Know. So the big idea introduced yesterday was that of a quantum amplitude, a complex number whose mode square gives you the probability for the outcome of some experiment, some measurement. And we have used the concept of a complete set of amplitude quantum amplitudes, so that if you knew these many all the amplitudes in a complete set, then you could calculate the amplitudes for any experiment that you might you might conceive any measurement that you might make.
And I made the point that quantum mechanics is all about going from the amplitudes in some complete set to calculating these other amplitudes for the outcomes of other experiments. So this is there is a there is a very powerful analogy here between so a knowledge of the state, a dynamical state of our system is encapsulated in the values taken by these complete sets of amplitudes, some series, some set of complex numbers.
And there's a very good analogy here between the way that we identify points in space and the coordinates of vectors. So we can use many different coordinate systems, many numbers to identify one in the same point in space. So the points in space are primitive notion in the sense the three numbers we use to identify them depend on well preference. And you might use their medical order systems.
We might use many different Cartesian quarters systems, we might use public borders we have and the corners we use to identify a given point depend on the problem we're trying to solve may be most efficient use. Furthermore, this may be most efficient to use a particular Cartesian coordinate system, whatever. So we want and we find it very useful to have the concept of a position vector. Oh what we understand to be taken away from ANZ you said the three numbers.
But it's more than a set of three numbers. It's really an equivalence class itself. It's freedom. It's because every different court and system, we have a different set of three numbers. On the same point we introduced the concept of a can of CI. So this symbol effectively characterises the symbol stands for the state of our system, the dynamic, the state of our system and you can think of it in symbolically stands for a one another. We have a new one, so it stands for a 1.334.1.1.
Right. So we don't know how many quantum amplitudes we need in order to characterise our system. So it just goes on to. But the power of the notation is, is the power that we get from position vectors. Instead of writing all this, if we write all this stuff down, then we are committing ourselves to a particular coordinates, to a particular coordinate system, if you like, to a complete particular set of complete amplitudes.
Whereas what we really want to do is focus on the dynamical state of our system. This is a dynamic side of our system. We we might even we might find it convenient to use the amplitudes to find the different possible energies. We might find it convenient to use instead the amplitudes with the different, different possible measurements of the momentum or the position or whatever. We leave that flexible by using by using, excuse me, that we have by using this symbol, said Kett.
And of course, that is sorry. That is the back end of brackets. We will have browse in a moment. Okay. Now we know what it is. We can if we have got two cats. Supposing this stands for. These are not another dynamical state of the system. And let it be defined. Let it be in some particular system. Let it be these numbers. B1, B2, B3, etc. then because we know what it is to add amplitudes, indeed, we know we're under orders to add amplitudes.
When something can happen by two different routes, it makes sense to define the object. We know what this object is. It is A1 plus B1 comma, a two plus B2, two comma and so on. So if you add two cats that that says the dynamical state of the system, which is described by the amplitude, the first amplitude being the sum of the amplitudes from the individual bits, the second amplitude being the sum of the amplitudes, the second amplitudes for the individual bits and so on.
Right. So just as you add two vectors, if you add two vectors, you add the X components and you add the way components and you add the Z components to make a new set of three numbers. That's what we do with Cat. So we know to add cats now and we also know what it is to multiply cats. We can define a new cat by primed being, which we write like this Alpha PSI, which is just some complex number. We define this to be the cat alpha, a one comma, alpha a two comma, and so on.
In other words, if you multiply a cat by some complex number alpha, what you mean is the dynamical state of the system that you would have, which has amplitudes, alpha times, the original alpha amplitude in every slot so we know how to add. These things are not to multiply these things by complex numbers. It follows that cats form a vector space. So you I guess you've been you've encountered this idea with in Professor Ashley's lectures, right?
That the elements of a vector space for a mathematician, they are nothing but objects which you can add and objects you can multiply by numbers are the real numbers or complex numbers at your discretion. So that can form part of a vector space. We'll call this vector space Big V. You from those lectures, I hope know that what you get. No, let's let's let's let's get those lectures. I hope you've met the idea of a basis. A set of basis. Cats. What is a set of basis? Cats. It's set of objects.
I well, you like this, which is such that any cat can be written as a linear combination. Whatever you need. It's a set of cat such that any cat does. For example, the dynamical state of our system can be written as a linear combination of these cats. Right. Then we have the idea of an adjoined space. I hope I'm just reminding you of stuff that you've already met. So if we consider the linea, we are going to be very interested in the linea complex. Valued. Complex valued.
