 So the big idea introduced yesterday was that of a quantum amplitude, a complex number whose mod square gives you the probability for the outcome of some experiment, some measurement. And we introduced 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 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 there is a very powerful analogy here between, so our knowledge of the state, the dynamical state of our system is encapsulated in the values taken by these complete sets of amplitudes. So it's some series, it's 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 of many sets of different numbers to identify one and the same point in space. So points in space are a primitive notion and the sets of three numbers we use to identify them depend on preference and you might use, there are many coordinate systems, we might use many different Cartesian coordinate systems, we might use polar coordinates we have and the coordinates we use to identify a given point depend on the problem we're trying to solve. Maybe most efficient to use spherical coordinates, maybe 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, are, what we understand to be, thanks for my wide comments there, it's a set of three numbers. But it's more than a set of three numbers, it's really an equivalence class of sets of three numbers. Because every different coordinate system would have different set of three numbers over the same point. So Dirac introduced the concept of a ket of psi, so this symbol effectively characterizes the, this symbol stands for the state of our system, the dynamical state of our system and you can think of it, it symbolically stands for a1, a2, a3, a4, dot, dot, dot, right? So we don't know how many quantum averages we need in order to characterize our system, so it just goes dot, dot, dot. But the power of the notation 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 coordinate system if you like, to a particular set of complete amplitudes. Whereas what we really want to do is focus on the dynamical state of our system. This is the dynamical state of our system, 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 for the 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 ket and of course that is, sorry, that is the back end of Brack ket, we will have bras in a moment. Okay, now we know what it is, we can, if we have got two kets, supposing this stands for, this is 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 are 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 a2 plus b2, 2 comma and so on. So if you add two kets, 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 y components and you add the z components to make a new set of three numbers, that is what we do with kets. So we know how to add kets now and we also know what it is to multiply kets. We can define a new ket psi primed being, which we write like this, alpha psi width is just some complex number, we define this to be the ket alpha a1 comma alpha a2 comma and so on. In other words, if you multiply a ket 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 amplitude in every slot. So we know how to add these things and how to multiply these things by complex numbers, it follows that kets form a vector space. So I guess you've been, you've encountered this idea with in Professor Esler's lectures, right? That the elements of a vector space are for a mathematician, they are nothing but objects which you can add and objects you can multiply by numbers, either real numbers or complex numbers at your discretion. So that kets form part of a vector space, we'll call this vector space big V. From those lectures, I hope, know that what you get, no, let's, let's, yeah, from those lectures, I hope you've met the idea of a basis, a set of basis kets, what is a set of basis kets? It's set of objects I, like this, which is such that any ket can be written as a linear combination to whatever you need. It's a set of kets such that any ket, for example, the dynamical state of our system can be written as a linear combination of these kets, right? Then we have the idea of an adjoint space. I hope I'm just reminding you of stuff that you've already met. So if we consider the linear, we are going to be very interested in the linear complex valued, complex valued functions on kets. Mathematician would say on V, functions on the elements of V. So, and you might imagine traditionally you would say, okay, f of psi 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 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're completely focused. The whole, all this mathematical apparatus 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 we, the thing is we don't actually use this notation. The notation we use is this, but these mean the same thing. Bracket opening, sort of angular bracket opening this way, f of psi, this thing here means the function f evaluated on psi 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 brar, the brar f. So we've got kets 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 kets form a 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 Esseless lectures. So the bras form the adjoint space, often called v primed. Why do they form a vector space? Because I know what it is to add two bras. If I, given, 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, what, in order to, in order to give meaning to this, I need to know what h does, what h does to any state of psi. I want to know a function is defined by the value it takes on any, on any possible argument. So I need to know what h of psi is, what number that is. And I define it to be f of psi plus g of psi. 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 bra h. 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 ket g primed, meaning alpha g, by the rule g primed of psi is alpha g of psi. Okay? So again, 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 psi. 