Which is where we were. I, I'd asked you to take on trust as a result that will be derived some way down the course. These results, which express the state in which you are the spin half particle like an electron, is certain to have a half plus a half for the answer to its spin along the unit vector n, which is given by the polar angles feature and fi this this particular state in which quote unquote its spin is a long end.
Somebody asked me about this at the end of the lecture, and I have to remind you of the health warning I gave in the first lecture, which is that we talk about the spin being along a certain axis, although the although even when you know the answer to the spin, the long Z is going to be a half and not minus a half. And you can only get those two answers a half or minus a half, even when you know it's plus a half you. That doesn't mean the spin is really along the Z axis.
There's still a substantial component of spin in the x y plane and you do not know its direction. So we use this loose talk. The spin is along some particular direction, like the Z axis. All this n meaning it's a shorthand for you are certain to measure plus a half if you measure the spin in this direction.
Right. So with that health warning, the state in which you are certain to to measure spin a half in the direction and is this linear combination of the state in which you're definitely going to get minus a half along the z-axis and this state in which you will definitely get plus a half along the Z axis.
And similarly, the state in which you are definitely going to get minus a half along the end direction, the unit vector n is this other linear combination of those same two states of well-defined spin along the z-axis. So I asked you to take that stuff on trust, and then we did some stuff with that. And I hope I persuaded you that these formulae are not completely implausible in the sense that what we did was we calculated the probability.
If it's definite, if we know the spin is plus a half along the Z axis, we calculated the probability that we found plus a half along the Nexus. We found that that probability was in fact simply cos squared theta upon two and this behaved in a reasonably plausible way in the sense that when it was one, when the end direction was the z-axis and it went to nothing, when the end direction was the minus IT axis and other such good stuff, then I wanted to show you this.
All I wanted to show you was what these formulae predict for what the state is of definitely having plus a half for the answer to what is your spin in the Z in the X direction? Right. It's easily done because we have the formula here. I was thinking I was trying some for some reason it went into my mind that I had to derive these formulae and we didn't have the bits on the table to do it. So all we have to do is plug into those formulae that features pi upon to find nought.
That is the that that by definition the polar coordinates makes and the x the unit vector in the x direction, which I'm calling x and put in pi. If you put in pi upon two, then you're looking at sine pi upon four cost pi upon four, one over root two and five being nothing means those two exponentials are nothing. So the state of having your the state of having your spin definitely down the X-axis in that sense. Right. And again, with that health warning.
So the strict statement being that we are guaranteed to get plus a half if we measure the spin down X turns out to be just the sum, essentially the sum of these two states. That's not very exciting. I think let's put in but now let's put in theatre is pipe on to phys pi upon to which by the definition of polar coordinates makes the univac to end the y direction.
Then what happens? Well, what happens is that those e to the IFIs upon to become each of the I PI's on four, and if I take the first e to the IP upon four out, then the second one. So this cosine is one of root two again, which we've taken out, but this one becomes e to the minus i pi upon four twice, i.e. to the minus minus i pi upon to which is which is actually minus. I'm slightly worried by this, but never mind assign is of no importance.
I thought it was a plus i. But it is looking like mine is light at the moment. So maybe maybe it is minus it's of no importance. What matters is that this state, which is physically quite distinct from this state, is also a linear combination of these two and the. And the. And the probability of if you are in this state, the probability of measuring your spin along Z to be minus is going to be a half because because this one of root two.
So the, so the, the complex number which comes here has the same modulus as the complex number which comes here. And ditto here that this complex number is the same as the complex number which appears there in modulus but different synthase. So that what the crucial point is that the right we are working with in a formalism where we're saying the state of my system can be written as a minus, minus plus, a plus plus.
And we understand that these things are the probability amplitudes to measure spin down on Z. Given that my system is in this state and this is the amplitude to measure spin up on Z, given that this is my state of my system, right? That's the formalism we're working with. And you might think that it's only the modulus of these complex numbers that matters physically, because the probabilities are obtained by doing most square of them. But this example is showing you that that's not the case.
The phases of these things are vitally important as well, that the very quantum mechanical thing, that the complex phase, the phase of the complex number encodes crucial physical information, is the spin of this particle more or less down the x axis or more or less down the y axis is controlled by the duration, by the phase of this animal relative to the phase of that animal. Let's do another example of a physical system which which is a two state system.
