 let's get started. So, let's see, you don't have any problem set and we'll have our intro problem set. We're going to type what we've noticed on the first problem. I think this said the sum over n, this is sum over a, provide, make that correct and make sure that it's clear. Okay. All right. So, last time we talked about a long last piece of mathematics that is going to be very important and this is the idea of special properties of operators that can commute with one another, or don't commute with one another. Okay. So, if we have two operators that commute with one another, then what we can say is that there exists a basis of vectors which are simultaneously, I think, vectors of these two operators. And that is to say that those two operators are simultaneously diagonalizable. Okay. Right? That's to say there is a common basis such that a representation of these two operators in that basis will be a diagonal. Now, of course, we talked a lot last time about all the subtle keys associated with degeneracies. And if there are degeneracies in the eigenvalues, then it's not guaranteed that a given one set of eigenvectors of one operator are also eigenvectors of the other. There'll be block diagonal associated in the blocks associated with the non-regenerative eigenvalues, but then you can always re-diagonalize within those blocks and therefore there exists always a common basis of eigenvectors for commuting operators. Okay. And then furthermore, what we discussed was this idea of a mutually commuting set of compatible operators. I think that's not quite right. This sort of says the same thing, you finish it. What I mean to say is a complete set of mutually commuting operators. So what do I mean by that? What I mean by that is let's suppose I have a set of operators that all commute with one another, okay, a mutually commuting. They're said to be complete if the state of the space of the Hilbert space can be specified uniquely as the common eigenvectors of these set of operators. And we gave the example of in angular momentum the magnitude square of the operator and some one component is a complete set of mutually commuting operators. That's to say every state is specified as a simultaneous eigenvector of those operators. I just told you one of those operators and I said find this eigenvectors, well that won't tell me uniquely what the state is because of degeneracies. But by specifying both of these I'm guaranteed there's only one state, okay. And that's going to be that concept will be important when we put the hooks on this abstract theory back into physics. The number of operators one needs in order to uniquely specify the state of the system as a simultaneous eigenvector is related to the number of degrees of freedom, physical degrees of freedom in the system. So for example, you know about the hydrogen atom. I think about the hydrogen atom forgetting about state for the moment and I ask you how do I specify the energy levels of the hydrogen atom? Well I need the principal quantum number. I need the orbital quantum number and the magnetic quantum number. NLM are the three quantum numbers that we need to spot. Why are they three? Because it's in three dimensions. The electron, if I fix the nucleus relative to the proton, there are three degrees of freedom associated with the motion of the electron. So we'll make that connection more people as time goes on. Alright, very good. So now we're finally at the state. We've spent this beginning part of our school year talking about some of the mathematics that we need in order to analyze quantum problems. And so now what we want to do is lay out the structure of the problem. So you know in some sense where the structure comes from and you know what are sort of the basic principles from which the structure of quantum theory arises is still an area of active research. Are there some sort of basic axioms, basic principles? Could you just say the rest of the structure kind of emerges? And we have theories about that. We have relativity for example, right? Relativity has a couple of basic possibilities. The field of light is the same in all inertial reference frames, etc. And we can kind of derive Lorentz transformations, etc. So quantum theory is still kind of grappling. What is really the essence from which quantum theory arises? Maybe we're getting a little bit closer. We'll see. We'll talk about that. But what do we demand from quantum theory? What does quantum theory need to do? Well what it needs to do, the goal is predict the outcome of experiments. That's what any physical theory should do. That's why we have physics, is to predict the outcomes of experiments. So you might say what's different about, why is, where is this quantum? Well it's not any different from all of physics. I mean to the degree to which we say that all of physics is fundamentally quantum theory and nothing else. Then everything else is sort of an ominominal out of fundamental quantum theory. Then what we want quantum theory to do is what we want physics to do, period, predict the outcome of experiments. So how are we going to do that? So the structure that we're going to use is, was really at some level way down at the beginning. So there's the so-called Copenhagen interpretation. I