Quantum Entanglements, Part 1 | Lecture 3
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Susskind, Leonard
Lecture 3 of Leonard Susskind's course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded October 9, 2006 at Stanford University. This Stanford Continuing Studies course is the first of a three-quarter sequence of classes exploring the "quantum entanglements" in modern theoretical physics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.
 This program is brought to you by Stanford University. Please visit us at stanford.edu. I want to make sure we're all on the same page with respect to some very elementary mathematics, complex numbers. It's one important concept. So I just want to go through them very quickly, extremely quickly. A complex number has a real part and an imaginary part. And you can plot it on a two-dimensional surface. The two-dimensional surface horizontal axis represents the real part of the function, not the function, just the real part of the number. And the imaginary part of the number is plotted on the vertical axis. And a point on this plane is an imaginary number. If this is x and the height is y, then the complex number is z, which is x plus iy. OK. We could also represent it another way. We can represent it in terms of a distance, which we can call r, and an angle that we can call theta. And now, just from elementary trigonometry, the distance x is r times the cosine of theta. So that's x is r cosine theta. And y is r times r times sine theta. So i times r sine theta. Or in other words, we can take the r-outs out of it, factor the r out, and write that z is equal to r times cosine theta plus i sine theta. The combination cosine theta plus i sine theta is called e to the i theta. e to the power i theta. So know that definition. e to the i theta is cosine theta plus i sine theta. The reason we call it e to the i theta is because it satisfies certain rules of exponentials. The rules of exponentials is that when you multiply exponentials, you add the exponents. e to the 3 times e to the 5 is e to the 8. 3 and 5 is 8. You can check using elementary trigonometry that if you have two angles, theta and phi, and you write e to the i phi, which is cosine phi plus i sine phi, and multiply them, you get something with a real and an imaginary part. The real part is cosine theta cosine phi minus sine theta times sine phi. Remember, i times i is minus i, as minus 1. So if you multiply these two, what you get is cosine theta. Well, I'll tell you what you get. Let's simplify it. You get cosine theta cosine phi minus sine theta sine phi. Anybody know what that is? cosine theta plus phi. Right. Cosine of theta plus phi. That's a high school trigonometry formula. The cosine of the sum of two angles, cosine cosine minus sine times sine. And then, plus the imaginary part, the imaginary part is cosine theta sine phi plus cosine phi times sine theta, and that's just sine of theta plus phi. And by definition, if e to the i theta is cosine plus i sine, then this is just cosine, then this is just e to the i theta plus phi. So from elementary trigonometry, we discover that this combination, cosine theta plus i sine theta, has all of the properties of the exponential of i theta. Now, all of trigonometry, anything you ever wanted to remember about trigonometry, couldn't remember, is all stored in this formula. For example, if you couldn't remember what the cosine of the sum of two angles is, all you need to do is multiply e to the i theta times e to the i phi in this form, and you'll discover that cosine theta plus phi is cosine theta cosine phi minus sine theta sine phi. So all of trigonometry, I don't know why they take a year to teach trigonometry in high school. And then at the end of it, they come here to Stanford and they don't know this formula. Another fact, we could check this in two ways. We could just use the formula for multiplying exponentials. e to the i theta, first of all, there's e to the i theta. That's cosine theta plus i sine theta. Remember what the complex conjugate of a thing is. The complex conjugate is just the same number except with the imaginary part changing sign. So it's the reflected number in the lower half plane. If the number is in the upper half plane, then it's reflected into the lower half plane or vice versa. The complex conjugate of e to the i theta is cosine theta minus i sine theta. But that's the same thing, that's e to the minus i theta. That's the same thing as cosine theta minus i sine theta. And if you remember that minus the sine of theta is the same thing as sine of minus theta, then you realize this is just e to the minus i theta. So the complex conjugate of e to the i theta, you just get by changing the sine of the i here. And what happens if you multiply these two? You get one. I think that it's changing the sine of the theta because that's what happens when you rotate the clockwise direction instead of the counterclockwise direction. Right. This is true, I don't know if everybody called that, but if you multiply e to the i theta times e to the minus i theta, you get one. So the class of numbers, which are the class of complex numbers, which are of the form e to the i theta, are numbers which have the special property that when they're multiplied by the complex conjugate, the result is one. And there's another way to say that. Another way to say it is that x square plus y square is equal to one. The class of numbers for which x square plus y square, remember, a number times its complex conjugate is just the sums of the squares of the real and imaginary part. And so the numbers which have the form e to the i theta, let's specialize to those, are numbers for which x square plus y square is equal to one. In other words, they're numbers on the unit circle, on the circle of radius one. They're simply called numbers, I don't know what they're called, unitary numbers. Of the form e to the i theta, you can call them unitary numbers. And they're numbers that have the property, that they lie on the unit circle. And they times their complex conjugate is exactly equal to one. Every complex number is a unitary number times a real number, a real positive number, and the real positive number is just a radius, or the distance to that point on the column. And that's all we need to know. That's all there is to complex numbers, there is nothing else. Nothing else I can think of offhand. I want to make sure everybody knows. Ah, the other word for a number which is of the form e to the i theta is that it's called a pure phase. The angle in here is called the phase angle, the phase, and the radius is called the modulus, I guess. The modulus means the size of it. So a number which has no r in front of it, or where r is one, number on the unit circle is called a pure phase, or just a phase. So when I swallow the chocolate chip wrong. When I refer to a pure phase, that means a number of cosine theta plus i sine theta. Okay, now. All right, now, the postulates of quantum mechanics. We want to now today go through the basic postulates of quantum mechanics. Now, what quantum mechanics is, is it's a calculus for calculating probabilities. After a while, you get some intuition for it, and maybe you get some pictures of what's going on. But think of it as a calculus, in other words, a calculational procedure for calculating probabilities. Probabilities of what? Probabilities for different measurements, different values of measurements. The things you measure are called observables. You prepare systems in certain states, not in California or New York, but in certain configurations that are called states. And those states are labeled by or described by vectors in a complex vector space. We've called them alpha, or A, I guess we've called them A. You can add them, you can subtract them, you can multiply them by numbers, and they form a complex vector space, number one. Number two, we normalize them. In fact, in the particular simple case that we studied, which was the two-level system, we could describe this as alpha times the upstate, for example, plus beta times the downstate, where alpha and beta are complex numbers. The coefficients here are such that alpha star alpha and beta star beta are the probabilities that if you were to measure up or down, you would measure up with probability alpha star alpha, down with beta star beta. And so, just to make the probability altogether equal to one, we require that the coefficients are such that the sums of the squares, or the sums of the, I'll call this just the square, it's the square of the, is just equal to one. All right, there's another way to write the same equation. It's that the inner product of a real state with itself is just equal to one. That is equal to alpha star alpha plus beta star beta. And so, this is the abstract way of writing that the total probability adds up to one. So states, state vectors are normalized, this is called normalized. Means that the length of the vector in some sense is equal to one, and altogether the sums of the probabilities are one. That's the basic posture with the states. This is the example of the two-dimensional space of states. Two-dimensional doesn't mean the world is two-dimensional, it just means that there are two independent possibilities, up and down, heads and tails. We discussed how you might measure whether a spin was up or down by putting it into a magnetic field and seeing if it emits a photon or not. So, given an arbitrary state of electron, we can measure its spin. If it gives off a photon, then we say it's down, I guess. And if it doesn't give off a photon, then we say it's up. And the relative probabilities or the probabilities for the two are just the coefficients here. Now, those probabilities, of course, do not completely determine the complex numbers here. Why not? Well, if I multiply any complex number by a pure phase, it doesn't change its magnitude. If I take a complex number and I multiply it by e to the i theta, where theta is any angle, I get a new number whose magnitude, whose alpha star alpha, is exactly the same as the original starting number. And so, knowing the probabilities is not enough to completely determine what these coefficients are. So, there's more information in these coefficients than just the probabilities. And we're going to find out what that additional information is. But for the moment, this is... And, of course, if there's more than two possibilities, up, down, and sideways, or up, down, and chocolate chip, then we add another one, gamma, chocolate chip, and whatever. Those are orthonormal vectors. And we're going to... And we represented them in a particular basis. The basis of up and down, we represented them as... Up was just one zero, and down is equal to zero one. They're orthogonal to each other. They're normalized, both of them, the sums of the squares of each individual one is one, so they're called orthonormal vectors. Orthogonal and normal. If we have a higher dimensional vector space with more possibilities, chocolate chip, then we might have three vectors altogether, zero, zero, one. They are still orthogonal and normalized. This zero should go here. Up and down or at right angle? We're going to have to remember the difference between these vectors, these abstract vectors which represent states, and vectors in real space, because that's going to come up today. Pointers. These are pointers, they're not vectors. And we'll label them slightly differently. All right, now, what about the quantities that you measure? They are called observables. They're the things that you measure. And by assumption, there may be more than one thing that you can measure about a system, and you can always reduce them to real numbers. The results of a measurement are always a collection of real numbers. You may combine them together, for example, into a complex number, but the results of real measurements, you know, pointers on a dial or whatever, are real numbers. And so observables are things which when you measure them, among other things, you get real numbers as a result. They are represented by matrices or operators. Strictly speaking, they're represented by linear operators, which you can take to just being matrices for our purpose. But a special class of matrices, the Hermitian matrices. Hermitian, H-E-R-M-I-T-I-A-N, Hermitian for Hermite, and Hermite was a mathematician. Hermitian matrices, and to define them, let's first define a couple of concepts about matrices. First of all, there's the concept of the transpose of a matrix. The transpose of a matrix, you have diagonal elements and off diagonal elements. And there's no reason why it has to be two by two. Let's take it to be three by three. There's M-1-1, M-2-2, M-3-3. And then there's the off diagonal elements. Let's just put one of them over here. Here's M-1-2. And here's M-2-1 and so forth. And there's other ones, other, they're all there. The transpose of a matrix is just, you act on the matrix to interchange rows and columns. All it means is you reflect it about the diagonal. You just imagine it's written on a piece of paper and you turn over the piece of paper so that M-1-2 replaces M-2-1 and so forth. The way to write that, the transpose is just written by putting a T up here. And it's just the diagonal elements are unchanged. But the off diagonal elements, M-1-2 becomes M-2-1 and M-1-2, they just interchange. And likewise over here and so forth. Okay, that's called the transpose. We can write that in the form, the transpose matrix, we can call it ij, the ijth element of it is just the j-i-th element of the original matrix and we can get the change of i and j, of rows and columns, it's called transpose. The next concept is the Hermitian conjugate. I'll write it out, Hermitian conjugate. And it's a kind of complex conjugation, but complex conjugation for matrices, Hermitian conjugate. It involves two operations. The first is to transpose and then the complex conjugate. And it's represented by a dagger. It's represented by a dagger. It's represented by, first of all, transposing, but then complex conjugating everything. All the elements, the diagonal elements as well as the off diagonal elements. And we write it by saying that the Hermitian conjugate, the dagger of a matrix, is you interchange rows and columns and then you complex conjugate. So let's do an example. Here's an example. 4 plus 2i, 7 minus i, 6 plus 4i, and 9 minus 2i. What's the dagger, the Hermitian conjugate of that matrix? Well, the diagonal elements, they don't move when you transpose. Transposing a diagonal element stays in the same place, but we have to complex conjugate it. So we get 4 minus 2i. Now, the element up here, first we have to transpose, what was this, 4? 4 minus 2i? Yeah, okay, we have to transpose, which means we put, we interchange rows and columns, and then complex conjugate. So then we get 4 plus 2i up here. This one complex conjugated. This one complex conjugated goes down here, 6 minus 4i, and the diagonal matrix element stays where it is, but it gets complex conjugated. 