amplitude to get the energy, the next energy above the amplitude, etc. So let's call these. Let's call this a zero. A one. Etc. So if you the idea here is that that for some systems, if you know the complex number, a zero whose mod square gives you the probability of if you would measure the energy that you've got, the possible value is zero.
And you also knew this number, a one whose mod square is this probability, and if you knew this number, whose mod square was the probability of getting the energy level and so on, right? In general, there'll be an infinite number of these. If you knew all of these amplitudes, you would completely know. You would have completely specified the nominal 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, which 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 that 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 and x, y and z and p, y and Z. Because then you've pin down where the thing is and how fast it's moving and when you know that build all done six numbers down because from that you could calculate the energy,
you calculate the angle, momentum, you know that. But in quantum mechanics, it's not 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 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 modes square give the probabilities. So knowing, specifying complete set of information is a 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 already. Maybe we do have time to just do this. Yeah. Okay, so let's have a look at this.
So I said that electrons and protons and neutrons and quarks, a huge number of an elementary particles have a gyros. So they have an intrinsic spin. They are gyroscopes, they have an intrinsic spin, and they're called spin half particles for reasons that will become apparent in a moment. Let's just use this where we will develop a 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.
Okay. So the total angular momentum of these particles is always the same. They spin at a certain rate so that they have an angle momentum that's root three quarters of h bar where each bar is Planck's constant over two pi. So that's the amount of 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 this the direction that this angular momentum points in changes.
So whereas the total angular momentum is this the Anglo mentum 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 each part. So. And and moreover, there is an amplitude. A plus is the. So let this be the amplitude. Two measure Jay-Z equals plus a half H bar, and obviously a minus will be the amplitude to measure that Jay-Z is minus the half edge bar.
Now, in an ordinary talk, what we say is what everybody says. And I will. You'll find me saying this, but it's immoral. I shouldn't. Is that if Jay-Z is plus a half edge bar, its spin is pointing upwards and you imagine it to be a little particle doing upwards. And when Jay-Z is minus one half bar, you say it's pointing downward. So this is a fundamental mistake because if you square, uh, a half h bar, you take the square root, you don't get that right.
This three indicates that actually this particle has a quarter of a bar. Sorry, there's a half 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 being having to 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 a little word of caution.
People get themselves into a real tangle by imagining that this means the spin is up and that means the spin is done. We all say that, and you'll find me saying that. But 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 Jay Z and counted up Jay Z or Jay Z and I found it down.
Okay, so the good news is that the set a plus, comma, a minus is a complete set of amplitudes. So what do I mean by that? What that means is, if I know those two complex numbers, if I know both 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. Yeah.
So. So from these, we will. Maybe. Maybe I want to write the formula down. I'm not sure I went with the notes. No, I don't think we do yet. We're not ready to write that down. We just want to make that statement that that it's a complete set of approaches in the sense that we will derive rules such that we can calculate. B plus, which is a function of a plus and minus, which is the and this is the amplitude. To measure j in some direction theatre to be.
And the theory consists. So what 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, you know, somebody is you want to know what's the probability of something happening given your current state of information, which is a complete set of amplitudes, down to a complete set of amplitude for something else to happen?
That's what the apparatus consists of and it's there to do. So I think that it probably is an appropriate moment to stop because the next section, uh. Requires a bit of space between which we shouldn't take. No.