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 adjoint space. And we're going to be working extensively with both the kets and the bras. The only other thing that we need to remind ourselves is that the dimension of the adjoint 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, what we prove. So, so if we're given a basis of kets 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 kets, 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 j will, label j is equal to the label, the label i, right? So, so this, this, this equation defines j, the bra j. The function, so that we're saying that, that for example, two, the function two belonging to the second ket 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 naught. So that is a perfectly good rule which defines the value that the function j takes in every element of the basis. And again, from Professor Estlitz lectures, I hope you're 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 of the basis, you know what it takes in every ket whatsoever. So there's one final thing that we want to do in this abstract area, we want to say supposing that psi is equal to the sum a i of, so we take a state of our system and we have it as a linear combination of the basis states, then we define a function, this is a 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 define a function on states and the function in question is defined by this rule that it's a i 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 psi at psi. 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, sum of i, and then I have to write 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 have a many we need and i is 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 outside. So I can write this as the sum of i and now j being one to whatever it is of a i star a j of i j. 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 i j. So it is nothing and less i equals j. So now let's do the sum over j, for example. As I do the sum over j, I will get nothing here except for that particular j which is equal to i and then this will become one. So this becomes the sum of a i star a i. In other words, it becomes the sum of a i star a mod squared which now, that's just mathematics. Now we're back to physics. This is an amplitude to find, this should be an amplitude a i, a quantum amplitude. And we're taking a sum of the mod squareds 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 states, I would like my states to have this normalization condition. This is proper normalization. Is that the state times its bra should come to one, not any other complex number, that particular complex number one. Okay, so that's the basic principles of direct notation. And now let's just talk about the energy. Let's have a look at this better understanding of what this physically means by having a look at the energy representation. So supposing we, in certain circumstances, for example, if you've got a particle that moves in one dimension, then it's possible in some, in some trapped in some well, then it is possible to, to characterize 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 idealized situation, but never mind trapped in a one dimensional potential well. I'm, we will see that, and I'm asserting for the moment that the AI form a complete set of amplitudes. AI mod squared is the probability of measuring the ith energy. The ith allowed energy, right? So the energy in this case when we have our particle trapped inside of 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 EI. There's a probability that if I would measure the energy, I would find the energy to be EI, that that's this mod square. And a complete characterization of the system, complete dynamical information is provided by knowing not only these probabilities, but actually the amplitudes themselves. So you can think of psi as a vector formed by these amplitudes. Now let's, let's write that psi, the state of our system, is equal, let's, let's be given some basis and let's write that it's equal to AI 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 E3. So that implies that A3 is 1 and AI equals 0 for I not equal to 3. So supposing we happen to know that the energy is E3, then, then the amplitudes must be like this. And what does, what does that mean? That means that psi, the state of our system is actually equal to 3 because on this, in this sum, there's only going to be one non-vanishing term and that will be A3, namely 1 times 3. So that tells us that this state 3 is actually the state of definitely being, having energy E3. 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 that psi is the sum I of AI times EI. This, this makes it clear what we've just established that the thing I is actually the quantum state of definitely being, having energy EI. So we've discovered the physical meaning of those abstract basis vectors. When we, 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, is placed 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 not as we solve the quantum analogues of Newton's laws of motion. It's also as we will find a very, 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 unrealizable in the real world. So this expresses a realizable state of affairs as a 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've measured 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 but you do not know the sign of this. You do not know how much spin is along x or along y. But a complete, so, so for s, so if we measure 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, the angular momentum h bar has dimensions of angular momentum. So the angular momentum, what this means is that the, if, if z is plus a half, that means the angular momentum in the z direction is plus a half h bar. But it turns out to be convenient to leave off the h bar when talking about the so-called spin of s z. Partly because you'll see that spin in quantum mechanics is, really has a slightly dimensionless being and partly because in, partly because writing, we don't want to write any more h bars than we have to. It's just, it's just economical. So, so physically there's a, the angular momentum is a half h bar but it's more convenient to write that s 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 formed 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 of half and this is the, the one where I'm certain to get minus a 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 a nice easy case, there are only two components to our ket, a minus and a plus. And just, in just the same way that I might in ordinary, in ordinary vectors write that r is equal to, is it, or 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 a bracket do I know. Where here I've got three