7 plus i. Alright, so that's the idea of Hermitian conjugation. Flip the rows and columns, and then complex conjugate. A Hermitian matrix, well first let's talk about symmetric matrices. Symmetric matrices, let's just go back to the idea of transpose without the complex conjugation. Just transpose. A symmetric matrix is one that is equal to its own transpose. In other words, when you transpose it, you get back the same thing. In order for that to be the case, all that you need is that the elements, the reflected elements are the same as each other. So it would say that m to 1 has to be equal to m12. We'll put the same m21 here. That's the idea of a symmetric matrix, and it just is exactly what it says. It's sort of symmetric with respect to flipping it or to reflecting it about the main diagonal. A symmetric matrix satisfies mij equals mji. That's a symmetric matrix, very simple concept. A Hermitian matrix is a little more complicated, and what it says, a Hermitian matrix is a matrix which is equal to its own Hermitian conjugate. So if you have a matrix and your Hermitian conjugate it, if it is equal to the original matrix, it's called Hermitian. It's just called Hermitian. That's the definition of a Hermitian matrix. Let's write a Hermitian matrix down. Write the formula for it. Yes, yes. There's the formula for it right here. mij is mij complex conjugate. Say it again. I think it's written correctly. mij is mji conjugate. It's correct. What it says, first of all, is that the diagonal elements are real. For example, this says if you interchange i and j on a diagonal element, you just get the same diagonal element, so it says that m11 is equal to m11 star. That means that m11 is real. So real numbers on the diagonal. Not all is the same real number. I just wrote real, real, real, real, meaning any real number in any one of these places. Then on the off diagonal elements over here, let's suppose that this off diagonal element over here is z, the complex number z, then the reflected off diagonal element has to be the complex conjugate. Then it's called Hermitian. So 2, 7, 1 plus i and 1 minus i is Hermitian. Real elements on the diagonal and complex conjugates when you reflect. This is a kind of reality property. A thing being its own complex conjugate for a number says that it's real. For a matrix, the concept of reality is not that the matrix elements are all real. Not that each matrix element separately is its own complex conjugate, but each matrix element is the complex conjugate of the reflected matrix element. We'll find out. We're going to see very shortly why this is a good definition. Okay, I'm going to write down some theorems. These are your homework to prove. They're very, very simple. But just they don't have names because they're too simple. Very elementary theorems. First of all, if you take two vectors, a and b, and you take the inner product of a with b. Now, remember, when you form the row vector b, you use the complex conjugate elements. This is equal to b star, b star 1, b star 2, complex conjugate times a1, a2. The first theorem is that if you interchange a and b, that will put the a's here and the b's here. What do you get? Is this correct? No. Complex conjugate. The way to think of it is that the row vector is like the complex conjugate of a certain column vector. And if you interchange the order of these, you're taking the complex conjugate of b and the complex conjugate of a, and if you multiply complex conjugates, it's just complex conjugates. So the first elementary theorem to check is that the inner product of b with a is the complex conjugate of the inner product of a with b. Now, it's extremely elementary. Anybody who's seen it, of course, is very familiar with it. Next statement. Now, this one is less obvious. Take any matrix, Hermitian or not Hermitian, and I'll, and, all right. Now, this is an operation that I haven't defined yet, but I'm going to define it for you right now. It's very simple. You have a matrix m and you multiply it by a. That gives you a new vector. This is a new vector. And you take that new vector and you take the inner product with b. And you can work out what this is in terms of the matrix elements of m and the entries that go into a and b. It's a very definite expression. First, multiply the matrix by a, or multiply a by the matrix. You'll be left over with a new vector that you can call a prime, and you take it in a product of b. This is related to, but not equal to, I'm going to write equal, but then we're going to change it, and we're going to change everything. A, put a on the left. That's kind of the complex conjugate of a. Put b on the right and put the Hermitian conjugate of m. Everybody has been complex conjugated, and all rows and columns have been interchanged. When you interchange a and b, you're interchanging a column vector with a row vector, and you're complex conjugating. The same for m. When you take a Hermitian conjugate, you interchange rows and columns, and you complex conjugate. This is a number, incidentally. This quantity is a number, a complex number in general. m on a gives you a vector. The inner product of a vector with another vector is a number. A simple number, it's not a matrix, it's not a vector, it's just a number. This equation is not right. What do I have to do to it? Complex conjugate it. I have complex conjugated a, I have complex conjugated b, and I have complex conjugated m, so I have to take the whole thing and complex conjugate it. That's a simple fact about complex conjugation of matrices and vectors, but let's suppose that our matrix is Hermitian. Let's suppose that the matrix in question is Hermitian. If it is Hermitian, I can erase this conjugation sign here. If it's Hermitian, it's equal to its own Hermitian conjugate, definition of Hermitian. So for a Hermitian matrix, and from now on, let's think about Hermitian matrices. For Hermitian matrices, for Hermitian matrices, this is called a matrix element, this is called the BA matrix element of m. The BA matrix element of m is the same as the AB matrix element of m, complex conjugated. Now, one last thing that will make the point for you, if a and b happen to be the same vector, if a and b happen to be the same vector, now let's let b equal a. Let b equal a. Look what this says. It says that the matrix element of m evaluated between sandwiched, between two states which are the same, and the state a on both sides is equal to its own complex conjugate. What does that say about AMA? That it's real. So if you have a Hermitian matrix, this was a theorem that I was going to ask you to prove, but I just proved it. If you have a Hermitian matrix, then we should give this object a name. It's called the expectation value. Why it's the expectation value will become, or what that has to do with anybody's expectations is something else. It just has a name. It's the expectation value of m in the state a. Now remember, just keep in mind, the way we're going to be describing states is through vectors. The way we describe observables is through matrices. So you can think of this basically as the average value of m when the state of the system is a. Yeah. The bracket notation for average, does it come from block and matrix? Yeah, sure. Of course, it comes from Dirac. So what do we learn? We learn that if we have a Hermitian matrix, its expectation value is real. Well, we haven't gone through the details yet, but that sounds like a good thing. If we have an observable, which is represented in which when you measure it, always gives you a real number, then obviously its average value should be real. And that's what we find, that for a Hermitian matrix, its expectation value in a given state is real. That's what makes the Hermitian matrix special, and what makes the Hermitian matrix natural candidates to have something to do with observable quantities, because their expectation values in any state doesn't have to, in many, many states, any one of them will satisfy this. Okay, any questions up till now? If there is a physical value in a particle or qm, where you, I think a unit time length is a mass, or you have an imaginary number of atoms, is there a value in the unit? Anything that you really measure is always a number. Now, you can decide to say, I will describe points, locations of particles on the surface of the Earth by giving a complex number. I'll give you an address in Manhattan, by giving you a, you know, Manhattan is laid out on the grid. I'll give you a complex number. But you don't have to do that. You can represent any observable quantity as a collection of real numbers. You can always represent the results of an experiment by a series of real numbers. And those real numbers are called observables. When they're complex, when you add them together in complex combinations, just because of the technical historical definition, we don't call it an observable. The observables are the collection of real numbers. The same question. What's a good way to think about measuring numbers when they show up in physical formulas? Like, is it like these things exist, but they're not observable? But then, how do we know that they exist if they're not observable? I think there's a better... I think the question that you really want to ask is why do we have to use complex numbers in quantum mechanics? That's, I think, what you really want to ask. Why is it that we're driven to complex numbers? This will become clear, but it has to do with time and reversibility. And we're not there yet. We're not ready for it. We could have thought about doing quantum mechanics with all real numbers and never having gotten complex numbers. We've missed some very important things, as you will see, as you will find out. But let's write down the postulates, examine their consequences, and then I will tell you what would go wrong if you restricted yourself to real numbers. It would have to do in particular with the evolution of state vectors. What we have not talked about at all is how things change with time. And it really has to do with how things change with time, where that I comes from. So, hold on, hold on to the question, and it'll come back again. I realize it's a different question than you asked, but I think it's the right question. What complex numbers are, is they're just a mathematical formalism for describing pairs of real numbers in certain ways. Why are we driven to use them in quantum mechanics? It has to do with time evolution, so we'll come to it. We're still doing mathematics, we cannot do this subject without mathematics. I mean, I can fake it. I can tell you lots of stuff about quantum mechanics and give you all sorts of mystical experiences about entanglement. And when you walk out of here, you will know no more about entanglement than you knew when you walked in, but you'll have this fuzzy feeling about it and you'll say, oh, isn't it wonderful, and it's a mysterious, but you won't have any idea what it is. So, if we want to really do it, honestly, we are forced to do this mathematics. We want to do better than that. We want to really do it right. In the simplest possible context, I mean, I'm giving you the simplest possible context to discuss quantum mechanics, basically the two-level system, and we'll do two-two-level systems, and then we'll talk about the entanglement of two-two-level systems, and we'll understand it in some detail, and you'll know what the words really mean. The next concept, and it's basically the last mathematical concept that we'll need for today, and possibly for quite a while, is the concept of eigenvalues and eigenvectors, particularly eigenvalues and eigenvectors of Hermitian operators. So, what is an eigenvalue and an eigenvector? Well, if I give you a matrix or an operator, a matrix or an operator, M, if it's a, in general, there may or may not be certain vectors that when I apply the matrix to them, don't change the vector except to multiply it by a number. In other words, there may or may not be, and typically there will be, and in fact, in the case of Hermitian matrices, there always are, certain vectors, we take a matrix, there will exist certain vectors which have the property, not every vector, there will exist certain vectors that are associated with that matrix, which are called its eigenvectors, which have the property that you just multiply by a number. The number depends on the vector. I will give you a very, very simple example right now. Supposing we have a diagonal matrix. In general, it will be called an eigenvalue and an eigenvector even if it's complex. But if M is a Hermitian matrix, then it follows that lambda is real. Let's prove that. Let's suppose we have a Hermitian matrix, we have found an eigenvector and an eigenvalue. Let's see what we learn by taking the inner product of this equation with A itself. We have M, A, A, and multiplying the equation on the left by A. Lambda is a number, it just comes on the outside, A, A. Now if you go back in your notes about five and a half minutes, remember that we proved that for any Hermitian matrix, A, M, A, its expectation value is real. We also proved that the inner product of A with itself is real. So we have an equation that a real number is equal to a real number times an eigenvalue. Well, that means that the eigenvalue is real. Or another way to write it is to divide it by A, A, and the ratio of two real numbers is certainly real. So the first statement is if we've managed to succeed and find an eigenvector and an eigenvalue of a Hermitian matrix, the eigenvalue will be real. Now I'm going to tell you the secret. The secret is that if M is an observable, then the values of it that you can measure when you do an experiment are its eigenvalues. That means the result of an experiment can measure M will be one of the eigenvalues of the matrix M. That's the significance of eigenvalues. They're the possible measurable values of the observable that's represented by M. Let me give you some examples. Oh, yes. The lambdas, the eigenvalues are the values that you measure. The eigenvector that's associated with that particular eigenvalue, that eigenvector represents a state where when you measure the quantity M with certainty, you get value lambda. So the eigenvalues and the eigenvectors go together. There are certain states for any given observable. There are certain states which when you measure that particular observable, you will with probability one get a particular answer and that answer is the eigenvalue. So let's just do an example or two. The easiest examples involve diagonal matrices. Here's a diagonal matrix. Let's just give it some letters. M11, zeroes, just two by two is good enough for us. M22, there's a diagonal matrix. What are its eigenvectors? Remember, they're vectors which when you multiply them by the matrix, you just get the original vector back times a number. I'll just tell you the answer. There are two eigenvectors of this matrix. One of them has a one in the upper place here. Let's check. What does this matrix do when it multiplies this vector? Well, we take the row, multiply it by the column. That puts in the upper place here M11. But