Functions on cats. Mathema a mathematician would say. On V functions on the elements of V. So as you might imagine, traditionally you would you would you would say, okay, f of SCI is a complex number. The complex number in question is going to be the amplitude. The reason why we care about these functions is because they're going to these complex numbers are going to be the all important amplitudes for something to happen, for something to be measured. Right.
And that's you know, we completely focus the whole all this mathematical power is only there to help us to calculate these amplitudes, because if we can calculate amplitudes, we can take the mod square and we then have a prediction for what some experiment is going to. A probabilistic prediction for what some experiment is going to is going to yield. Okay. So so we're interested in these complex valued functions. I'm just I'm just saying that they're going to turn out to be the amplitudes.
I'm not establishing that at this point. And the thing is, we don't actually use this notation, and the temptation we use is this. But these mean the same thing, a bracket opening, sort of angular bracket opening this way F of CI. This thing here means the function F evaluated on its side means that it is a complex number. It is going to be interpreted as an amplitude for something to happen.
And this gives us the idea of saying that f which so this thing is a function, a linear, complex valued function is called the bra and the bra f. So we've got cats which define dynamical states of our system and we've got bras which are functions on the dynamical states of the system which extract the all important amplitudes, the cats form of vector space because it's a vector space, it must have bases like that up there.
And the bras also form a vector space, as I hope you've discovered in in Professor Isserlis lectures. So the brass. Form the Adjoint space. Often called V primed. Why do they form a vector space? Because I know what it is to add to bras. If I give an if you give me a bra F and a bra G, I can form a new bra. Let's call it H for originality. Right. What do you want? In order to. In order to give meaning to this, I need to know what H does. What H does to any state website.
I want to know. Function is defined by the value it takes on any on any possible argument. So I need to know what age of CI is, what number that is, and I define it to be efficaci plus skip CI, which of course is a perfectly well defined expression because this is a complex number. This is a complex number and we all know how to add complex numbers. So this is the definition of the function of the of the branch.
So I know what it is to add two functions. And of course, I know what it is also to multiply a function by some constant thing. So I define the g primed meaning alpha g. By the rule. G primed of upside is alpha g of upside case of gain. This is perfectly well-defined because. That's just a complex number. And so this multiplication is well defined. So now I know what g primed what value it takes in every CI.
So this is so this is the point that this is, this is the basic principle that establishes that the functions, the linear, complex valued functions on a vector space form, a vector space, the adjoined space. And we're going to be working extensively with both the cats and the pros. The only other thing that we need to remind ourselves is that the dimension of the adjoined space is equal to the dimension of the space itself. And so if we. And how do we how do we define this?
We have a chorus where we prove so. So if we're given a basis of case I for each one of these, we define a a bra and we do it as follows. We say that. The bra j is the object is the function on the on the parts such that this complex number j i is equal to delta i j so in other words, it's nothing if if j the label j is not equal to the label I and it's one if the label label J is equal to the label, the label I write. So, so this do this this equation defines.
J. The for all j the funk so that we're saying that that for example to the function to belonging to the second cat in our basis is defined. This is a function and it's defined such that two on two is one and two on anything else equals nought. So that is a perfectly good rule which defines the values that the function j takes in every element of the basis. And again from Professor S this lectures.
I hope you are aware and can show that if you know what a function takes in every element of the basis, a linear function takes in every element. The basis you know what it takes in every cat whatsoever. So there's one final thing that we want to do in this abstract area. We want to say, supposing Abassi is equal to the sum I. Of of. So we take a state of our system and we have is a linear combination of the basis states then we define a function.
This is the funny part, right? So so far I hope I think everything's been I hope everything's been fairly straightforward. But now I'm saying associated with the state of our system. I want to find a function on states and the function in question is defined by this rule that it's a complex conjugate.
Times I. The bra i. So given that my state of my system is a certain linear combination of the basis states, I'm saying that the function associated with that state of the system is a certain linear combination of the functions, these functions which are associated with the basis states. Why do we do that? One reason we do that is in order that we can evaluate this important number of science. I. So let's have a look at that number.
That is the sum. I write this out as a sum, a I star I sum of I. And then I have to write the this one out as a sum, a j of j. So I'm summing over J. These are just dummy labels, right? So I'm entitled to call one J and one I. So it's a sum over J is one to how many we need and i's one to have a many we need. This is a this is a linear function, right? We're evaluating this linear function on this dirty great sum. But because it's a linear function, the dirty great sum can be taken out side.