Let's talk about polarised light. This is an example which enables us to connect back to classical physics in an interesting way. So let's do classical physics. We know all about polarised light. Well, actually you may not quite because it may be part of upcoming IMAG course, but you will recognise enough of it. I think I can write the electric field. Supposing we have a we have the y direction this way, the x direction this way.
Suppose we have polarised light with electric vector in this direction. With that angle being thetr, then we can say the electric vector is equal to some number in front of cost theta times x plus sine theta times e y. Times cos omega t. Now supposing we. So we've got we understand we're writing down the electric vector of a electromagnetic wave, a plane, polarised electromagnetic wave that's travelling in the Z direction. Okay. In, in some plane, this is what it looks like.
It oscillates with some frequency, angular frequency omega right now supposing we stick in some this beam comes along and it hit some Polaroid. And let's imagine that the Polaroid blocks the electric vector. So Polaroid blocks one of the polarisations. Let's let's orient a piece of Polaroid so it kills the oscillations parallel to the Y axis and let's only through the oscillations parallel to the X axis.
So after Polaroid. We're going to have that e is equal to e nought e nought cos theta cos omega t e x will just, it just, it just wipes that out. So the intensity of the intensity of the radiation, the energy that it carries is going to be looking like in a squared times crossword theatre. Yeah. And they'll be we should really do a crossword, make a T average value, which is in fact a half. But I don't think we're really interested in that.
The crucial thing is that the intensity of the lights that gets through is going to be moderated by the square of the cosine of that angle. Right. The angle between the electric vector of the of the wave and the direction that the Polaroid leapt through. That's what classical physics teaches us. How would we express this? So let's now think about this from a quantum mechanical perspective. What classical physics says is a is an electromagnetic wave.
Quantum mechanics says it's a stream of photons, and each photon encounters that Polaroid on its own, on its lonely and some. And either it's killed by that Polaroid turned into something else destroyed, or it's allowed through. It can't be half allowed through or square feet or allowed through. It's either allowed through. It's not allowed through. So how does that how does that look? What we say is the state of our incoming photon.
We can write as a linear combination. We can say that this is equal to cost. The letter of. A state in which it is going to get through because in some senses electric field is down the x axis plus sign feature of a state that is not going to get through. So this is the state of certainly gets through. And this is the state of certainly blocked. Well, we were taking the position that the Polaroid is making a measurement on the photon. So what was the probability? Gets through.
Well, it's equal to this is an amplitude, right? It's equal to the amplitude for getting through mode squared, which is equal to cos squared feature. And therefore the number of photons that gets through is proportional to cost squared theta, but the number of photons is the amount of energy that gets through. Right. So it should be the intensity of the light goes like cos squared teacher. And quantum mechanics recovers our classical result.
We can go further than that because we know that if we think about circular polarisation. So we know that classically. We can write the electric field of a circularly polarised radiation. So and so in plain polarised radiation, the electric field just oscillates up and down some definite direction. In circular polarisation at a given place, the electric field always has the same value and it rotates in its direction.
So now it's pointing in the x axis and it is pointing in the y axis, nice pointing the minus x axis, etc. and it can go round clockwise or anticlockwise, etc. How do we write that? How do we write that classically? Well, we can write that. It's that it's in order to have a route to air. And then I would write the real part of the neatest way to write it is the real part of. The. X plus i e y times e to the i omega t and that's all inside this real operator.
Let's think about that for a moment, because what does that give me? This X meets that cost plus I sign. So we find when this read operator works and we're looking at x times cos omega t and this I e y meets cos omega t plus I sign omega t so this, this, this eye and the eye that's sitting inside here make the real part of this minus sign omega t so this is looking like x cos omega t minus e y he's a unit vectors sign scientific a kitty.
And so that's what this is. That's what we get from this notation. So this indeed is a is a is a circularly polarised beam. The mod square of this electric field is is is going to be in short squared over two and it's in fact right hand polarised in the complex in the plane. It's going to go around that away because Y is going to start going to become negative first the component because of this minus sign.