how about the lower place? Zero times one plus M22 times zero. Zero. But this vector is just M11 times one zero. So you see what's happened. The matrix times the vector gives you a number times the original vector. In other words, in particular, it doesn't change. Yeah, it just gives you the back the original vector times a numerical number. In this case, the numerical number happens to be M11. So the vector one zero is an eigenvector of this matrix with eigenvalue M11. There's another eigenvector and the other eigenvector is to put a zero here and a one here. Let's see what we get. In the top place, we get M11 times zero plus zero times one is zero. And then we get zero times zero plus M22 times one, M22. And that's just M22 times zero, one. So we found another eigenvector, matrix times vector equals number times the same vector. And this time, the eigenvalue is M22. So the two vectors, zero one and one zero, are both eigenvectors, but with two different eigenvalues, M11 and M22. You can see where this is going. For example, if I'm interested in up, down, and the spin of an electron, I might describe it by a matrix with a one and a minus one. Then the eigenvalues would just be one and minus one. The eigenstates would be one zero. That's the correct equation. So the state with the electron up is an eigenvector of a certain observable, which is represented by a matrix. And the other eigenvector is zero one, and that is equal to minus zero one. The minus comes from this minus sign over here. So the two states of the electron, up and down, are eigenvectors of the spin operator, the spin operator, which is usually called sigma three. This is definition, sigma three is just this. They're eigenvectors with two different eigenvalues, and you see the pattern that the value that you measure, either plus one or minus one, is the eigenvalue. The eigenvector is the state in which the eigenvalue is the desired measured quantity. Any questions? And in fact, these are normalized. Room, yeah, rule. We normalize vectors, yes. Let me give you another example, which is less, a little bit less trivial. Incidentally, if it were three by three or n by n matrices, the same pattern would be there that if the matrices are diagonal, then the diagonal entries are the eigenvalues, and the eigenvectors are just vectors with zeros everywhere except in one place, a one. So you can check that out for yourself, experiment around the diagonal matrices. Let me give you one other example. The other example involves an off-diagonal matrix. These matrices, incidentally, will play an important role in the ones we're talking about, but from time being, they're just arbitrary matrices. It's Hermitian. It has real diagonal elements. Zero is a real number, and the off-diagonal elements are complex conjugates of each other. One is its own complex conjugate. It also happens to be symmetric, but it's both Hermitian and symmetric. Now let's see if we can find the eigenvectors. Rather than to do any algebra, I'm going to show you what they are, and then we'll see that they are eigenvectors. The two eigenvectors in this case, okay, now what's wrong with this eigenvector? It's too big. It's not normalized. One squared plus one squared is two. But here's a statement. If a vector is an eigenvector, it doesn't matter what its normalization is. If you double it or triple it or anything else, it's still an eigenvector. But if you wanted to normalize it, you would write one over square root of two, one over square root of two. It doesn't really matter in checking whether it's an eigenvector or not, but let's normalize it just to remind ourselves to always normalize vectors. What is this matrix times this vector? We take the zero, one, and we multiply it by these two elements. And what do we get? We get zero times, well, let me come on the side, one over square root of two plus one over one over square root of two. One over square root of two, and likewise in the bottom. We got back exactly the same vector. So this is an eigenvector of this operator with eigenvalue plus one. What about zero, one, one, zero? Can I find another eigenvector? Yes, I can. The reason I can find it is because I know the answer. But it's not hard to find. It's not hard to find. I just want to illustrate the idea of eigenvectors. Another time I can tell you how you would go about finding them, but I do want to illustrate it. The other possibility is one over square root of two minus one over square root of two. In other words, instead of having the same element here and here, we have opposite sign. Let's check it out. Zero times this, zero, one times this is minus, minus one over square root of two. And then here we have one over square root of two plus zero. So we have one over square root of two. And that's the same thing as minus the vector that we started with. We have a minus sign here and the opposite sign here. It is minus the original vector. So we found another eigenvector, and this time the eigenvalue is minus one. We found two eigenvectors with eigenvalue plus one and eigenvalue minus one. I defy you to find any other eigenvector. There are none. There are none, and that's a theorem. There are no more eigenvalues or eigenvectors. Right. Right, right, right. You can always multiply an eigenvector by a number, but it's still an eigenvector, but no more, no more, which are not just numerical multiples of the same eigenvector. Right. All right, I'm now going to prove a fundamental theorem of quantum mechanics. Very fundamental about Hermitian matrices. Am I going too fast? Well, if I am, yell out. What did you call the spin matrix? This one, say it again. Yeah, yeah, yeah. Okay, sigma three, and this is definition, one, one, sorry, one minus one, zero, zero, and it has eigenvalue plus one and minus one. Sigma one is equal to this one, one, zero, zero. I'll show you two interesting things, or, well, let's just say, one interesting about both of these. The square of each one of these matrices, does everybody know how to square a matrix? How to multiply a matrix by itself. Good. The square of each of these matrices is the same, and it's just plus one. The square of this matrix, you just square the elements, one squared and minus one squared are both plus one. The square of sigma one, let's square sigma one, one, one, zero, zero, and multiply it by one, one, zero, zero. Alright, the first element in the upper left-hand corner is the inner product, this times this, the one times the one gives us a one, then the zero times one plus one times zero, zero, one times one plus zero times zero, now sorry, one times zero plus zero times one is zero, and one times one plus zero times zero is one. So both sigma one squared and sigma three squared are both equal to one. Now that's kind of an interesting fact. Notice that it's also true of the eigenvalues of the matrices. They were both plus and minus one. If a square of a thing is one, it stands the reason that it's either plus or minus one. So for matrices, in particular two by two matrices like this, but matrices in general, if the square of a matrix is the unit matrix, this matrix is called, is the unit matrix, it's called one. Alright, it's just a unit matrix with ones on the diagonal. Sigma three squared, sigma three squared is the same as sigma one squared equal to one. And it's not independent of the fact that the square of each of the eigenvalues is equal to one. It means that the measured values that you can measure are such that the square is equal to one. And it means that you can only measure plus one or minus one. There's one more matrix I'll write down, and it's called sigma two. And it's equal to minus ii. We will come back to the sigma matrices. In fact, we'll probably come back to them tonight. We found the eigenvectors of sigma one and sigma three. We have not yet found the eigenvectors of sigma two. And I will tell you in due time what their significance is. Yeah. You can see all this geometrically. It's a very simple geometric interpretation. Okay. If you're going to say that, then tell us what it is. Oh, okay. I was going to write something up, but I can put it on the internet. Tell me how. Basically, if you look at the unit vectors, if you look at the unit vector, well, think of e1 and e2, like the z1 and 1, 0, as the unit vectors. If you look at a matrix, if you look at the column vectors, those are the images of the unit vectors. I know what you're saying. So forget it. So one of them is reflected, you're changing e1 and e2. That's called sigma one. And so the diagonal is fixed. And also the anti-diagonal is fixed. And when you're reflecting in the main diagonal, so the normal to it is also reflected. Those are eigenvectors, those directions, I think, and that's what you got. And then the, that's one of the square root of two things, those are pointing in the diagonal directions. So they're fixed. And then the other one is you're sending, you're keeping the e1 fixed, and e2 is going into its negative. So you're reflecting in the x-axis. So the vertical vector is an eigenvector going into its negative. And also the squares of these things, if you apply them twice, you come back to where you say it. Not so completely obvious for this one, but the square of this one is also one. Yeah, well then I take the i out in front, and I have i times something very obvious. Right. Sigma 2 squared is also one, not minus one, but one. They all have that property. They're all, in that sense, they're similar. That also proves incidentally that the eigenvalues of sigma 2 are also plus or minus one. It doesn't tell us what the eigenvectors are, but it tells us what the eigenvalues are. All right, let me prove to you now the following interesting theorem. Suppose, and this theorem is significant, highly significant. Supposing you have an observable, and it has more than one eigenvector, in general, will. The number of eigenvectors that it has is usually equal to the dimensionality of the matrix. It is the dimensionality of the matrix. So if you have three by three matrices in a Hermitian, they will typically have three eigenvalues and three eigenvectors. Four by four will have four eigenvectors and four eigenvalues. But here's the theorem. If I have a Hermitian matrix M, and it has two different eigenvalues, two different eigenvalues, that means that there's an eigenvector A with eigenvalue lambda A, and there is an eigenvector B with a different eigenvalue lambda B B, then if lambda A and lambda B are different, what it says is that A and B will be orthogonal to each other. Before I prove it, let me say what it means. What it means is that if there is any observable distinct quantity that you can measure, which will distinguish between two distinctly different possible measurable answers that you can get, for some observable, then that's equivalent to saying that the eigenvectors, remember the eigenvectors are the states in which you definitely measure lambda A. The eigenvectors are the states where if you make a measurement of M, you will get lambda A. B is the state in which if you measure M, you will definitely get lambda B. These states which correspond to distinctly different measurable quantities, its distinctly different measurable values of the same quantity, they are orthogonal. The basic idea is orthogonal states are distinguishable, measurably distinct, and you can distinguish between them by measuring appropriate observables. For example, we already have some examples. The two examples that we had, one of them was 1, 0 and 0, 1. These were the two eigenvectors of sigma 3. Notice that they're orthogonal because they don't share any elements, this times this plus this times this is 0. The orthogonality is the thing that tells you that in the unique experiment, you can distinguish between them. You distinguish between them by measuring the quantity which is represented by the Hermitian operator that has too many words but you get the idea. Let me prove to you that if this is so with two different lambdas, then A and B must be orthogonal to each other. This is not very hard, we just have to juggle a few definitions. We first of all multiply this equation, the first equation on the left, by B. So we'll rewrite it then as BmA equals lambdaA times BA. So I'm going to multiply both sides by B, by the vector B. I'm going to do a similar thing here. I'm going to multiply both sides by A. So we get AmB equals lambdaB times AB. I would like to switch. Let's see, so what do we have to do next? This always confuses me. We have the two equations. Now let's take this equation. We want to conjugate one of these equations. We want to take the complex conjugate of one of these equations. Which one should we choose? The second one. This is BmA complex conjugated is what it is. Remember since n is permission, we don't have to complex conjugate it. That's equal to lambdaB star times BA. Now I forgot what to do next. Yes? Yes? Yeah, now we'll just conjugate everything. Oh, did I do something? Which? Did I forget to conjugate something? What are we doing? Sorry, this is conjugated, right? Yeah, good. But before we do that, let's complete. What I've done here is said if I interchange, let's see, this one, if I interchange A and B, and at the same time take the Hermitian conjugate of M, which I don't have to do because it's Hermitian, and take the complex conjugate, I get the complex conjugate of the right-hand side. Now just complex conjugate the whole equation. Let me just throw away the star. Throw away the star. Now I have two equations. Let's get rid of the middle one here. One of them says BmA equals lambdaA times BA. The other one says BmA equals lambdaB times BA. It will not surprise you that the conclusion is there are two possible conclusions. The conclusion is that lambdaA equals lambdaB. But I told you already, I'm talking about two eigenvalues which are known to be different. So by assumption lambdaA and lambdaB are not the same. If lambdaA and lambdaB are not the same, there's only one way this can be true. In other words, and it's that BA is zero. In other words, let's subtract these two equations. We subtract the two equations. On the left side we get zero. On the right-hand side we get lambdaA minus lambdaB times BA. BA. If a product is equal to zero, that means one or the other factor is equal to zero. The product of two things can only be zero if one or the other factor is equal to zero. One possibility is that lambdaA is equal to lambdaB. True, this is true. It's a possibility. But I explicitly said let us consider two different eigenvalues, two different values of the same observable. If they're different, then lambdaA minus lambdaB is not equal to zero, and the only possibility is that B and A are orthogonal to each other, that inner product of B with A is equal to zero. So it follows for emission matrices, if I have two distinct eigenvalues, I can immediately conclude that these are orthogonal vectors. Let's check it out for the eigenvectors of sigma2. Remember where are the eigenvectors of sigma2? Here they are. Here they are right here. No, I'm