So I can write this is the sum of I and now J being one two, whatever it is of a I star, a j of i j and there I've used the linearity of the function I and now I use the fact that this is by definition of this function delta right? J So it is nothing and less I equals. J So now let's do the sum over. J For example, as I do this summer over J I will get nothing here except for that particular j which is equal to Y, and then this will come become one.
So this becomes the sum of a I star, a I in other words, it because the sum of I mild squared, which now that's just mathematics. Now we're back to physics. This is an amplitude to find this is this should be a this should be an amplitude a I a quantum amplitude. And we're taking a sum of the mode squares of the amplitudes. So this is the sum of the product, sorry of the probabilities. So that should be one because the probabilities should all add up to one.
So my my states, I would like my states to have this normalisation condition. This is proper normalisation. Is that any of the state times its bra should come to one. Not any other complex number. That particular complex number one. Okay. So that's that's the basic principles of direct notation. Now, let's just talk about the energy. Let's let's have a look at this better understanding of what this physically means by having looking at energy representation.
So supposing we in certain circumstances, for example, if you've got a particle that moves in one dimension, then it's then it's possible in some in some trapped in some. Well, then it is possible to to characterise the dynamical state of the system simply by giving the amplitude to measure. The possible values of the energy. So a complete set. So so this is this is not always the case. But for a one dimensional particle, a particle trapped.
This is a very idealised situation, but never mind trapped. In a one dimensional potential. Well. We will see that. And I'm asserting for the moment that the a I form a complete set of amplitudes. Where? A mod squared is the probability of measuring the ice energy. The ice allowed energy, right? So the energy in this case, when we have our particle trapped inside a potential well, has a discrete spectrum.
Remember, we introduced the idea of a spectrum. Those are the possible values of your measurement. You can only measure a discrete set of numbers. They're called EEI. There's a probability that if I would measure the energy, I would find the energy to be I that that's this mod square and a complete characterisation of the system. Complete dynamical information is provided by knowing not only these probabilities, but actually the amplitudes themselves.
So you can think of of PSI as a vector formed by these amplitudes. Now, let's let's write that upside. The state of our system is equal. Let's let's be given some basis and let's write that it's equal to i. I summed over I. So out of these complex numbers, which we know and some basis, any basis we can, we can write a symbol like this that's just a repeat of what we've already done. And now let's ask ourselves, what are the meaning, what's the physical meaning of these states?
These are this is expressing my actual state of the system as a linear combination of some states of the system that we've conjured out of nowhere. Right. But each one of these is, according to our formalism, corresponds to a complete set of amplitudes. It's it's a state of the system. Now, let's find out what these ones mean in this context. Suppose. We know. The energy is actually a three. So that implies that a three is one and a equals nought four, not equal to three.
So supposing we happen to know that the energy is three. Then. Then the amplitudes must be like this. And what is that? What does that mean? That means Ixi. The state of our system is actually equal to three. Because on this. In this sum, there's only going to be. One non vanishing term, and that will be a three, namely one times three. So that tells us that this state three is actually the state of definitely being having energy three.
And similarly for all the other ones. So a better notation or a clearer notation is. To write to rewrite that in a clearer notation is a cy is the sum I of a i times e i. This this makes it clear what we've just established that the thing is actually the quantum state of definitely being having energy. I. So we've discovered the physical meaning of those abstract basis vectors. When when these are the amplitudes to measure the different energies.
And this is called the energy representation, right? This is the energy representation. This is when we express the state of our system as a linear combination of states of well-defined energy. This representation is and is playing an enormously important role in quantum mechanics, because it's how we it's by going to this representation for mathematical reasons. Going to this representation is how we solve the time evolution equation as we solve the quantum analogues of Newton's Laws of Motion.
It's also as we will find a very, uh, a very abstract representation in the sense that and this may surprise you, no physical system ever has well-defined energy. So these quantum states are, in fact realisable in the real world. So this expresses a realisable state of affairs, this linear combination of states that you can never actually find anything in. But it's it's of enormous technical and mathematical importance.
Let's talk now about something and we'll we'll we'll we'll we'll come back to the energy representation later on. But now let's move straight on to another illustration, which is back to spin a half. So I said that elementary particles are these tiny gyros that the the the rate at which they spin never changes, but the direction in which the spin is oriented does change.
I made the point yesterday that the though you can know for certain the result of measuring the spin in one particular direction, for example, the component of the spin parallel to the z-axis, you cannot know the direction in which the thing is spinning, because even when you measure the component parallel to the z-axis with precision, you're, you're in deep ignorance about the about the value of the spin parallel to the x axis or the y axis.