Right. And similarly, if we wrote so this so let's call that E plus for know e subscript plus for the electric field associated with the right hand circular polarised beam. Correspondingly, we would have e minus four left hand. So that can be polarised. Joanie would be this. We get the we get a change in the sense of rotation just by changing that plus II to a minus side. These are check. That's true. So this is left hand polarised.
How would we do this quantum mechanically? What we would do is we would say we would say that there's a state. Plus, which is equal to the state that has this electric vector in the X direction, plus times the state which has this electric vector in the Y direction and doesn't get through the pedal.
Right. And this does get through the Polaroid. And we would say so this would be a state of the right hand polarised state of our photon is a linear combination of when I should have a one of a root two outside here. That's that too, basically. So so a state of circular polarisation of a photon is a linear combination of two plain polarised states and similarly we would have that minus is equal to the left hand polarised state would be.
And we will be able to make statements like if we want a kind of statement we could make is we could add these two equations. And we would discover that being polarised in the X direction is one over two of being right hand polarised plus being left hand polarised.
And this is also a result that we have in classical physics that if you have a plain polarised beam, you can consider it to be a linear superposition of, if you like, an interference pattern, whatever, between two circularly polarised beams of opposite senses of polarisation. But there's a different but this has a different meaning sort of emotionally, right?
This is saying that a particular state of one photon, of a particular photon is this linear superposition of its two other possible states. Something else that you learn from this. I mean, another thing that should be pointed out is that think in classical physics we were using, I was using I hear it up there as a sort of handy way to reduce algebra, etc. There was a real operator sitting in front of it.
The electric field was totally real, and then the appearance of the square root of minus one was merely as an as a shorthand, as a trick, as a device for compressing the algebra in the quantum mechanical case. This I is is I there's no real operator. There's no nonsense with that. This is inherently a complex animal. Now maybe it's time to move cross. Let's say a little bit about measurement. We've already encountered these ideas, really.
But let's let me take you back to what we did yesterday with the energy representation. What I said was, look, supposing I rise up PSI in terms of some basis vectors. I because we had we had agreed that the quantum state of a system, a cat was an inhabitant of a vector space. Vector spaces have bases. Therefore, any cat can be written as a linear combination of basis vectors. Supposing these happen to be physically the amplitudes.
To measure a particular value of the energy, say I. Then I hoped I persuaded you that the physical meaning of this either basis factor is the state in which you are certain to measure eei. Because. Because. If. If abassi. Is is the state in which we are certain to measure this energy.
Then what does that mean? It means that a I is one, an AJ and every other a has to be nothing AJ not equal to I. And so we can look into this expression here under those circumstances, under those circumstances of, say, on the left here becomes II, the sum collapses just to I. And that tells us that I is actually in the state of which you are certain to measure, know. So that's how we understood the meaning of these things had. Now suppose that website is some general thing.
It's some general stage. In other words, loads of these eyes are nonzero. So it's some superposition, linear combination of a non-negligible number of these states, a well-defined energy. So suppose initially. That aside, it's not a state of well-defined energy. It is a sum a i. E i. With lots of. Non-zero. I. Fine. Now, suppose we measure the energy. If we measure the energy according to our conception?
Well, obviously, if we measure the energy, we are going to find one of the allowed values, one of the values in the spectrum of the energy. We are going to come up with one of these I's, shall we call it E k. So we do a measurement. So we measure. E and find it k. Having found E.K. We know what the energy is. We know it's weak.
Therefore, we know the state of our system. So now Asi equals e k. So after we've made the measurement of size different from the from the cap than it was before we made the measurement, it's changed into this, which is just one of the terms that occurred in this series. So. So this some run over many of these I's and one of the I's was K. It just happened when we made the measurement. Bingo. This is the one that popped out.
But having made the measurement, we know what the energy is. It's EAC, so the system is definitely in the state EAC. So the original wave function is of state. Quantum state is changed into a different quantum state. On making the measurement and this different quantum state looks simpler than that one. And what we what people conventionally say is that this quantum state is a result of our measurements, has collapsed into this quantum state. So this is the collapse. Of the quantum state.
Traditionally known as the collapse of the wave function. We haven't yet met wave functions. But it's the same it's the same phenomenon. Now, it's an extremely interesting question. What's really happening? This is a fundamental, absolutely non-negotiable piece of this theory. The matters discussed rather more in Chapter six of the book and at some point, say, in the vacation, I would urge you to read that. And you will find that it is. It's.