sorry, sigma1 is what I meant. Sigma1. Let's check it out for sigma1. Notice that the entries here are real, and we don't have to complex conjugate anything. All the entries here are real. To take the inner product of two vectors, all of whose entries are real, all we have to do is take the product of the first two entries, sorry, the product of this entry with this entry, and add to it the product of this one with this one. So we get 1 over square root of 2 times 1 over square root of 2, that's a half, and then 1 over square root of 2 times minus 1 over square root of 2, that's minus a half, a half plus minus a half is zero, this vector is orthogonal to this vector. So there's an example of the orthogonality of eigenvectors with different eigenvalues for Hermitian matrices. As I said, the significance of this is deep, and it says that whenever there is an observation that you can do that will uniquely distinguish between two possible states of a system or two possible values of an observable, the states associated with them are orthogonal to each other. Or you can say it the other way, if two states are orthogonal, it means that there exists some measurement you can do which will uniquely tell you which one of the two is realized in your system. So that's one of the fundamental theorems of quantum mechanics, and let's take a little break. The eigenvectors of sigma 2 are so that you can check them yourself. The two eigenvectors of sigma 2 are 1i, stick a 1 over square root of 2 to normalize it in front of it, and the other one is 1 minus i, 1 over square root of 2. One of them has eigenvalue plus 1, and the other has eigenvalue minus 1. So you can check that yourself. It's a little exercise. Please do it. And you see that sigma 1, sigma 2, and sigma 3 are all similar to each other and that they have the same eigenvalues, plus 1 and minus 1. They all square and give 1. They have another interesting property that we'll come to in a moment, I think. But let me tell you what their physical meaning is. They are observable quantities, and with respect to thinking about the electron spin. Now, I told you the first time, or was it the second time, the electron spin classically in classical physics is like a magnet. It has a north pole and a south pole, and it's a pointer. It points along an axis. If we were talking about, and it has what's called a magnetic moment, and that magnetic moment is a, I hesitate to use the word vector, but the kind of vector that points in ordinary space, I guess we called it a pointer, right? A magnetic moment. Or we can call it the spin. It's the same, apart from a numerical constant, the spin and the magnetic moment are the same thing. It's a pointer in space. As such, it has components. It has components in the vertical axis, for example, as vector has a component which is about over here in the vertical direction. It has a component in the horizontal direction over here and a component over here in the other horizontal direction. And those components characterize the pointer. They are measurable quantities for a magnet. You could decide to measure the x component of the magnetic moment. You could decide to measure the y component, or you could decide to measure the z component. You might think, classically, you'd be right, that you could measure all three of them simultaneously. You could do an experiment to measure all three of the components of the magnetic moment simultaneously, and in that way figure out exactly where the magnetic moment is pointing. Let's save that question whether you can measure all of them simultaneously for an electron or not, but the answer is no. But you can measure any one of them, the x component, the y component, or the z component. How do you do it? Suppose I wanted to measure the x component. The x is this way. I put it in a big magnetic field, and I check whether or not it emits a photon. If it emits a photon, then it had one component or the other component. And the only possible answers are minus one and plus one. In other words, you'll either see a photon or you won't see a photon. So the components of the electron spin have only two possible values, the x component, the y component, and the z component. And what's more, we're going to see in a minute that the component in any old direction, if you measure it, will only have two possible values. Now, that's a little hard to think about, but let me tell you right now what sigma one, sigma two, and sigma three are is they represent the observable values of the components of the electron spin along the three axes of space, the three axes of ordinary space. I'll show you how that works and how we can construct the component along any direction in a moment. But notice that they do have sort of very similar properties, same eigenvalues. So if you measure the possible values that you can get in an experiment, for sigma one, you get one minus one. For sigma three, you get one and minus one. For sigma two, you get one and minus one. That's all you can ever get when you actually measure them. And we're going to see the same is true along any axis. But before we do, let me tell you the last, I think it's the last postulate of quantum mechanics. And here is the probability interpretation. Suppose you prepare a system in a state, a particular state, and let's see, let's call that state B. Somebody gave you an electron or whatever it happens to be in a particular quantum state B. Now, there is some observable. Let's call it M, some observable, M that has an eigenvalue lambda A, which means that if you measure M, you can get the value lambda A, or you may not get the value, you may get some other lambda. But if you measure M, you get one of its eigenvalues. And the associated eigenvector is A. And you can see that there's a difference between them just to indicate. M is an observable, meaning to say it's a Hermitian operator. Lambda A is one of the possible values that you can measure. And the associated eigenvector is lambda A. B is just any old state, any old state, whatever, not particularly an eigenstate of M, it could be or not. It's just any old arbitrary state. Then we could ask the question, if the system is prepared in B and we measure M, what's the probability that the answer will be lambda A? Lambda A are the possible values that you can measure. M is the thing you're measuring. And A is the eigenvector of M with eigenvalue lambda A. The answer is that the probability is given by the expression, the inner product of A with B times its complex conjugate. One way, times its complex conjugate, the easy way to write the complex conjugate is just to write it at time B A. Which is just the same as multiplying this by its complex conjugate. Remember that when you multiply a number by its complex conjugate, you always get a real positive number. If A and B are normalized, it's an easy theorem to prove that this is less than one, less than or equal to one. If A and B are unit length, then the inner product between them has a magnitude less than one. So this number is always less than or equal to one, and it is the probability, it's the probability, again I'll state what it is, it's the probability that if you start with an arbitrary state A, a system prepared in an arbitrary state A, and you measure the measurable quantity M, this is the probability that you get lambda A. I'm prepared to say it. I'm prepared to say it wrong. Sorry, what did I say? Did I say it wrong? You said it was prepared to state A. Oh, sorry. Let me say it again. I thought I said it the right way, but I probably said it wrong. It's prepared in the state B, arbitrarily prepared in the state B, and the probability that you measure lambda A will be the inner product between B and A squared, times its complex conjugate, a real positive number given by the square, or I call it the square, but I mean times the complex conjugate, of A with B. That's the probability. Now notice it's sort of symmetric in A and B. It is symmetric in A and B, but for the moment think of it asymmetrically. You prepare the system in A, you prepare the system in B, and you measure M and ask what the probability for lambda A is. That's the basic postulate of quantum mechanics, always assuming that all of your vectors are normalized. Always assuming that your vectors are normalized, that your state B is normalized, and that the eigenvector A is normalized. Is that the cosine squared of the phase angle between them? If these were real vectors in a real vector space, a two-dimensional vector space, it would be the cosine of an angle. Since they're both complex vectors, they have an angle between them. Is that equal to the cosine squared of that angle? What do you mean they have a cosine angle between them? No such concept. What is true is that if these are just ordinary three-dimensional vectors, real vectors, and ordinary three-dimensions, that the dot product of two vectors, two unit vectors, is equal to the cosine of the angle between them. So it's kind of like the cosine of the angle, but it's not a real thing, not real. In general, this won't be real. That's why we multiply it by its complex conjugate. Yeah. By the theorem, it just proves that B is the eigenvector of the different eigenvalues that are part of the equation. Good, good, good, good, good, excellent. If B happens to be one of the eigenvectors of M with a different eigenvalue, then A and B are orthogonal. And so what it says is that if you prepare the system in one eigenvector corresponding to a different eigenvalue, then the probability that you get A is zero. Good, perfect. So if you prepare it in one eigenvector and you measure the probability that you get a different eigenvalue is zero. Good. So that's part of the probabilistic interpretation here. And all it says is if a quantity is definitely equal to one thing, it's surely not equal to the other thing, but expressed in terms of inner products. A little more complicated, but yes. Yeah, we're going to do many, many such things. But this is the simplest of the things we could do. And for the moment, this is our basic quantum postulative probabilities. Let's give an example. What's the probability? Let's take the case where sigma three is equal to one. Let's see, let's say it this way. Here we prepare B is equal to one zero, which means, by the way, that sigma three is equal to plus one. But now we're not going to measure sigma three. We're going to measure sigma one. We're going to measure sigma one. And we're going to ask what's the probability that when we measure sigma one that we get plus one or minus one, as the case may be. So let's ask what's the probability that if we prepare, here's the experiment, we prepare the system with the spin up by putting it in a big magnetic field pointing upward. And then we measure sigma one, which is the component of the spin along the x-axis. What's the probability that we get plus one? What's the probability that we get minus one? You can guess, but we'll work it out. So what is A? A is an eigenvector of sigma one with eigenvalue, let's say, plus one. So A is equal to one over root two, one over root two. What's the inner product between these? Quick, quick, quick, quick. One over square root of two, right? One over square root of two, what's the square of it? One half. So A, B is one over square root of two, and times its complex conjugate, that's usually written this way, times its complex conjugate, its absolute value squared, which is the same as multiplied by its complex conjugate, is one half. So what this tells us is if we prepare an electron in the upstate and we measure the x component of spin, we have a probability one half for getting plus, and it will also be true that we will have probability one half for getting minus. To see that, what's the eigenvector that's associated with minus one for sigma one? We put a minus in here. It doesn't change the answer. Still one half. So there's an example, and this is a characteristic example. You prepare a system, in this case the system is prepared with the spin vertically up, and it's measured in the horizontal direction. Let's do the opposite. We're going to get the same answer. Let's do, we start B now. This time we first prepare the electron horizontally, and we measure the vertical component. It's going to be the same in a product. The horizontal, this time B, just all we've done is interchange B and A. I forgot which was which now, B and A. We've prepared the system with its spin pointing, with its x component positive, and we've measured the z component. It's just exactly the same thing, one half. Let's do a more complicated case. I've told you what the eigenvectors of sigma two are. So let's ask the most complicated question we can ask. What's the probability that if we prepare the state with sigma one equal positive one? That means B is equal to one over square root of two, one over square root of two. And this time we measure sigma two, not sigma three, but sigma two. So this time B is one of the two eigenvectors of sigma two. Let's take this one, one over square root of two, I over square root of two. Sorry, this is A now, this is A. The eigenvector of sigma two. So we start with the spin pointing along the x-axis, and we measure it along the y-axis. We start with the pointing along the x-axis. That's an eigenvector of sigma one or along the one axis. And now we measure it along the two axis. Okay, can somebody compute the inner product of these two? I think I can. The inner product of these two, we first of all have to complex conjugate. So don't forget to complex conjugate. But it's one over the square root of two times one over the square root of two is one half. And then plus, but we have to complex conjugate, so that means minus I over two. One over square root of two times one over square root of two is one half with an I. This is the inner product of A with B. I don't know if it's A with B or B with A, but it doesn't matter. What's the complex conjugate of this with itself? Let's work it out. We have to multiply it by one over two plus I over two. Quick answer? One. Not one. What was that? Mm-hmm. Okay, the imaginary parts cancel. We have a one half times a one half with an I, so we have a minus one half times a one half with a minus I. So the imaginary parts cancel. The real parts are one half times one half is one quarter plus one quarter, one half. Okay? So we see if we, the way to say this now is if we line up the spin in any direction by a strong magnetic field, and then we measure the spin in a perpendicular direction, we have a half problem in any, in fact, in any perpendicular direction, we have a probability of a half that it's this way and a half that it's this way. So any polar, it's called the spin polarization, any spin state, if we start it out with a strong magnetic field and freeze it into some direction, if we then remove the magnetic field and measure an orthogonal component to spin, it will have equal probability of being up and down. What about the components of spin in some arbitrary direction? Let's discuss the components of spin along