real numbers, b x, b y and b z, which are the components of b in some particular coordinate system. So here I'm saying the state of our electron could 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 in so far as I've only got two components, a minus and a 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 a plus is, 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. Okay, that's what I claimed. And now 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 n, n is a unit vector and it's in the direction theta and phi, 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 writable as a linear combination of this state and this state, right? And I, I'm now going to say that that is sine, I better just check that I'm getting this right, yep, sine theta upon 2 e to the i phi on 2 of minus plus cos theta upon 2 e to the minus i phi on 2 plus. Now we will derive, or at least you will derive 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 or half. Correspondingly, there is a minus object, which turns out to be cos theta over 2 e to the i, whoops, phi over 2 minus, minus sine theta over 2 e to the minus i phi over 2 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 calculate something useful, we need to know what the bras are that belong to those, right? So, so these are the kets. 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 that the bra n comma 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 2 e to the minus i phi over 2 of the bra minus plus cos theta over 2 e to the plus i phi over 2 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. Cost these are on 2. E, I need to concentrate. E to the minus i phi over 2. So there's a bit of practice in taking her mission, taking an adjoint, calculating the adjoint that belongs to a, belongs to a vector, a ket. Now what do 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 means, in that case what we will know is that the state of our electron is actually plus. Let's just suppose we 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 realize that I have left out, can we just cycle back to the energy representation where I should have pointed something out. What I should have pointed out was, from this expression here, well, perhaps we better to be done, we better to 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 bra Ej, so if I do Ej times this equation, what that means is that I'm going to evaluate the function Ej, Ej on both sides of the equation, then, then what am I going to discover? I'm going to discover that Ej up psi is equal to Ej. Why is that? Because, well, Ej up psi is obviously what appears on the left. What appears on the right is Ej times all this stuff, but Ej being a linear function, Ej pops inside here and meets that. These are two basis vectors, so they have delta ij for their Ej on this Ei produces delta ij. So when I, when I do a, so I get a delta ij, when I do the sum over i, all that survives is Ej. 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 a state of my system, it enables me to recover the amplitude for measuring Ej out of the state of the system. The rule is to get the amplitude for something, take the state of your system and bra through by the bra associated with the result, the interesting result of your measurement. In this case, Ej. So the amplitude to find that the energy is Ej is just Ej braed 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 N. So what I need to do is to calculate this by that principle. So what I do is I take that N plus thing, this thing and I knock it into, I bra 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 cos theta over 2 E to the i phi over 2. So that's the amplitude to measure, this complex number is the amplitude to measure that the spin is along the vector N where theta and phi are the angles which define N, which means that the probability of measuring plus on N is simply cos squared phi over 2. Does that make sense? If, sorry, theta over 2, right? Because this goes away when we take the mod square. The, does this make sense? When theta is not, when theta is not, 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 theta is pi, which means that N is pointing in the direction of the minus z axis, we should get the probability zero because that's the probability to find that it's pointing down the minus z axis which is the same as the probability that we get minus along the plus z axis. And when theta is pi, lo and behold, we're looking at cos squared of pi, cos of pi upon 2 squared which is zero. So this does behave in a sensible way. Let's, let's put theta equal to pi upon 2 and phi equal to naught. What does that imply? It implies that N is equal to E 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, it's not the probability, the amplitude, then I'm looking at, I'm looking at cos pi upon 2 upon 2. So cos pi upon 4, which is 1 over root 2, and I have an E to the i, nothing, E to the nothing, so that's just that, right? So guess what? If the spin, if, if we are guaranteed the answer plus a half for s z, what's the probability of measuring plus along 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 along z, doesn't really, rather than minus along z, doesn't really help us to say anything about x. So we really have total uncertainty, because if the probability to be plus on x is a half, the probability to minus on x must also be a half. Let's, let's put theta equal to pi by 2 and phi equal to pi by 2. 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 1 upon root 2, but now we have E to the i pi on 4. So the amplitude is now genuinely a complex number, whereas in the x case it was a real number. But it mean, but the probability for getting plus on y is still a half. It's 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 theta over 2, well strictly speaking E to the i stuff, but that's, well E to the i pi over 2, that's it. Actually let's just make this n minus, right? Then now I can, now I can from this formula deduce the x and the y ones. Is what I want to do, I have the x minus plus, I have to put theta upon 2 in here, this is going to be minus 1 over root 2, and I'm going to have the y minus, y minus plus is going to be, so in that case I'm going to be, I have a 1 over root 2 here, I have minus 1 over root 2, and then here I'll have an E to the i pi upon 4.