It's. This piece of the theory is fundamentally unsatisfactory. It's clearly not right, but nobody knows how. There are various proposals, including many worlds and all sorts of things for fixing it, but none of them really. There is no known satisfactory fix. There is no consensus, there is no really persuasive fix. Consequently, different people say, Well, I think the fix is probably something like this. Somebody else is to fix it.
Something else like this. The fundamental principles that I think everyone will agree on is a it's non-negotiable. It's absolutely essential for the working of the theory that we do some such collapse to that. When you make a measurement, there are logically there are two possibilities. Logically, it's just a thought process. Right? Okay. I was I wrote that down because frankly, I didn't know what the energy was. So so that covered my bases. And, you know, it was probabilistic.
There were many possible values of the energy, etcetera that were. And I stuck in some amplitudes to reflect my, my uncertainty. And having made a measurement, I discovered what the energy was. And so it's this now everything's okay. We, we've discovered something. So I've updated my information and the, and the state vector is merely reflecting my improved information is a subjective change, not a real change. That interpretation proves to be untenable. There really is a change. It's there.
At the moment we are operating chapter six. Only Chapter six introduces an apparatus that deals for muddle and uncertainty, which is kind of worrying because in real life, in real physics, there's always masses of of of genuine uncertainty and genuine muddle. But we are not we are operating in an ideal world at the moment, in which there is total clarity, there is complete information, there is nothing left to chance beyond what is in inherent.
I mean, so this is so so this is a completely well-defined state of the system. It changes into some other completely well-defined state of system. It actually objectively changes. And here we have a crucial we have a crucial thing that is being added in quantum mechanics to classical physics, which is the concept that when you make a measurement, you disturb the system that you're measuring.
I think this is totally reasonable. It's it's obviously an abstraction that classical physics makes that you can that you can make measuring instruments of arbitrary delicacy so that you can have these. So when a measuring instrument interacts with the system, the measuring instrument in classical physics is affected. You know, the needle moves overall, light glows or whatever, but the system carries on blithely, you know, without without being changed in any way.
It's clear there's action and reaction. If the instrument is affected, the system is affected. And since we're concerned with systems, quantum mechanics is about systems which are very small. It's very natural that the impact on the system should be kind of substantial.
So it's totally reasonable that the that we should be working with the theory where every measurement is, is associated with the disturbance of the system and leaves the system in a configuration different from the one that it found it in. So that's not the problem. The problem is that the theory doesn't describe the process of getting from here to here. But but that's a topic which I can't discuss at this stage or indeed in this course. It's you can find something about it in chapter six.
It's all highly of syllabus. I want, however, to point out something else, which is that we we started with a basis up there, remember, we started with AI and the mathematicians already taught us to associate with AI the cat AI a broader AI such that j loops j ai equals Delta II j So in our physical example, this maps into e j e i equals delta i. J. So this is this this was just mathematics. But it has it has a deep physical meaning as follows.
I think I made the point yesterday that if you that if you want to know what a K is, you the way to find it is to do e k ci. That's broadly speaking, why we introduced these bras. We introduced these bras because we wanted out of an object like website to extract the amplitude for something to happen. Because, you know, amplitudes are the things whose modes square make predictions. And we, you know, that's what we take down to the lab to test against nature.
So. So let's ask ourselves in this context, let's look at this formula. This is the amplitude to find energy. K If the system is in the state of PSI. So what's this? This is the amplitude to find e j if the system is in the state e i Well, if the energy is e i it can't be e j can it? If j is different from my. So the reason this thing vanishes when I is not equal to J is because it reflects the exclusive well. It reflects the fact that if your energy is e i it's the i it's not e j.
So this, this, this all functionality condition. With a logical necessity. The mathematician has given it to us, but we need it for physical reasons. We need and its associated with we all because it's a requirement of our fundamental principle. This gives us the aptitude for making. So of the details they cover at this point is is suppose we've got a sci fi is able to form a high quality on some basis. I like the energy space, Michael. Okay.
I suppose we have some other thing. We have some other quantum state, which is the J. Yeah. So these these two states are two different states because they have different they're associated with different attitudes. This is associated with a set of amplitudes, i.e., numerical values II and this state is associated with numerical amplitudes. B i b j whatever. And let's calculate the number phi assign.