some other axis. We, maybe it's a good time to stop now and, well, let's, let's do it. And I promise to do this again, yeah. Regarding orthogonal and this case, this case versus classical, if you have two vectors that are 90 degrees apart, you would say that... Are we talking about vectors or pointers? Pointers, there are 90 degrees apart, we would often say this orthogonal, and the projection of one and the other equals zero. Yeah, okay, good, good, good, good, good. So let's not, so right, so we have to, distinguish vectors from pointers. Right, so here we have a situation where we have, where we have a pointer which has been forced to point in some direction and we measure the pointiness in a, what's called perpendicular direction, all right? That's not, all right, good. Or let's just say, yeah, okay, good, good, or better yet, let's look at the eigenvector which corresponds to the pointiness plus one in the x direction and compare it with the eigenvector that corresponds with the pointiness in another direction. They're not orthogonal eigenvectors, they're not orthogonal vectors in the, in this sense. Yeah, yeah, right, comes out to one half instead of zero, right, good. What is the thing which has zero probability of pointing up like this? Yeah, the one that's pointing down. The one that's pointing down in this sense is orthogonal to the one pointing up. So we have to, so let's use the word perpendicular to mean ordinary space and orthogonal, okay, here's our vocabulary. Pointers instead of vectors, vectors mean states, pointers mean pointers. Orthogonality means states being orthogonal, being distinctly different from each other, measurably different, and perpendicular means perpendicular at 90 degrees in ordinary space, okay, good. Independent means something a little different. Independent doesn't mean a perpendicular. So let's stick with the word. No, independent simply means it's not pointing in the same direction. And the set of vectors, there's a concept of independence, linear independence with, which is not, which is different than orthogonal, mutual orthogonality. So independence and orthogonality are quite different for linear independence anyway. Okay, now let's come to classical pointers. I'm interested in something, I'm interested now in the component of the pointer along an arbitrary direction, along an arbitrary direction in real space. What direction? Let's pick a direction. How do we pick a direction? We pick a direction by picking a set of components of a unit vector. We pick a unit vector and pick a set of components in, let's call it a unit vector, so unit pointer, my goodness. A unit pointer, we take a unit pointer. Now this is in real three-dimensional space. One, two, three, one is the same as X. Two is the same as Y. Three is the same as Z. Same thing, X, Y, Z. And now I pick some arbitrary pointer. So some arbitrary pointer is pointing along that direction. And let's take it to be a unit vector, which means it's one unit in length. It's described by a set of three components. The sums of the squares of the components have to add up to one. But here are the three components. Let's call this vector N. And a unit vector is usually described by putting a little hat over it like that. N hat means a unit vector. This is a standard notation for a unit vector. A unit pointer, my goodness, please correct me, don't allow me to do that. A unit pointer is indicated by a little hat. And it has a set of components. The components are in X and Y and in Z, or in one and two and in three. Those are the components. And they're basically this component, this component, and this component. You add them up and you get the pointer that's pointing in some arbitrary direction. These are the components of the pointer. Now what about the component of the spin along the direction N? How do we calculate that? That the component of the spin, or the component of the spin pointer, along the arbitrary direction N is given, let's call it the spin. Now we're thinking classically. We're thinking about a classical magnetic moment, a classical spin. The dot product, sigma dot N, is the component of sigma along the direction of N. That's what a dot product is. It's the component of one vector along the axis of another vector. Sorry, pointer. What is sigma dot N? It's the first component, or the X component, called sigma one times N one plus sigma two times N two plus sigma three times N three. Well this gives us an interesting candidate in the quantum mechanics for the operator, which is the component of spin along a direction N. Namely, we take the operator sigma one, multiply it by N one, add two with the operator sigma two times N two, add two with the operator sigma three, and multiply it by N three. That's a candidate, and it is, it is the correct candidate, which we can call sigma dot N, but it's an operator. Why is it an operator? The N's are numbers. They're just numerical numbers that multiply here. Sigma's are matrices. If we multiply a matrix by a number, we just multiply all the entries by the same number, and then we add them up. We can write this down actually, given three N one N two and N three. We can add them up. Let's do it. N one times sigma one, what's sigma one? Sigma one is one one, and if I multiply it by N one, it looks like this. That's sigma one times N one. What about sigma two times N two? Well, sigma two is minus i, i, and now we multiply it by N two. Sigma three times N three. We take N three, which is one minus one, and we multiply it by N three. So that's just N three and three. Zero zero. Now we add them up. What do we get? On the diagonal, these have no diagonal elements. This has diagonal. So we get N three, N three, minus N three. We get N one minus i N two, and N one plus i N two. There's three components, N one N two and N three. The sums of the squares should be equal to one because it's a unit vector. And here is the operator that corresponds to measuring the components of spin along the direction N. So what's the experiment? The experiment is we take an electron and we put it in the magnetic field pointing along the N direction, and we see what we get, up or down. This is the operator that corresponds to that. N three on the diagonal, minus N three, N one minus i N two. Is this her mission? Yes, it's her mission. Real, real, and this one's the complex conjugate of this. So it's her mission. Is it square equal to one? What's that? Any unit vector. Well, I don't expect you to be able to see it offhand. So what I'm going to do is I'm going to show you an important property and let you check it yourself. An important property of these sigma matrices, one that you can check, but I'll derive it's importance and then allow you to check it by yourself. Let's take this sigma one plus sigma two, sigma one N one plus sigma two N two plus sigma three N three. Leave it in this form, not in this form, leave it in this form and square it. What do we get? Well, let's write it out. Sigma one N one plus sigma two N two plus sigma three N three times sigma one N one plus sigma two N two plus sigma three N three. Let's multiply. We're going to get a whole slew of terms. The first set of terms are going to be things like sigma one squared times N one squared. When I multiply this out, I'll get sigma, what's sigma one squared? One. So from the N one squared sigma one squared terms, we'll get N one squared. What about from the sigma two times sigma two term plus N two squared? What about from the sigma three N three term plus N three squared? But what's that? That's one. All right, now let's look at the thing which multiplies N one times N two. Here's the thing which is going to multiply N one times N two, N one N two. And what's it going to contain? It's going to contain sigma one sigma two. But then there's another term like that which is sigma two sigma one. All right, you've got one term which is sigma one on the left, sigma two on the right. And the other term has sigma two on the left, sigma one on the right. Why do I bother writing sigma one, sigma two, and sigma two, and sigma one instead of just calling them both sigma one sigma two? Because when you multiply matrices, the order counts. The order counts when you multiply matrices. Matrices in general have different products when you multiply them in different orders. The technical term is that they don't commute. So it's important that these things all, when you multiply them by themselves, they give you one. Yeah, these are, each of the square of these gives the identity. I know, but you just kind of prove that for the, for the sigma dot N. We're going to prove that. All right, then we have N two and three times sigma two, sigma three plus sigma three, sigma two. And then another term similar, which is N one times N three. All right. Well, this adds up to one, which means the unit matrix. This is the unit matrix one. What's this doing here? This is bad. We don't want that there. We want the square of the component of the spin in any direction to be the same as in every other direction. There's nothing special about any of these directions. Well, try it out. Multiply, let's multiply, let's multiply sigma, this you go home and do yourself. Prove to yourself that sigma one times sigma two is the opposite, is the negative of sigma two times sigma one. They're anti-commute. They're anti-commute. This is a special property. They anti-commute, which is to say that when you change the order, it changes their sign. This is something to check. You sit down and do it numerically. Sigma one sigma two plus sigma two sigma one is zero. Sigma two sigma three plus sigma three sigma two is zero. We have five minutes. We have three minutes. Let's do one case. One case for fun, let's say. Let's do, I've avoided imaginary numbers by doing sigma one and sigma three. And one times sigma three, what's that? That's equal to one one zero zero times one zero zero minus one. And that's equal to, let's see, this time this is zero. This time this is minus one. This time this is one. And this time this is zero. Did I get that right? I think I did. Now let's do it the other way. One zero zero minus one zero one one zero. All right? And the first element one times zero plus zero times one that's zero again. But now let's go to one times one is one. And now we go down to this corner at zero times zero minus one zero. So you notice when you multiply them in opposite order, you get exactly the same thing except for a minus sign. When you add these, sigma one sigma three and sigma three sigma one, add them together you get zero. Same is true for sigma one sigma two and sigma two sigma three. So these are zero. So what do we learn? We've learned that with this particular choice of matrices, the square of any component of sigma along any axis, any axis whatever, is one. That means the eigenvalues of such a matrix are plus and minus one. So any matrix where did I write it? I have it written down. Here it is. The eigenvalues of any matrix of this form are plus and minus one. This we'll check next time. But what it says is the possible measurable values of the spin along any axis of plus or minus one. This is a very weird thing in quantum mechanics. You can only get plus or minus one if you measure it along any axis. But then take another axis. You can also only get plus or minus one and plus or minus one. It's the probabilities which reflect whether two ends are close. We'll come to it next time. That's enough for this time. I've probably saturated you. I've certainly saturated myself. The preceding program is copyrighted by Stanford University. Please visit us at stanford.edu.
 This program is brought to you by Stanford University.  Please visit us at stanford.edu.  I want to make sure we're all on the same page with respect to some very elementary mathematics,  complex numbers.  It's one important concept.  So I just want to go through them very quickly, extremely quickly.  A complex number has a real part and an imaginary part.  And you can plot it on a two-dimensional surface.  The two-dimensional surface horizontal axis represents the real part of the function,  not the function, just the real part of the number.  And the imaginary part of the number is plotted on the vertical axis.  And a point on this plane is an imaginary number.  If this is x and the height is y, then the complex number is z, which is x plus iy.  OK.  We could also represent it another way.  We can represent it in terms of a distance, which we can call r, and an angle that we can call theta.  And now, just from elementary trigonometry, the distance x is r times the cosine of theta.  So that's x is r cosine theta.  And y is r times r times sine theta.  So i times r sine theta.  Or in other words, we can take the r-outs out of it, factor the r out, and write that z is equal to r times  cosine theta plus i sine theta.  The combination cosine theta plus i sine theta is called e to the i theta.  e to the power i theta.  So know that definition.  e to the i theta is cosine theta plus i sine theta.  The reason we call it e to the i theta is because it satisfies certain rules of exponentials.  The rules of exponentials is that when you multiply exponentials, you add the exponents.  e to the 3 times e to the 5 is e to the 8.  3 and 5 is 8.  You can check using elementary trigonometry that if you have two angles, theta and phi,  and you write e to the i phi, which is cosine phi plus i sine phi, and multiply them,  you get something with a real and an imaginary part.  The real part is cosine theta cosine phi minus sine theta times sine phi.  Remember, i times i is minus i, as minus 1.  So if you multiply these two, what you get is cosine theta.  Well, I'll tell you what you get.  Let's simplify it.  You get cosine theta cosine phi minus sine theta sine phi.  Anybody know what that is?  cosine theta plus phi.  Right.  Cosine of theta plus phi.  That's a high school trigonometry formula.  The cosine of the sum of two angles, cosine cosine minus sine times sine.  And then, plus the imaginary part, the imaginary part is cosine theta sine phi plus cosine  phi times sine theta, and that's just sine of theta plus phi.  And by definition, if e to the i theta is cosine plus i sine, then this is just cosine,  then this is just e to the i theta plus phi.  So from elementary trigonometry, we discover that this combination, cosine theta plus i  sine theta, has all of the properties of the exponential of i theta.  Now, all of trigonometry, anything you ever wanted to remember about trigonometry, couldn't  remember, is all stored in this formula.  For example, if you couldn't remember what the cosine of the sum of two angles is, all  you need to do is multiply e to the i theta times e to the i phi in this form, and you'll  discover that cosine theta plus phi is cosine theta cosine phi minus sine theta sine phi.  So all of trigonometry, I don't know why they take a year to teach trigonometry in high  school.  And then at the end of it, they come here to Stanford and they don't know this formula.  Another fact, we could check this in two ways.  