You only know it does have spin in those directions that you do not know the sign of this. You do not know how much spin is a long X or a long y, but a so so for s. So if we measured the spin along the Z axis and I'm going to say that this is now plus or minus a half a half. Now, yesterday I had an H bar here. In some sense I was using a slightly different notation, but I had an H bar there. I want to look at the angular momentum. H Bar has dimensions of angular momentum.
So the angular momentum, what this means is that the if said is plus a half. That means the angular momentum in the Z direction is plus a half H bar, but it's turns out to be convenient to leave off the bar when talking about the so-called spin of said. Partly because you'll see that spin in quantum mechanics is. Really has a slightly dimensionless being. And partly because partly because writing we don't write any more, because we have to.
It's just it's just economical. So that so physically there's the angular momentum is a half edge bar, but it's more convenient to write that as Z this abstract thing, the spin is plus a half or minus a half. So what do we have? We have two states. We have a we have a complete set of states. Followed by plus and minus. Okay, so this is the state in which I am certain.
If I measure the spin parallel to the z-axis that I'm going to get the value a half, and this is the one where I'm certain to get minus the half. And the statement that's a complete set is to say that any state of my electron or whatever could be written as a plus plus. Actually, maybe it's better to write it this way. A minus minus plus. A plus plus. So since this is an easy case, there are only two components to our cat A minus and a plus.
And just in just the same way that I might in ordinary in ordinary vectors write that oh is equal to is all the vector a let's say B perhaps it's better b is equal to b x e x plus b y e y plus b z e z. Don't need to bracket, do I know? Where? Here, I've got three real numbers B, B, Y and Z, which are the components of B in some particular coordinate system. So here I'm saying the state of our electron can be written as a linear combination of this basis vector and this basis vector.
So these kind of map across here. But this is a simpler case insofar as it only got two components A minus and de plus rather than three components. So that's the analogy. Okay. Now we need to anticipate a formula. So what I what I claimed was earlier was that if you know what a minus and pluses are, what those amplitudes are, to find the spin in the Z direction, either up or down, then you can calculate the amplitude to find the spin in any other direction,
either parallel to that direction or anti parallel to that direction. That's what I claimed. And I'm going to quote a result which which we will arrive at later. But we have to take it on trust for the moment. So the state if we if we have a unit vector n. So so let and. And it's a unit factor. And it's in the direction theatre and. Fine. Right. These are regular polar coordinates which are defining a direction by by pointing to a place on the unit sphere.
And let n be the unit vector that points in that direction. Then I make the following assertion that the state of being plus along the vector n so can be. So this is a state of. This is a state of my electron. So if it's true that that's a complete set, it must be right. It is a linear combination of this state and this state. Right. And I I'm not going to say that that is sine I better just check that I'm getting this right. Yep. Science teacher upon to e to the i fi on to.
Of minus. Plus costs are upon to each of the minus i. On to. Plus, now we will derive or at least you will drive in a problem this formula. We will show that it's why it's true. At the moment, we're just asserting that it is true. So this this is a complex number, right?
And this is a minus. This is a complex number. And this is a plus for that particular for the for the for the quantum state, of having your spin, of being certain that if you measure the spin along this direction, you get the answer plus a half. Correspondingly, there is a minus object. Which turns out to be, cos these are over to each of the eye loops. Five over two minus minus sign feature over to E to the minus I fly over to plus.
So it has it's made, of course, it's this is naturally another linear combination of this and this basis vectors. And now we just have different a because it's a different state, it has different a minus and different a plus. Now we in order to to to calculate something useful, we need to know what the bras are that belong to those. Right? So. So these are the kits. I will I will want to do something with the bras in a moment. So let's calculate what the bras are. So we have the bra in common.
Plus, the rule is that we take the complex conjugate of of whatever comes in front of this, and then we change this into a bra. That was the rule we agreed on. So this is going to be sine theta over to e to the minus i fi over two of the bras minus plus cos these are over to each of the plus I high over two times the bra plus. So that's, that's the bra that belongs to that. And I want the bra that belongs to the other thing cos these are on to. The need to concentrate each of the minus five to.
So there's a bit of practice in taking her mission, taking in adjoint, calculating the adjoin that belongs to a belongs to a vector, a cat. Now what we want to do. So let's calculate. Let's suppose. Let's suppose that we've just measured. The spin and we found the spin on the Z direction. And the result of that measurement was plus a half that in that case, what we will know is that the state of our electron is actually plus let's just suppose we've made the relevant measurement,
and that's the bottom line. So what we want to find now is the amplitude that if I would measure the spin along n, I would find that it was plus on n. Now. I now realise that I have left out. Can we just cycle back to the energy representation? Why? I should have pointed something out. What I should have pointed out was. From this expression here. Well, perhaps it'd be better to be done. We better be done here. Let us point out at this point.