mean as we know the reason that this is a challenge anyway is why we use funny words like interpretation. We don't have the Hamilton interpretation of classical mechanics. I mean classical mechanics is classical mechanics. The term interpretation means there's something which still doesn't quite sit well with us or in some ways feels incomplete about it. We'll of course see where those things are. But what the Copenhagen interpretation is, is an operational interpretation. What I mean by operational, it doesn't in some sense give you a philosophical or deep nature of the world interpretation. It just says this is what you do. We're going to try to connect that to how we should think about these things as we go on. So what are these operational tools, these operational tools that we're going to develop? So the first tool we have is something we call the state. So this is our first tool. And what is the state? The state is a mathematical tool or mathematical object from which can calculate the probability for the outcome that one can do. So really we should sort of say the state of the system. So there's some physical thing we're calling the system and to it we assign a state. State's a mathematical object and we use it to calculate the probability for any measurement. Now there's a particular subclass of states which you may or may never have heard about before, which I'm going to call pure states. Pure states should not be confused and is not the same word as island state. That's not what I mean. There are some people of my generation of older who were taught to have some confused language about pure states and mixed states and island states and non-igond states. That's not what this means. Nipping that in the butt, pure state doesn't mean island state of some operator. So what do I mean by a pure state? A pure state is if, so these are states that we assign to a system if we have maximum possible information. This term about the word information is going to keep popping up in our description of quantum theory because in some sense these postulates really are, there's very little physics at the moment that I'm talking about. This stuff, I mean, you don't have to have no any physics. You don't have to have no any physics. You don't have to have no physics. You don't have to have no physics. This stuff, I mean, you don't have to have no physics. Anything on this board. Where the heck is the physics? So there's a piece of quantum theory in some sense that is information theory. It's sort of in some sense independent of the platform that we're talking about. And that's a very interesting development of the last, you know, 20, 30 years. The relationship between quantum theory and information theory. Let me say something close to my heart. But let me just emphasize what this means. It means that there's, I've compared the system as well as physics allows me to. It doesn't say that it's an eigenstate of position or it's an eigenstate of momentum or it's an eigenstate of energy. It just says I've compared it in some way with devices that weren't noisy in any way. It was at zero temperature. Okay. It's the equivalent in classical mechanics as a point in face space. A point in face space is a pure state. That means that's the maximum possible knowledge you can have about a classical system. It's that point in face space and then it undergoes some trajectory. If I don't know where it is, of course it's some certain level in face space. Okay. But pure state is the equivalent in quantum mechanics of a point in face space. And if the state is prepared in a pure state, then what the structure of quantum mechanics says is that to this, we assign up to a pure state. Every pure state is assigned a vector in Hilbert space. The dimension of that Hilbert space, we haven't yet talked about what that is. Let's just assume it has some dimension. So let's just say we define, so this pure state, we assign a vector and we typically call it psi. Okay. So this is some vector in Hilbert space. So every pure state is assigned a vector in Hilbert space. This is like, as I say, a point in face space. All right. And what this is, is as we remind ourselves, this is the mathematical object from which we calculate all probabilities of outcomes of any measurement we can do. What else can we say about this state? Well, really what we have is really there is an equivalence class. So that is to say psi and some constant times psi. This is the complex number. Remember we're talking about Hilbert space. Our scalars, our complex, are the same state. So we really should say instead of a vector, we really should say a ray. Moreover, so in order to restrict this equivalence class, as you know, we take as our convention to restrict this. We pick within this class of states to represent the state, to choose it to be a unit vector, having one. Okay. So if it's not normalized, we can just divide by its norm and make it normal one. Okay. As we've discussed, that doesn't nail it down completely because this state is normalized, so is this. So in some sense, there's still an ambiguity, there's still an equivalence class of vectors in Hilbert space, all of which represent the same state, even if I restrict it to be normalized. So this overall phase is