So we have to take the complex. We have to. But we have to make the Brahe out of that and use our rules that we introduced yesterday. So this is the sum j of b j complex conjugate. J Times the sum a i. I. But when I'm image j we get a Delta ii j which means this. And then when we conduct the sum over I, we get nothing. Except there's only one term that contributes because of the delta, right? J And that's when I equals J. So this becomes the sum b.j star AJ.
So that tells us how to work out this complex number in terms of these quantum amplitudes. That turns out to be very useful. The thing I want to say at the moment is supposing I worked out the other thing I worked out fi, sorry, ABC onto fi instead of fi onto a sci fi, then everything would be the same here except that this will be in a star and that would be a b AI and we would be looking at the sum over j of a j star PJ.
But this. Is the complex conjugative this by the rules of of complex because this is just a sum of complex numbers. Right. So we know what the rules of complex conjugation are. So this is the sum PJ Star, a J Star, which is Phi PSI Star. So this is an important equation to remember that sci fi is equal to Phi PSI complex conjugated. We'll need that many times. And. I think that's all we want there. Let's now introduce the next topic, which is operators.
And that connection to observable things we can measure. So what we're interested in linear operators. What does that mean? I guess you probably know, but let me just write it down anyway. So if Q is a linear operator. Well, first of all, Q is an operator. What does that mean? That means it turns cats into cats. Give it a cat. It produces a cat. That is to say, fi, if I do Q The operator on ABC, the cat, I get another cat fi.
Right. But that's what an operator is. It's something which turns cats into cats. What's a linear operator? If I have Q on a linear combination of alpha of psi plus beta, say of chi. So this is just to any old two cats. I take alpha times one and beta times the other because I know I'm allowed to do that. Well, what is that? That's equal to Alpha. Q Operating on up upside plus beta of. Q Operating on CHI.
That's the linearity property. So we're only going to be interested in these linear operators. Now let me write down an operator. There's an operator. A very, very, very important operator. Like this. If we have a basis of tests, I. I can form this creature here. This is the cat. I somehow multiplying the bra. I just like that. The first thing I have to do is persuade you that this is an operator, right? So I say, let's consider this, and I need to persuade you that this is an operator.
How do I do that? I show you how it operates. Right. So long as I if you know how this operates on any cat, then it's an operator. So let's have a look at this. Supposing I do. I am, Sy. What does that give me? It gives me the sum of. I, i, i. Sy Now, this is a complex number. It's even an interesting, complex number with emotional appeal because it's the quantum amplitude for something. But let's not worry ourselves about that at the moment. This is a complex number.
This is a cat. So this is a linear combination of cats. Ergo, it is a cat. It is something we can call it Fi if we want to. All right. So that means that AI does turn up sci fi into some state fi. It is an operator. Now let us. Let us replace Abassi with. So let's replace this up sci with its expansion some A.I. so I can write this is the sum I on this is going to work on the sum over j of a j. J. So this is another way of writing up SCI.
We've done it time and time again. Now this AI is going to meet that J and produce a Delta II j. When I do this, some over every term will vanish, except the term where j equals I. And the. And then when J does equal, I will have I on I, which is one. So this is equal to the sum i. I which is a sign. So I. This operator, this thing here is not only an operator, it's the identity operator. Because it turns upside, any upside gets turned back into itself.
So we have that this thing here is the identity operator. And I've told you nothing about what these things here are, except they form a basis or a complete set of cards. And we use this representation of AI sometimes called a resolution of the identity. It's not a phrase, an expression I will use, but we use this representation of the identity operator time and time and time again.
It's tremendously valuable. Now let's introduce a sexier operator, in fact, the most important operator in the universe. We're going to introduce H, which is by definition the some e i e i e ii. So these are the states of well-defined energy, and the numbers II are the possible energies that the spectrum of the energy operator, this operator here is call the Hamiltonian. After William Rowan Hamilton, who lived in the first half of the 19th century and introduced the classical analogue of this.
And I think it's I hope it's clear from what I did above that this is an operator. Let's just say if we if we do age sci, we will get this stuff. Times E upside. This is a complex number which multiplies this real number times some cats. So we will get back a cat and it's obviously an operator associated with the energy and the general scheme is going to be forever. With everything that we can measure. We're going to associate an operator and we're going to do it in just this way.