We could just use the formula for multiplying exponentials.  e to the i theta, first of all, there's e to the i theta.  That's cosine theta plus i sine theta.  Remember what the complex conjugate of a thing is.  The complex conjugate is just the same number except with the imaginary part changing sign.  So it's the reflected number in the lower half plane.  If the number is in the upper half plane, then it's reflected into the lower half plane  or vice versa.  The complex conjugate of e to the i theta is cosine theta minus i sine theta.  But that's the same thing, that's e to the minus i theta.  That's the same thing as cosine theta minus i sine theta.  And if you remember that minus the sine of theta is the same thing as sine of minus theta,  then you realize this is just e to the minus i theta.  So the complex conjugate of e to the i theta, you just get by changing the sine of the i  here.  And what happens if you multiply these two?  You get one.  I think that it's changing the sine of the theta because that's what happens when you  rotate the clockwise direction instead of the counterclockwise direction.  Right.  This is true, I don't know if everybody called that, but if you multiply e to the i theta  times e to the minus i theta, you get one.  So the class of numbers, which are the class of complex numbers, which are of the form  e to the i theta, are numbers which have the special property that when they're multiplied  by the complex conjugate, the result is one.  And there's another way to say that.  Another way to say it is that x square plus y square is equal to one.  The class of numbers for which x square plus y square, remember, a number times its complex  conjugate is just the sums of the squares of the real and imaginary part.  And so the numbers which have the form e to the i theta, let's specialize to those, are  numbers for which x square plus y square is equal to one.  In other words, they're numbers on the unit circle, on the circle of radius one.  They're simply called numbers, I don't know what they're called, unitary numbers.  Of the form e to the i theta, you can call them unitary numbers.  And they're numbers that have the property, that they lie on the unit circle.  And they times their complex conjugate is exactly equal to one.  Every complex number is a unitary number times a real number, a real positive number, and  the real positive number is just a radius, or the distance to that point on the column.  And that's all we need to know.  That's all there is to complex numbers, there is nothing else.  Nothing else I can think of offhand.  I want to make sure everybody knows.  Ah, the other word for a number which is of the form e to the i theta is that it's called a pure phase.  The angle in here is called the phase angle, the phase, and the radius is called the modulus, I guess.  The modulus means the size of it.  So a number which has no r in front of it, or where r is one,  number on the unit circle is called a pure phase, or just a phase.  So when I swallow the chocolate chip wrong.  When I refer to a pure phase, that means a number of cosine theta plus i sine theta.  Okay, now.  All right, now, the postulates of quantum mechanics.  We want to now today go through the basic postulates of quantum mechanics.  Now, what quantum mechanics is, is it's a calculus for calculating probabilities.  After a while, you get some intuition for it, and maybe you get some pictures of what's going on.  But think of it as a calculus, in other words, a calculational procedure for calculating probabilities.  Probabilities of what? Probabilities for different measurements, different values of measurements.  The things you measure are called observables.  You prepare systems in certain states, not in California or New York,  but in certain configurations that are called states.  And those states are labeled by or described by vectors in a complex vector space.  We've called them alpha, or A, I guess we've called them A.  You can add them, you can subtract them, you can multiply them by numbers,  and they form a complex vector space, number one.  Number two, we normalize them.  In fact, in the particular simple case that we studied, which was the two-level system,  we could describe this as alpha times the upstate, for example, plus beta times the downstate,  where alpha and beta are complex numbers.  The coefficients here are such that alpha star alpha and beta star beta are the probabilities  that if you were to measure up or down, you would measure up with probability alpha star alpha,  down with beta star beta.  And so, just to make the probability altogether equal to one,  we require that the coefficients are such that the sums of the squares,  or the sums of the, I'll call this just the square, it's the square of the, is just equal to one.  All right, there's another way to write the same equation.  It's that the inner product of a real state with itself is just equal to one.  That is equal to alpha star alpha plus beta star beta.  And so, this is the abstract way of writing that the total probability adds up to one.  So states, state vectors are normalized, this is called normalized.  Means that the length of the vector in some sense is equal to one,  and altogether the sums of the probabilities are one.  That's the basic posture with the states.  This is the example of the two-dimensional space of states.  Two-dimensional doesn't mean the world is two-dimensional,  it just means that there are two independent possibilities, up and down, heads and tails.  We discussed how you might measure whether a spin was up or down  by putting it into a magnetic field and seeing if it emits a photon or not.  So, given an arbitrary state of electron, we can measure its spin.  If it gives off a photon, then we say it's down, I guess.  And if it doesn't give off a photon, then we say it's up.  And the relative probabilities or the probabilities for the two are just the coefficients here.  Now, those probabilities, of course, do not completely determine the complex numbers here.  Why not?  Well, if I multiply any complex number by a pure phase, it doesn't change its magnitude.  If I take a complex number and I multiply it by e to the i theta, where theta is any angle,  I get a new number whose magnitude, whose alpha star alpha, is exactly the same as the original starting number.  And so, knowing the probabilities is not enough to completely determine what these coefficients are.  So, there's more information in these coefficients than just the probabilities.  And we're going to find out what that additional information is.  But for the moment, this is...  And, of course, if there's more than two possibilities, up, down, and sideways,  or up, down, and chocolate chip, then we add another one, gamma, chocolate chip, and whatever.  Those are orthonormal vectors.  And we're going to...  And we represented them in a particular basis.  The basis of up and down, we represented them as...  Up was just one zero, and down is equal to zero one.  They're orthogonal to each other.  They're normalized, both of them, the sums of the squares of each individual one is one, so they're called orthonormal vectors.  Orthogonal and normal.  If we have a higher dimensional vector space with more possibilities, chocolate chip,  then we might have three vectors altogether, zero, zero, one.  They are still orthogonal and normalized.  This zero should go here.  Up and down or at right angle?  We're going to have to remember the difference between these vectors, these abstract vectors which represent states,  and vectors in real space, because that's going to come up today.  Pointers. These are pointers, they're not vectors.  And we'll label them slightly differently.  All right, now, what about the quantities that you measure?  They are called observables.  They're the things that you measure.  And by assumption, there may be more than one thing that you can measure about a system,  and you can always reduce them to real numbers.  The results of a measurement are always a collection of real numbers.  You may combine them together, for example, into a complex number,  but the results of real measurements, you know, pointers on a dial or whatever, are real numbers.  And so observables are things which when you measure them, among other things, you get real numbers as a result.  They are represented by matrices or operators.  Strictly speaking, they're represented by linear operators, which you can take to just being matrices for our purpose.  But a special class of matrices, the Hermitian matrices.  Hermitian, H-E-R-M-I-T-I-A-N, Hermitian for Hermite, and Hermite was a mathematician.  Hermitian matrices, and to define them, let's first define a couple of concepts about matrices.  First of all, there's the concept of the transpose of a matrix.  The transpose of a matrix, you have diagonal elements and off diagonal elements.  And there's no reason why it has to be two by two.  Let's take it to be three by three.  There's M-1-1, M-2-2, M-3-3.  And then there's the off diagonal elements.  Let's just put one of them over here.  Here's M-1-2.  And here's M-2-1 and so forth.  And there's other ones, other, they're all there.  The transpose of a matrix is just, you act on the matrix to interchange rows and columns.  All it means is you reflect it about the diagonal.  You just imagine it's written on a piece of paper and you turn over the piece of paper  so that M-1-2 replaces M-2-1 and so forth.  The way to write that, the transpose is just written by putting a T up here.  And it's just the diagonal elements are unchanged.  But the off diagonal elements, M-1-2 becomes M-2-1 and M-1-2, they just interchange.  And likewise over here and so forth.  Okay, that's called the transpose.  We can write that in the form, the transpose matrix, we can call it ij,  the ijth element of it is just the j-i-th element of the original matrix  and we can get the change of i and j, of rows and columns, it's called transpose.  The next concept is the Hermitian conjugate.  I'll write it out, Hermitian conjugate.  And it's a kind of complex conjugation, but complex conjugation for matrices, Hermitian conjugate.  It involves two operations.  The first is to transpose and then the complex conjugate.  And it's represented by a dagger.  It's represented by a dagger.  It's represented by, first of all, transposing, but then complex conjugating everything.  All the elements, the diagonal elements as well as the off diagonal elements.  And we write it by saying that the Hermitian conjugate, the dagger of a matrix,  is you interchange rows and columns and then you complex conjugate.  So let's do an example.  Here's an example.  4 plus 2i, 7 minus i, 6 plus 4i, and 9 minus 2i.  What's the dagger, the Hermitian conjugate of that matrix?  Well, the diagonal elements, they don't move when you transpose.  Transposing a diagonal element stays in the same place, but we have to complex conjugate it.  So we get 4 minus 2i.  Now, the element up here, first we have to transpose, what was this, 4?  4 minus 2i?  Yeah, okay, we have to transpose, which means we put, we interchange rows and columns,  and then complex conjugate.  So then we get 4 plus 2i up here.  This one complex conjugated.  This one complex conjugated goes down here, 6 minus 4i,  and the diagonal matrix element stays where it is, but it gets complex conjugated.  7 plus i.  Alright, so that's the idea of Hermitian conjugation.  Flip the rows and columns, and then complex conjugate.  A Hermitian matrix, well first let's talk about symmetric matrices.  Symmetric matrices, let's just go back to the idea of transpose without the complex conjugation.  Just transpose.  A symmetric matrix is one that is equal to its own transpose.  In other words, when you transpose it, you get back the same thing.  In order for that to be the case, all that you need is that the elements, the reflected elements are the same as each other.  So it would say that m to 1 has to be equal to m12.  We'll put the same m21 here.  That's the idea of a symmetric matrix, and it just is exactly what it says.  It's sort of symmetric with respect to flipping it or to reflecting it about the main diagonal.  A symmetric matrix satisfies mij equals mji.  That's a symmetric matrix, very simple concept.  A Hermitian matrix is a little more complicated,  and what it says, a Hermitian matrix is a matrix which is equal to its own Hermitian conjugate.  So if you have a matrix and your Hermitian conjugate it,  if it is equal to the original matrix, it's called Hermitian.  It's just called Hermitian.  That's the definition of a Hermitian matrix.  Let's write a Hermitian matrix down.  Write the formula for it.  Yes, yes.  There's the formula for it right here.  mij is mij complex conjugate.  Say it again.  I think it's written correctly.  mij is mji conjugate.  It's correct.  What it says, first of all, is that the diagonal elements are real.  For example, this says if you interchange i and j on a diagonal element,  you just get the same diagonal element,  so it says that m11 is equal to m11 star.  That means that m11 is real.  So real numbers on the diagonal.  Not all is the same real number.  I just wrote real, real, real, real,  meaning any real number in any one of these places.  Then on the off diagonal elements over here,  let's suppose that this off diagonal element over here is z,  the complex number z,  then the reflected off diagonal element has to be the complex conjugate.  Then it's called Hermitian.  So 2, 7, 1 plus i and 1 minus i is Hermitian.  Real elements on the diagonal and complex conjugates when you reflect.  This is a kind of reality property.  A thing being its own complex conjugate for a number says that it's real.  For a matrix, the concept of reality is not that the matrix elements are all real.  Not that each matrix element separately is its own complex conjugate,  but each matrix element is the complex conjugate of the reflected matrix element.  We'll find out.  We're going to see very shortly why this is a good definition.  Okay, I'm going to write down some theorems.  These are your homework to prove.  They're very, very simple.  But just they don't have names because they're too simple.  Very elementary theorems.  First of all, if you take two vectors, a and b,  and you take the inner product of a with b.  Now, remember, when you form the row vector b,  you use the complex conjugate elements.  This is equal to b star, b star 1, b star 2,  complex conjugate times a1, a2.  The first theorem is that if you interchange a and b,  that will put the a's here and the b's here.  What do you get?  Is this correct?  No.  Complex conjugate.  The way to think of it is that the row vector is like the complex conjugate  of a certain column vector.  And if you interchange the order of these,  you're taking the complex conjugate of b and the complex conjugate of a,  and if you multiply complex conjugates, it's just complex conjugates.  So the first elementary theorem to check is that the inner product of b  with a is the complex conjugate of the inner product of a with b.  