unphysical. And what it means by unphysical is that it in no way affects our prediction of outcomes of measurements. That's what we mean by physical. If it affects our prediction of an outcome of an experiment we can do, then we say it's a physical effect. It's like, as you know, the gauge in intellectual magnetism is a mathematical thing. We have different gauges of, say, the vector and scalar potential to get n, but they don't affect any of the predictions of the trajectories of charges because the forces depend on e and d. So it's a mathematical thing, but it's not a physical thing. Okay. Now, even though the overall phase of the state is unphysical, it doesn't mean that all phases are unphysical. For example, let's think about the simple example we've been studying for the last week or so, if we look at spin one half, and let's say I have some pure state. So that state I can expand, say, in a superposition of the basis vectors spin up and spin down along the z-axis. Okay. Now, these are complex numbers. So alpha has some magnitude and some phase and data has some magnitude and some phase in general. Now, the overall phase doesn't matter. It's unphysical. So that means that this state is equivalent to, well, let's say, first of all, let me just factor out this, the phase of this. Just factor out that one phase. This is unphysical. I can set it to be this phase to be zero and set this factor to be one because it doesn't matter. The overall phase can be whatever I like. However, this phase matters. It's very physical. Yeah, I knew that going on. So let's go to that case. All right. So this is what should fit, right? Thank you. Let's suppose, in other words, what I mean is very physical means that there's a very different physical effect. You're seeing the different interpretations of, I mean, so there's a different measurement outcome that I can see depending upon what that mathematical value is. If this equals this, then the state is this. Let's also, let's let alpha equal data, and if alpha equals data, what does it have to be to normalize? But what are the square roots? Exactly. Because we have to, let's say the case normalization says that alpha squared plus beta squared is one. So let's, so if this is true, then this is the state. And what is that state? From your homework, you might have seen that particular state. It's the state that's been up along the x axis. Okay. So if I were, for example, to measure this along, and the probability that I'll find this been down along x is zero. Because this is been up along x. However, if alpha is beta is one over two, and say the difference between these phases is pi over two, then this state is what? Well, it's one over two plus i. And what is that state? Y. Yeah, that's been up along y. That's a state that if I were to measure it along x, I could get spin down along x. So there's a very different physical measurement, a very different outcome. So there's this relative phase between the probability amplitudes in a basis matrix. But the overall phase doesn't. How many parameters, real parameters does it take to specify a pure state of spin one half? So a pure state. So a pure state of spin one half. Look at this expression over here. We need to specify the relative phase. That's one parameter. And then I have alpha and beta. But alpha and beta are not independent. The magnitude of alpha and the magnitude of beta are not independent. Because we have normalization. These are two parameters. First we have psi, beta and psi alpha are dependent on each other. Or they're combined to write a variable, the relative phase. They're combined into one, we variable the relative phase. So it's specified by two parameters. And the way we saw that is that we have two complex numbers. So that's two times two real parameters. Because the complex parameter has a real and negative part. A minus one for the overall phase. Minus, we take away one parameter because one of the amplitudes is defined in terms of the other by normalization. So the number of real parameters that this is four minus two is two. In fact, what that parameter is, as we will see next time, is just what is the direction on the sphere that the state is pointing. And that's specified by two angles. It's another way of specifying this. Because we see here, this guy is spin up along x, this guy is spin up along y. Every pure state of spin one half is spin up along some direction. So all of them, do you tell me what direction it is? I know what the state is. So next stage is spin down? No. The term mixed should not be confused with superposition. Again, pure does not mean eigenstate. All of these are pure states. Of course, they're eigenstates of something. But it just means that I have the maximum possible information. The term mixed state is from several points. And we'll get to that in the next lecture. So these pure states cannot be done. They can. Because spin up along some direction is spin down along minus So if I have, say, spin up along x, spin up along z, this is spin up along minus z. Which is the same thing as down along z. So what is the two degrees of freedom of being? Is it like the direction you're measuring and then where it is along that? Or could I give you an example of the electron in the