But let's just focus on this particular one for a moment, because it is the most important operator in quantum mechanics. And let's work. Let's see one thing that we can do with this. Supposing we work out a possi h abc. So what is this? This is a number. A complex number. Why? Because h operating on up side makes some state, shall we call it fi and ABC. The bra working on fi produces a number. So we can see that. We can say straight off that this is some complex number.
Let's find out which complex number by putting in for ABC its expansion some a i e ii. So what we're going to do now is replace both of these by their expansions. So on the left here, I'm going to have a j star e e j, so that is ci the bra. Then I have h and then I have. E.J. Sorry. Then I have the some a i d i. An ancient self. Oh dear. This is getting complicated. Is the sum over k of e.
K. K. Sorry. Okay. Okay. So every term in this expression here has been expanded in terms of the basis states, the states of well-defined energy states where measurement of energy is certain to yield a particular result. Now, what happens with summing over every every everything j is being summed over, eyes being summed over and case summed over. This e k oops. Yes, right. Sorry. Let's, let's, let's work on this. EAC is going to. This is a, this EAC is a linear function on this.
So this passes through the I by the linearity of this function and meets this and produces me a delta k i. So when I sum over I I get nothing except when I is k. So that is going to this sum is going to collapse. And in the next line we're going to be looking at the sum over j of a j star e j oops. Which way it's pointing is pointing this way sum over k e k e k and the action here is that is that this sum of I is going to make that into an arc. So that's looking this way and that is an arc.
Because of the all fog analogy, that and that. And now we repeat this trick. We say when we do this, e j is going to pass through this by linearity and etcetera, meet that and produce a delta g k so you will get nothing on this sum except when J is equal to K. So this is going to become the sum over K of E k. This is this a style. The J is going to be the only term that's going to survive. When this meets, this is when J equals K. So this is going to be coming a k star.
This A.K. Star is going to meet with that a k to produce a k mod squared. But a k mod squared is the probability of getting this energy. So this is equal to the sum of peak. The probability of getting e k times the value e k in other words, is equal to the expectation value of the energy. So that's one reason why these operators why operators associated with observables in this way are so useful.
If you want to know your state is upside and you want to know what the expectation value of the energy is. You take h the operator associated with energy and you squeeze it between the bra upside and the cat upside. So we. Can you give me any observable. Call it. Q For that observable, we've we have agreed there will be a spectrum, so there will be numbers. Q II the spectrum, the possible values that the measurement can return, and there will be a complete set of states.
Q I mean, these are the states for which the results. Of observing Q. Certain. What can I do? I form the operator. Q We might give it a hand just to stress that this is an operator, which is a mathematical object, whereas. Q is a sort of concept like momentum or, or angular momentum or position or something. We got a natural operator which is defined to be the summer of a K of CUC, Chuck.
Q Okay. And by exactly the logic up there, we will find that the expectation value of the observable Q is going to be upside. Q Upside. So that's just repeating what I've done for the energy operator. The Hamiltonian making the point that it's going to carry through for any observable. Now there's more than we can do. More than that we can now ask ourselves, okay, let's look at this. What is Q operating on? Q I mean, what does.
Q Is an operator? What does it do to one of the special states associated with with that observable? Well, by definition, this is the sum over K of tuk tuk. Q Okay. Q I. Right. So this is this is merely the definition of the operator. Q There is mobile operating on this is going to produce a Delta K II. So when we do this sum of a K, we get nothing except when K is equal to Y. So what we get is only one term in this series survives.
And the answer is it's Q IQ II. So what Q does to this state of well defined Q of the worldwide value of the observable is it turns it it it makes them a scale model of what we what we started with with the scaling factor. Q II So in the mathematical language, which I guess you've met, this says that that. Q I. It's an iron cat.
Of Q you have met that right you've you've met eigen cats eigenvectors whatever of operators and the eigenvalue so that the states have well-defined observable which are really the primitive physical thing all turn out to be eigen states of this operator that we've introduced and the eigenvalues. And the eigenvalues are the possible values of the elements of the spectrum, the possible numbers you can get if you measure the observable Q. Well, that probably is the right moment to stop.