A very simple fact that if I if I multiply this equation through by the bras e j so if I do e j times this equation, what that means is that I'm going to evaluate the function. E.J., E.J. on both sides of the equation then. Then what am I going to discover? I'm going to discover that, E.J. Upside is equal to AJ. Why is that? Because while Egypt sighs, obviously what appears on the left, what appears on the right is E.J. times all this stuff.
But E.J. being a linear function. E.J. pops inside here and meets that. These are two basis factors. So they have Delta, E.J. for their E.J. on this EEI produces Delta AJ. So when I, when I do a so I get a Delta IJA when I do the sum of Ry, all it survives is age. Now, this is a fabulously I should have pointed this out. It's an obvious equation, but it's fabulously important.
And it tells us really why we're interested in these animals here, because it means that given the state of my system, it enables me to recover the amplitude for measuring E.J. out of the state of the system. The rule is to get the amplitude for something. Take the state of your system and browse through by the bra associated with the result. The interesting result of your measurement. In this case. E J So the amplitude to find that the energies E.J. is just E.J.
Brought into the state of our system. So when I come back to this problem here, I want to know the amplitude. To measure plus on end. So what I need to do is to calculate this by that principle. So what I do is I take that in plus thing this thing and I knock it into, I bring it into plus that will produce me a minus plus here which vanishes and a plus plus here, which is the number one. So I simply extract this. So this turns out to be costs feature over to each of the I find over two.
So that's the amplitude to measure this this complex number is the amplitude to measure that the spin is along the vector n where C4 and phi the angles which define n, which means that the probability of measuring plus on n is simply cost squared phi over two. Does that make sense if sorry feature of two.
Right. Because this this goes away when we take the mode square the does this make sense when theta is nought when theta is nought n coincides with the z axis and therefore the probability has to be one because we already know that it's certainly pointing down the z-axis. And guess what? It is one when theta is let's say that theatre is pi, which means that n is pointing in the direction of the minus it axis.
We we should get the probability zero because that's the probability to find that it's pointing down the minus it axis, which is the same as the probability that we get minus along the plus it axis. And when seat is pi, lo and behold, we're looking at cost wed of cost of pi upon two squared, which is zero. So this does behave in a sensible way. Let's let's put three to equal to pi upon two and phi equal to nought.
What does that imply? It implies that N is equal to x the unit vector in the x direction. So n becomes the x direction. What does that give me? That gives me that a that gives me that the probability for being plus on x. Given that I'm plus on Z is the probability the amplitude. Then I'm looking at I'm looking at costs of pi upon to upon to so cost pi upon four, which is one over two. And I have an E to the I not for each of the nothing. So that's just that, right?
So guess what? If the spin, if we are guaranteed the answer plus a half for as I said, what's the probability of measuring plus a long X? The answer is a half because it's the square. The probability is the square of this. So p x plus is in this case equal to a half, which seems pretty reasonable because in some sense, knowing that the spin is along, it has a component plus a long Z doesn't really rather than minus along.
It doesn't really help us to say anything about X. So we really have total uncertainty because it's the probability to be plus on x is a half the probability to minus one. X must also be a half. Let's let's put seats are equal to pi by two and phi equal to pi by two. That implies that n is equal to e y the unit vector in the y direction. What do we get then? Then we find that the amplitude y plus. Plus is. Still one upon two. But now we have e to the i. Pi on for.
So the amplitude is now genuinely a complex number, whereas in the case it was a real number. But it means that the probability for getting plus on Y is still a half the same as it is on X, which again has to be the case by symmetry if you think about it. If we calculate the corresponding negative amplitudes, let's calculate x minus. The probability, let's find the amplitude that it's pointing minus on x. So then we have to take that n minus thing and bang it into plus.
And what survives is the minus sign. Caesar over to. Well. Strictly speaking each of the stuff. But that's well each of the I fi over to. That's it. Actually, let's just make this an minus. All right. Then I know I can, from this formula, reduce the X and Y ones. That's what I want to do. I have the X minus plus. I have to put the feature upon two in here. That's going to be minus one over two. And I'm going to have that y minus. Y minus plus is going to be.
So in that case, I'm going to be have a one of a root two. Here I have minus one over root two and then here I'll have an each of the IP upon four.