orbit? I can visualize three degrees of freedom in space. Well, of course, this, as you know, spin is not a orbital degree of freedom. Let's just say it's not associated with physical space. So this is an abstraction. There's no way to think about those degrees of freedom in terms of our standard. It's a new degree of freedom. Alright, so now I have a question for you. Give me a pure state in the d-dimensional numbers. These are the two-dimensional numbers. But suppose I have d-dimensions at spin 1. Something like that. That has, as you know, three dimensions. How many real parameters do you take to specify a pure state? You might guess that addresses this too. It's a good conduction, but it's not quite right. You should think about this argument. So generally we might say the state is expanding. We can expand it in a basis, right? So let's just think about it. I have d basis vectors, and I have d complex. 2 times d is the number of complex amplitudes, right? However, I still have the overall phase. I can subtract that, take that out. I don't need to specify that, so I can factor that out. And I also have normalization, which means that all of these guys have to add to 1, which means that one of the magnitudes can be expressed in terms of all the others. Can we move 2 times 1? So it's 2 times 2. This is the overall phase of the relevant, and this is normalization. So for the case of spin 1.5, d was 2. This was 2. What if d was 3? Yeah. And you only want to have the constant, like alpha and delta. If it was an alpha, you know beta or delta, you would set me one more degree of normalization, right? What I'm saying is if I, suppose I have, let's say I have, as you said, a state which is, let's call it, these three cases that you're dealing with, too. Okay? How many real parameters do I need? Well, I can factor out the overall phase. So let me do that. In case that I want to be the phase of this. So this becomes beta e, beta minus alpha. So that's equivalent because I just got rid of one phase. That's the same state. That's over. So right now I have 1, 2, 3, 4, 5 parameters, just as I say. But that's not quite true because I can just say, instead of alpha, I can write this as the square root of 1 minus beta squared minus gamma squared. So I have instead 1, 2, 3, 4 parameters, right? Which is 2 times 3 minus 2. So you have to refocus on two of the classes, right? What I'm saying is this is how many parameters you need if it's a first. Good. So that's the state. Of course, there's another piece, and this is, in some sense, not really, but we write it from the original Kopenhagen interpretation as an important point to make. Really isn't. Does it really affect this operational definition of how we calculate the probabilities and outcomes of variables, but I'll state it anyway. And that has to do with observables. So what do we need by observables? So what the Kopenhagen interpretation says, I'll state it in a slightly different way than my typical statement says. Every permission operator is associated with a physical quantity one can measure. It's sometimes stated the other way around. Anything that we can observe is associated with the Hermitian operator. That's not quite true. That is to say, there are observables that are not for which there is no permission operator. You didn't know that. For example, when I'm observing the time, maybe it's wrong, but I'm observing the time, but there is no mathematical object that considers the reason for this problem. That doesn't mean that we can't observe it. It means that somehow we can talk the wrong thing, that every measurement is somehow associated with eigenvectors that are permissible of their numbers. That's not true. But it is true that every machine operator is associated with a measurement like you did. And examples of this are the standard physical quantities that were familiar with from classical mechanics, like energy, momentum, position, angular momentum. But of course there were new physical quantities and we alerted to this, which are not, don't have this space-time kind of classical property like a spin. They were new kinds of things. I should also note there are observations one can do for measurements associated with non-remission operators. For example, okay, take a look at me. You just measured my position of momentum at the same time. You did it. Well, maybe not precisely, but you looked at me, you saw where I was, and you saw how fast I was going. So who says you can't measure position of momentum at the same time? You just did it. It's just not associated with the usual kind of measurement you learn about. You want to recognize what you did. Everything we do is quantum mechanical. Let's see. I'll remind myself where I was going next. So this says that the eigenvalues, permission operator, let me call it operator capital K, the eigenvalues, the eigenvalues, let's call that set, little bit, permission operator, correspond to the possible, call them sharp or fine-grained, one times, one observes. Okay, what do I mean by that? So if one somehow, in a way that we'll try to explain, does a measurement that the device is designed to measure the observable A, and we want to know what outcomes that measurement device can find, well, of course that