Now, it's extremely elementary.  Anybody who's seen it, of course, is very familiar with it.  Next statement.  Now, this one is less obvious.  Take any matrix, Hermitian or not Hermitian,  and I'll, and, all right.  Now, this is an operation that I haven't defined yet,  but I'm going to define it for you right now.  It's very simple.  You have a matrix m and you multiply it by a.  That gives you a new vector.  This is a new vector.  And you take that new vector and you take the inner product with b.  And you can work out what this is in terms of the matrix elements of m  and the entries that go into a and b.  It's a very definite expression.  First, multiply the matrix by a, or multiply a by the matrix.  You'll be left over with a new vector that you can call a prime,  and you take it in a product of b.  This is related to, but not equal to,  I'm going to write equal, but then we're going to change it,  and we're going to change everything.  A, put a on the left.  That's kind of the complex conjugate of a.  Put b on the right and put the Hermitian conjugate of m.  Everybody has been complex conjugated,  and all rows and columns have been interchanged.  When you interchange a and b,  you're interchanging a column vector with a row vector,  and you're complex conjugating.  The same for m. When you take a Hermitian conjugate,  you interchange rows and columns,  and you complex conjugate.  This is a number, incidentally.  This quantity is a number, a complex number in general.  m on a gives you a vector.  The inner product of a vector with another vector is a number.  A simple number, it's not a matrix, it's not a vector,  it's just a number.  This equation is not right.  What do I have to do to it?  Complex conjugate it.  I have complex conjugated a,  I have complex conjugated b,  and I have complex conjugated m,  so I have to take the whole thing and complex conjugate it.  That's a simple fact about complex conjugation  of matrices and vectors,  but let's suppose that our matrix is Hermitian.  Let's suppose that the matrix in question is Hermitian.  If it is Hermitian, I can erase this conjugation sign here.  If it's Hermitian, it's equal to its own Hermitian conjugate,  definition of Hermitian.  So for a Hermitian matrix,  and from now on, let's think about Hermitian matrices.  For Hermitian matrices,  for Hermitian matrices,  this is called a matrix element,  this is called the BA matrix element of m.  The BA matrix element of m is the same as the AB matrix element of m,  complex conjugated.  Now, one last thing that will make the point for you,  if a and b happen to be the same vector,  if a and b happen to be the same vector,  now let's let b equal a.  Let b equal a.  Look what this says.  It says that the matrix element of m  evaluated between sandwiched,  between two states which are the same,  and the state a on both sides is equal to its own complex conjugate.  What does that say about AMA?  That it's real.  So if you have a Hermitian matrix,  this was a theorem that I was going to ask you to prove,  but I just proved it.  If you have a Hermitian matrix,  then we should give this object a name.  It's called the expectation value.  Why it's the expectation value will become,  or what that has to do with anybody's expectations is something else.  It just has a name.  It's the expectation value of m in the state a.  Now remember, just keep in mind,  the way we're going to be describing states is through vectors.  The way we describe observables is through matrices.  So you can think of this basically as the average value of m  when the state of the system is a.  Yeah.  The bracket notation for average,  does it come from block and matrix?  Yeah, sure.  Of course, it comes from Dirac.  So what do we learn?  We learn that if we have a Hermitian matrix,  its expectation value is real.  Well, we haven't gone through the details yet,  but that sounds like a good thing.  If we have an observable, which is represented in which when you measure it,  always gives you a real number,  then obviously its average value should be real.  And that's what we find, that for a Hermitian matrix,  its expectation value in a given state is real.  That's what makes the Hermitian matrix special,  and what makes the Hermitian matrix natural candidates  to have something to do with observable quantities,  because their expectation values in any state doesn't have to,  in many, many states, any one of them will satisfy this.  Okay, any questions up till now?  If there is a physical value in a particle or qm,  where you, I think a unit time length is a mass,  or you have an imaginary number of atoms,  is there a value in the unit?  Anything that you really measure is always a number.  Now, you can decide to say,  I will describe points, locations of particles on the surface of the Earth  by giving a complex number.  I'll give you an address in Manhattan,  by giving you a, you know, Manhattan is laid out on the grid.  I'll give you a complex number.  But you don't have to do that.  You can represent any observable quantity as a collection of real numbers.  You can always represent the results of an experiment by a series of real numbers.  And those real numbers are called observables.  When they're complex, when you add them together in complex combinations,  just because of the technical historical definition,  we don't call it an observable.  The observables are the collection of real numbers.  The same question. What's a good way to think about measuring numbers  when they show up in physical formulas?  Like, is it like these things exist, but they're not observable?  But then, how do we know that they exist if they're not observable?  I think there's a better...  I think the question that you really want to ask is why do we have to use complex numbers in quantum mechanics?  That's, I think, what you really want to ask.  Why is it that we're driven to complex numbers?  This will become clear, but it has to do with time and reversibility.  And we're not there yet. We're not ready for it.  We could have thought about doing quantum mechanics with all real numbers  and never having gotten complex numbers.  We've missed some very important things, as you will see, as you will find out.  But let's write down the postulates, examine their consequences,  and then I will tell you what would go wrong if you restricted yourself to real numbers.  It would have to do in particular with the evolution of state vectors.  What we have not talked about at all is how things change with time.  And it really has to do with how things change with time, where that I comes from.  So, hold on, hold on to the question, and it'll come back again.  I realize it's a different question than you asked, but I think it's the right question.  What complex numbers are, is they're just a mathematical formalism  for describing pairs of real numbers in certain ways.  Why are we driven to use them in quantum mechanics?  It has to do with time evolution, so we'll come to it.  We're still doing mathematics, we cannot do this subject without mathematics.  I mean, I can fake it.  I can tell you lots of stuff about quantum mechanics  and give you all sorts of mystical experiences about entanglement.  And when you walk out of here, you will know no more about entanglement  than you knew when you walked in, but you'll have this fuzzy feeling about it  and you'll say, oh, isn't it wonderful, and it's a mysterious,  but you won't have any idea what it is.  So, if we want to really do it, honestly, we are forced to do this mathematics.  We want to do better than that.  We want to really do it right.  In the simplest possible context, I mean, I'm giving you the simplest possible context  to discuss quantum mechanics, basically the two-level system,  and we'll do two-two-level systems, and then we'll talk about the entanglement  of two-two-level systems, and we'll understand it in some detail,  and you'll know what the words really mean.  The next concept, and it's basically the last mathematical concept that we'll need for today,  and possibly for quite a while, is the concept of eigenvalues and eigenvectors,  particularly eigenvalues and eigenvectors of Hermitian operators.  So, what is an eigenvalue and an eigenvector?  Well, if I give you a matrix or an operator, a matrix or an operator, M,  if it's a, in general, there may or may not be certain vectors  that when I apply the matrix to them, don't change the vector except to multiply it by a number.  In other words, there may or may not be, and typically there will be,  and in fact, in the case of Hermitian matrices, there always are,  certain vectors, we take a matrix, there will exist certain vectors  which have the property, not every vector, there will exist certain vectors  that are associated with that matrix, which are called its eigenvectors,  which have the property that you just multiply by a number.  The number depends on the vector.  I will give you a very, very simple example right now.  Supposing we have a diagonal matrix.  In general, it will be called an eigenvalue and an eigenvector even if it's complex.  But if M is a Hermitian matrix, then it follows that lambda is real.  Let's prove that.  Let's suppose we have a Hermitian matrix, we have found an eigenvector and an eigenvalue.  Let's see what we learn by taking the inner product of this equation with A itself.  We have M, A, A, and multiplying the equation on the left by A.  Lambda is a number, it just comes on the outside, A, A.  Now if you go back in your notes about five and a half minutes,  remember that we proved that for any Hermitian matrix, A, M, A, its expectation value is real.  We also proved that the inner product of A with itself is real.  So we have an equation that a real number is equal to a real number times an eigenvalue.  Well, that means that the eigenvalue is real.  Or another way to write it is to divide it by A, A, and the ratio of two real numbers is certainly real.  So the first statement is if we've managed to succeed and find an eigenvector and an eigenvalue of a Hermitian matrix,  the eigenvalue will be real.  Now I'm going to tell you the secret.  The secret is that if M is an observable,  then the values of it that you can measure when you do an experiment are its eigenvalues.  That means the result of an experiment can measure M will be one of the eigenvalues of the matrix M.  That's the significance of eigenvalues.  They're the possible measurable values of the observable that's represented by M.  Let me give you some examples.  Oh, yes.  The lambdas, the eigenvalues are the values that you measure.  The eigenvector that's associated with that particular eigenvalue, that eigenvector represents a state where when you measure the quantity M with certainty, you get value lambda.  So the eigenvalues and the eigenvectors go together.  There are certain states for any given observable.  There are certain states which when you measure that particular observable,  you will with probability one get a particular answer and that answer is the eigenvalue.  So let's just do an example or two.  The easiest examples involve diagonal matrices.  Here's a diagonal matrix.  Let's just give it some letters.  M11, zeroes, just two by two is good enough for us.  M22, there's a diagonal matrix.  What are its eigenvectors?  Remember, they're vectors which when you multiply them by the matrix, you just get the original vector back times a number.  I'll just tell you the answer.  There are two eigenvectors of this matrix.  One of them has a one in the upper place here.  Let's check.  What does this matrix do when it multiplies this vector?  Well, we take the row, multiply it by the column.  That puts in the upper place here M11.  But how about the lower place?  Zero times one plus M22 times zero.  Zero.  But this vector is just M11 times one zero.  So you see what's happened.  The matrix times the vector gives you a number times the original vector.  In other words, in particular, it doesn't change.  Yeah, it just gives you the back the original vector times a numerical number.  In this case, the numerical number happens to be M11.  So the vector one zero is an eigenvector of this matrix with eigenvalue M11.  There's another eigenvector and the other eigenvector is to put a zero here and a one here.  Let's see what we get.  In the top place, we get M11 times zero plus zero times one is zero.  And then we get zero times zero plus M22 times one, M22.  And that's just M22 times zero, one.  So we found another eigenvector, matrix times vector equals number times the same vector.  And this time, the eigenvalue is M22.  So the two vectors, zero one and one zero, are both eigenvectors,  but with two different eigenvalues, M11 and M22.  You can see where this is going.  For example, if I'm interested in up, down, and the spin of an electron,  I might describe it by a matrix with a one and a minus one.  Then the eigenvalues would just be one and minus one.  The eigenstates would be one zero.  That's the correct equation.  So the state with the electron up is an eigenvector of a certain observable,  which is represented by a matrix.  And the other eigenvector is zero one, and that is equal to minus zero one.  The minus comes from this minus sign over here.  So the two states of the electron, up and down, are eigenvectors of the spin operator,  the spin operator, which is usually called sigma three.  This is definition, sigma three is just this.  They're eigenvectors with two different eigenvalues, and you see the pattern  that the value that you measure, either plus one or minus one, is the eigenvalue.  The eigenvector is the state in which the eigenvalue is the desired measured quantity.  Any questions?  And in fact, these are normalized.  Room, yeah, rule. We normalize vectors, yes.  Let me give you another example, which is less, a little bit less trivial.  Incidentally, if it were three by three or n by n matrices,  the same pattern would be there that if the matrices are diagonal,  then the diagonal entries are the eigenvalues,  and the eigenvectors are just vectors with zeros everywhere except in one place, a one.  So you can check that out for yourself, experiment around the diagonal matrices.  Let me give you one other example.  The other example involves an off-diagonal matrix.  These matrices, incidentally, will play an important role in the ones we're talking about,  but from time being, they're just arbitrary matrices.  It's Hermitian. It has real diagonal elements.  Zero is a real number, and the off-diagonal elements are complex conjugates of each other.  One is its own complex conjugate.  It also happens to be symmetric, but it's both Hermitian and symmetric.  Now let's see if we can find the eigenvectors.  Rather than to do any algebra, I'm going to show you what they are,  and then we'll see that they are eigenvectors.  The two eigenvectors in this case,  okay, now what's wrong with this eigenvector?  It's too big. It's not normalized.  One squared plus one squared is two.  But here's a statement.  If a vector is an eigenvector, it doesn't matter what its normalization is.  