measuring device may have some intrinsic noise on itself. It's not necessarily going to give a particular value because devices are often imperfect in some way. But if it can give the sharpest possible values, the most fine-grained values it can find are the eigenvalues. So what I mean, one of the things, and I mean, elaborating on these postulates in ways that you perhaps have not seen in your earlier courses, making them a little bit more refined and modern, frankly, not every measurement that one does is of this sort. I mean, when I just, you know, did my little job down there, maybe made some estimate, but it wasn't a sharp measurement of my momentum and my position. It was uncertain in some way to you. So not every measurement you do is of this sort. But if you do try, if you do try to devise a device to make the sharpest possible measurement you can make, the values you will find are the eigenvalues of the observable. And those, in this case, are new numbers. And now we come to the important rule, which is what we want to do, what we say is the goal here is to predict the outcomes of experiments. So the next thing is what is, so we do an experiment somehow. We haven't said what this is exactly. There's some device that does this thing. And it finds one of the eigenvalues. And we want to know what's the probability of finding a particular measurement outcome. And this is the whole story. What we have is the Born rule. So suppose we assign psi as the state of the system. Suppose it's an eigenvalue, an operator in the emission operator, capital A, associated with a non-degenerate eigenvector, the probability to observe upon that little piece of A is equal to this complex amplitude state. That's the Born rule. Where the Born rule comes from remains a mystery. But it was motivated as we saw and as I would like you to review in the first lecture, where we talked about probability amplitudes as related to wave amplitudes and fractions of intensities. But it was motivated from that and then became the rule that worked. And we to this day still trying to see where this really comes from. Now, I'd like to say about this. So remember, psi itself is expandable in the basis of the eigenvectors of the emission observable. Because the emission observable, because the observable is an emission operator and we know that all emission operators, their eigenvectors are on basis, it means that I can always expand any vector in that basis. So what that tells me is that this projection is just this amplitude. So the probability to see the outcome A is this probability amplitude squared. So in other words, the same thing. Moreover, when normalized, we assumed psi is normalized. Which means that if it's normalized, the sum of all the bifurcation theorem, the sum of all these guys is one. Which is to say this total probability to find something is one. If it wasn't normalized, we could use the same rule, but we just have to divide this that we just automatically make this a probability. If a probability, remember all probabilities are numbers between 0 and 1 and the total probability has to add to it. Very good. So the probability of some outcome happening, something's going to happen if this measurement is better than a variable. So I made this point here about non-degenerative eigenvectors in my particular way of writing down the Born rule in its most elementary form. But we're going to stick increasingly sophisticated about the Born rule. What if A is a degenerative value? So let's say that there is a set of eigenvectors of all the means of A, i where i goes from one to some degeneracy factor, bigger than one, such that all of these have the same... What is the probability... So again, suppose the state I assigned to the system is the ket of psi. And I want to know what is the probability of finding measurement outcome of observable capital A this. I want to know what is the probability of finding that. Well, the answer is you can't tell me because I haven't given you enough information. It depends on what the measuring device does. If I have degenerative eigenvectors, it means that there are actually other quantum markers, other mutually commuting observables that I would need to completely specify which one of these vectors in this degenerate set I'm talking about. So the measuring device doesn't distinguish between the different other eigenvalues that are necessary to specify which state I'm talking about. Then my... So let me write these things down. To fully specify Ua I'm talking about, we need to specify... So it's supposed to be like which one of these guys I'm talking about if I wanted to take that inner product according to the one way. I need to specify the eigenvalues of a mutually commuting set. But suppose... So there's two possibilities I might consider. If the device doesn't measure the other observables, well, it doesn't know which one it got. In that case, the total probability would be the sum over all the possible probabilities associated with that outcome. So in this case, we're saying the device, in principle, for example, again, I'll come to your question in a moment, so you know when we just finish this stuff. Let's think about angular momentum. If I have a device which measures the magnitude of this angular momentum that doesn't tell me what direction the angular momentum was in, then I have the sum over the probabilities of all the states, all the lm's with the same l, but all the possible n's. Because I add the probabilities because those different alternatives are, in principle, distinguishable, but I don't know which one happened. On the other hand, if the device measures also a compatible observable b, which commutes with a, then the probability depends on both the particular outcomes. Let's just say there were two. Let's say the complete set is specified by two observables. So it's not enough. The born rule, as it's typically stated, is stated for what we are really, really typically stated for the case where you have a non-degenerative vector. But if you have degeneracies, it's already subtle. Yeah, Steven, you have a question. There are two lines about the devices. It looks like you have to stop one and start a new one without attention of thought. To fully specify which bits you have, you need to specify something. You need to specify the other eigenvalues of a mutual theory set. So now the born rule, as written here, are examples of projected measurements. Why do I call them that? Call them that because there's a relationship between this born rule and the projection operator. So let's take another look at the born rule. So let's look at the non-degenerative case. We said the probability of finding outcome A is equal to this. I'll always assume my states are normalized unless it's stated otherwise. Which is equal to this amplitude times its complex tangent. Side star side. Everyone remember side star side. And this is the projection operator that projects me onto that direction in Hilbert space. So another way of writing the born rule is this. In this case, because it's a non-degenerate eigenvalue, this is a one-dimensional protector. Which is a projector onto a direction in Hilbert space. On the other hand, if I had the degeneracies and if, for example, not doesn't resolve the probability we set over here, I'd sum over all the probabilities for all the degenerate eigenvectors that all have that same. Which, again, I can write side star side. I can take this because it's linear, the inner product is linear, I can take it outside to sum. And this, again, is a projector, but it's a projector, not onto a 1D space, but a projector onto the space of dimension G associated with that degeneracy. Notice, the solution of the identity is to say any observable A, we can resolve the identity in terms of projections onto all the spaces associated with the different eigenvalues. Right? That could be a sum of 1D projectors or multiple dimensional projectors depending on the degeneracy. Because of that, this is another way of saying that this is a way of decomposing the probabilities. As I say, if I look at the inner product of this, this is 1, if this is normalized, and this is equal to the sum over A, so we have, we can associate each one of these objects is the probability of seeing the measurement outcome. So by having a resolution of the identity, we have a way of assigning probabilities. Because the probabilities are numbers that add to 1, and moreover, these are positive or non-negative numbers. How do we know that? Yeah, that's true, but how do we see that from this? This is a projection operator. How do we know that this inner product is always positive? If the observables are from our mission, they're always real, right? Yeah. But really, I'm just asking about the properties of projection operators themselves. I mean, we kind of see this here. Yeah. I mean, you have your A on the side. Right. We'll give you every one of these things as positive. Now, what I want to say that is the following. This I'm just going to mention because we should, although we won't get into it too deeply. Yes. Not every measurement is projected. Projective measurements are measurements that are fine-grained. They tell you, are you this eigenvalue? What's the probability of finding it? As we said, sometimes your measuring device can't be that sharp. It doesn't know whether or not it had that eigenvalue. So we want a mathematical structure in quantum mechanics that allows us to sign probabilities to measurement outcomes that are you this eigenvalue? Yes or no? Maybe they want to say, are you this eigenvalue within the uncertainty of an error bar? We should be able to do that. Why would we do that not the matter? What we do, what's required is a following. We need some way of breaking down probabilities as positive numbers that add to one. And what's required for that is we need a resolution of the identity in terms of what I'll call a positive operator. Let me explain this. So let's assume that there is some set of operators, I'll call them E, I'll say they're positive. What does that mean for an operator to be positive? Such that the sum of these guys adds to one. An operator is said to be positive with its Hermitian and all its eigenvalues. So suppose I have a set of such operators whose eigenvalue satisfies that. Then I can describe the generalized born rule as the following. The probability of outcome A is that instead of having the projector, I have this positive operator. That's the born rule. The measurement that one can do in the physical world is defined by a set of positive operators that add to one. You may never have seen this before. Most businesses have it, but they should because this is what really is true. Rarely, it's almost impossible to do such a measurement that's really protective. That would require a lot of resolution in your detector. This kind of thing has a very heavy-handed name that's arcane, but which still stands. And it's called a PODM, which stands for Positive Operator Value Measure. It has to do with measure theory and blah-dee-blah, but it doesn't matter. It's the thing previously known as the Positive Operator Value Measure. It's called a PODM, so no one even knows what it stands for anymore. But a PODM... So, alright. The difference between a PODM generally and a projected measurement is that the projected measurements have this connection to observables. The way we did this, as we said, there's an observable. We want to measure angular momentum. We try to measure it. We find one of its eigenvalues and the probability we find that eigenvalue is associated with the overlap of the eigenvector. General measurements... I'm not saying what these a's are. They're measurement outcomes. Look at examples of this, as we go. But they're not necessarily the eigenvalues of a Hermitian operator. They're just the possible outcomes of the device. And if it satisfies this property, then I can assign probabilities. And the fact of the matter is any PODM, there exists a way of measuring these outcomes and they occur to the probability. So if you have, mathematically, write down a set of positive operators and add to one, I can go in the lab or tell my friend, Professor Pacera, to go in the lab and design a device that can measure these measurement outcomes and they will occur in his device with this probability. So let me spend five minutes then simply reviewing a couple of things that you know about. Because what we found here is that one of the things that's strange in quantum theory, unlike classical theory, is that measurement outcomes are generally random. We can only assign a probability for what we're going to see. We can't say for sure. Even when we have a pure state, I mean in classical mechanics we have uncertainty too. If I even want to have mathematical possible knowledge about the system, every measurement I do isn't necessarily determined. We can only come back rather than that's weird. What it means if we're going to analyze things, we have to think a little bit about statistics. So if I have a random variable, a, which occurs with probability, then what can we say statistically? Well, we can look at things like the mean value, which is the sum over all the values weighted by the probability. Right? Now, if this is quantum mechanics and we're talking about, for example, projective measurements of observables, then this probability is equal to that. The absolute value of A. Right? Again, my familiarity I can take outside of the sum and what is this object? Do I recognize it? If it was non-degenerative, you might recognize it. What is this in the bracket? This is the diagonal representation of something. So if I told you, what is this operator? Yeah? Well, what effect? This is the observable A. Right? So this mean value is this. Sometimes the one has the expectation value or the expected value. So the expected value or the expectation value of this observable is given by that. Nothing you know is re-derived. We also have, of course, uncertainty, but it's the fluctuation we'll see in a measurement about so this is defined as the square root of the variance. The variance is given by the difference between the expected value or what is it this way? My definition is this. So I look at the average value of these what are these called? The residuals. How different is the value from its mean? Square. That's the variance. This is respect to a state. I'm not going to write down this state. So I can write this out. This is this square minus twice this like that. This is a number so it can come outside and minus two a and a plus a square or a familiar form that you know the variance is equal to the expected value of the square minus the square of the potential value. So finally what we wanted to say because you need this for your homework but we'll talk about it more we derived it is we have the uncertainty principle. The uncertainty principle states the following. If the product of the uncertainties for two observables in a given state must be greater than or equal to a half the expected value this is of course in a given state. What does this say? It says that if I have two observables that don't commute if this number is not zero then in general I cannot be simultaneously certain about the value I will measure for a and b. If this right hand side is non-zero they don't commute and the expected value of that is non-zero. For that state there is no measuring device that can simultaneously sharply determine the value of a and b. Of course we know that in the case of position and momentum we have x and p your commutator is on h-bar and this is the famous principle delta x times delta p is greater than h-bar over 2 but this version of it is more general. With that next time let's go over