If you double it or triple it or anything else, it's still an eigenvector.  But if you wanted to normalize it, you would write one over square root of two,  one over square root of two.  It doesn't really matter in checking whether it's an eigenvector or not,  but let's normalize it just to remind ourselves to always normalize vectors.  What is this matrix times this vector?  We take the zero, one, and we multiply it by these two elements.  And what do we get?  We get zero times, well, let me come on the side,  one over square root of two plus one over one over square root of two.  One over square root of two, and likewise in the bottom.  We got back exactly the same vector.  So this is an eigenvector of this operator with eigenvalue plus one.  What about zero, one, one, zero?  Can I find another eigenvector?  Yes, I can.  The reason I can find it is because I know the answer.  But it's not hard to find.  It's not hard to find.  I just want to illustrate the idea of eigenvectors.  Another time I can tell you how you would go about finding them,  but I do want to illustrate it.  The other possibility is one over square root of two minus one over square root of two.  In other words, instead of having the same element here and here,  we have opposite sign.  Let's check it out.  Zero times this, zero, one times this is minus, minus one over square root of two.  And then here we have one over square root of two plus zero.  So we have one over square root of two.  And that's the same thing as minus the vector that we started with.  We have a minus sign here and the opposite sign here.  It is minus the original vector.  So we found another eigenvector, and this time the eigenvalue is minus one.  We found two eigenvectors with eigenvalue plus one and eigenvalue minus one.  I defy you to find any other eigenvector.  There are none.  There are none, and that's a theorem.  There are no more eigenvalues or eigenvectors.  Right.  Right, right, right.  You can always multiply an eigenvector by a number,  but it's still an eigenvector, but no more, no more,  which are not just numerical multiples of the same eigenvector.  Right.  All right, I'm now going to prove a fundamental theorem of quantum mechanics.  Very fundamental about Hermitian matrices.  Am I going too fast?  Well, if I am, yell out.  What did you call the spin matrix?  This one, say it again.  Yeah, yeah, yeah.  Okay, sigma three, and this is definition, one, one, sorry, one minus one, zero, zero,  and it has eigenvalue plus one and minus one.  Sigma one is equal to this one, one, zero, zero.  I'll show you two interesting things, or, well, let's just say,  one interesting about both of these.  The square of each one of these matrices, does everybody know how to square a matrix?  How to multiply a matrix by itself.  Good.  The square of each of these matrices is the same, and it's just plus one.  The square of this matrix, you just square the elements,  one squared and minus one squared are both plus one.  The square of sigma one, let's square sigma one, one, one, zero, zero,  and multiply it by one, one, zero, zero.  Alright, the first element in the upper left-hand corner is the inner product, this times this,  the one times the one gives us a one,  then the zero times one plus one times zero, zero,  one times one plus zero times zero, now sorry,  one times zero plus zero times one is zero,  and one times one plus zero times zero is one.  So both sigma one squared and sigma three squared are both equal to one.  Now that's kind of an interesting fact.  Notice that it's also true of the eigenvalues of the matrices.  They were both plus and minus one.  If a square of a thing is one, it stands the reason that it's either plus or minus one.  So for matrices, in particular two by two matrices like this,  but matrices in general, if the square of a matrix is the unit matrix,  this matrix is called, is the unit matrix, it's called one.  Alright, it's just a unit matrix with ones on the diagonal.  Sigma three squared, sigma three squared is the same as sigma one squared equal to one.  And it's not independent of the fact that the square of each of the eigenvalues is equal to one.  It means that the measured values that you can measure are such that the square is equal to one.  And it means that you can only measure plus one or minus one.  There's one more matrix I'll write down, and it's called sigma two.  And it's equal to minus ii.  We will come back to the sigma matrices.  In fact, we'll probably come back to them tonight.  We found the eigenvectors of sigma one and sigma three.  We have not yet found the eigenvectors of sigma two.  And I will tell you in due time what their significance is.  Yeah.  You can see all this geometrically.  It's a very simple geometric interpretation.  Okay.  If you're going to say that, then tell us what it is.  Oh, okay.  I was going to write something up, but I can put it on the internet.  Tell me how.  Basically, if you look at the unit vectors, if you look at the unit vector,  well, think of e1 and e2, like the z1 and 1, 0, as the unit vectors.  If you look at a matrix, if you look at the column vectors,  those are the images of the unit vectors.  I know what you're saying.  So forget it.  So one of them is reflected, you're changing e1 and e2.  That's called sigma one.  And so the diagonal is fixed.  And also the anti-diagonal is fixed.  And when you're reflecting in the main diagonal, so the normal to it is also reflected.  Those are eigenvectors, those directions, I think, and that's what you got.  And then the, that's one of the square root of two things,  those are pointing in the diagonal directions.  So they're fixed.  And then the other one is you're sending, you're keeping the e1 fixed,  and e2 is going into its negative.  So you're reflecting in the x-axis.  So the vertical vector is an eigenvector going into its negative.  And also the squares of these things, if you apply them twice, you come back to where you say it.  Not so completely obvious for this one, but the square of this one is also one.  Yeah, well then I take the i out in front, and I have i times something very obvious.  Right. Sigma 2 squared is also one, not minus one, but one.  They all have that property.  They're all, in that sense, they're similar.  That also proves incidentally that the eigenvalues of sigma 2 are also plus or minus one.  It doesn't tell us what the eigenvectors are, but it tells us what the eigenvalues are.  All right, let me prove to you now the following interesting theorem.  Suppose, and this theorem is significant, highly significant.  Supposing you have an observable, and it has more than one eigenvector, in general, will.  The number of eigenvectors that it has is usually equal to the dimensionality of the matrix.  It is the dimensionality of the matrix.  So if you have three by three matrices in a Hermitian, they will typically have three eigenvalues and three eigenvectors.  Four by four will have four eigenvectors and four eigenvalues.  But here's the theorem.  If I have a Hermitian matrix M, and it has two different eigenvalues, two different eigenvalues,  that means that there's an eigenvector A with eigenvalue lambda A,  and there is an eigenvector B with a different eigenvalue lambda B B,  then if lambda A and lambda B are different,  what it says is that A and B will be orthogonal to each other.  Before I prove it, let me say what it means.  What it means is that if there is any observable distinct quantity that you can measure,  which will distinguish between two distinctly different possible measurable answers that you can get,  for some observable, then that's equivalent to saying that the eigenvectors,  remember the eigenvectors are the states in which you definitely measure lambda A.  The eigenvectors are the states where if you make a measurement of M, you will get lambda A.  B is the state in which if you measure M, you will definitely get lambda B.  These states which correspond to distinctly different measurable quantities,  its distinctly different measurable values of the same quantity, they are orthogonal.  The basic idea is orthogonal states are distinguishable,  measurably distinct, and you can distinguish between them by measuring appropriate observables.  For example, we already have some examples.  The two examples that we had, one of them was 1, 0 and 0, 1.  These were the two eigenvectors of sigma 3.  Notice that they're orthogonal because they don't share any elements,  this times this plus this times this is 0.  The orthogonality is the thing that tells you that in the unique experiment,  you can distinguish between them.  You distinguish between them by measuring the quantity which is represented by the Hermitian operator  that has too many words but you get the idea.  Let me prove to you that if this is so with two different lambdas,  then A and B must be orthogonal to each other.  This is not very hard, we just have to juggle a few definitions.  We first of all multiply this equation, the first equation on the left, by B.  So we'll rewrite it then as BmA equals lambdaA times BA.  So I'm going to multiply both sides by B, by the vector B.  I'm going to do a similar thing here.  I'm going to multiply both sides by A.  So we get AmB equals lambdaB times AB.  I would like to switch.  Let's see, so what do we have to do next?  This always confuses me.  We have the two equations.  Now let's take this equation.  We want to conjugate one of these equations.  We want to take the complex conjugate of one of these equations.  Which one should we choose?  The second one.  This is BmA complex conjugated is what it is.  Remember since n is permission, we don't have to complex conjugate it.  That's equal to lambdaB star times BA.  Now I forgot what to do next.  Yes?  Yes?  Yeah, now we'll just conjugate everything.  Oh, did I do something?  Which?  Did I forget to conjugate something?  What are we doing?  Sorry, this is conjugated, right?  Yeah, good.  But before we do that, let's complete.  What I've done here is said if I interchange,  let's see, this one, if I interchange A and B,  and at the same time take the Hermitian conjugate of M,  which I don't have to do because it's Hermitian,  and take the complex conjugate,  I get the complex conjugate of the right-hand side.  Now just complex conjugate the whole equation.  Let me just throw away the star.  Throw away the star.  Now I have two equations.  Let's get rid of the middle one here.  One of them says BmA equals lambdaA times BA.  The other one says BmA equals lambdaB times BA.  It will not surprise you that the conclusion is there are two possible conclusions.  The conclusion is that lambdaA equals lambdaB.  But I told you already, I'm talking about two eigenvalues  which are known to be different.  So by assumption lambdaA and lambdaB are not the same.  If lambdaA and lambdaB are not the same,  there's only one way this can be true.  In other words, and it's that BA is zero.  In other words, let's subtract these two equations.  We subtract the two equations.  On the left side we get zero.  On the right-hand side we get lambdaA minus lambdaB times BA.  BA.  If a product is equal to zero,  that means one or the other factor is equal to zero.  The product of two things can only be zero  if one or the other factor is equal to zero.  One possibility is that lambdaA is equal to lambdaB.  True, this is true.  It's a possibility.  But I explicitly said let us consider two different eigenvalues,  two different values of the same observable.  If they're different, then lambdaA minus lambdaB is not equal to zero,  and the only possibility is that B and A are orthogonal to each other,  that inner product of B with A is equal to zero.  So it follows for emission matrices,  if I have two distinct eigenvalues,  I can immediately conclude that these are orthogonal vectors.  Let's check it out for the eigenvectors of sigma2.  Remember where are the eigenvectors of sigma2?  Here they are.  Here they are right here.  No, I'm sorry, sigma1 is what I meant.  Sigma1.  Let's check it out for sigma1.  Notice that the entries here are real,  and we don't have to complex conjugate anything.  All the entries here are real.  To take the inner product of two vectors,  all of whose entries are real,  all we have to do is take the product of the first two entries,  sorry, the product of this entry with this entry,  and add to it the product of this one with this one.  So we get 1 over square root of 2 times 1 over square root of 2,  that's a half, and then 1 over square root of 2 times minus 1 over square root of 2,  that's minus a half,  a half plus minus a half is zero,  this vector is orthogonal to this vector.  So there's an example of the orthogonality of eigenvectors  with different eigenvalues for Hermitian matrices.  As I said, the significance of this is deep,  and it says that whenever there is an observation that you can do  that will uniquely distinguish between two possible states of a system  or two possible values of an observable,  the states associated with them are orthogonal to each other.  Or you can say it the other way, if two states are orthogonal,  it means that there exists some measurement you can do  which will uniquely tell you which one of the two is realized in your system.  So that's one of the fundamental theorems of quantum mechanics,  and let's take a little break.  The eigenvectors of sigma 2 are so that you can check them yourself.  The two eigenvectors of sigma 2 are 1i,  stick a 1 over square root of 2 to normalize it in front of it,  and the other one is 1 minus i, 1 over square root of 2.  One of them has eigenvalue plus 1, and the other has eigenvalue minus 1.  So you can check that yourself.  It's a little exercise. Please do it.  And you see that sigma 1, sigma 2, and sigma 3 are all similar to each other  and that they have the same eigenvalues, plus 1 and minus 1.  They all square and give 1.  They have another interesting property that we'll come to in a moment, I think.  But let me tell you what their physical meaning is.  They are observable quantities, and with respect to thinking about the electron spin.  Now, I told you the first time, or was it the second time,  the electron spin classically in classical physics is like a magnet.  It has a north pole and a south pole, and it's a pointer.  It points along an axis.  If we were talking about, and it has what's called a magnetic moment,  and that magnetic moment is a, I hesitate to use the word vector,  but the kind of vector that points in ordinary space,  I guess we called it a pointer, right?  A magnetic moment.  Or we can call it the spin.  It's the same, apart from a numerical constant,  the spin and the magnetic moment are the same thing.  It's a pointer in space.  As such, it has components.  It has components in the vertical axis, for example,  as vector has a component which is about over here in the vertical direction.  It has a component in the horizontal direction over here  and a component over here in the other horizontal direction.  And those components characterize the pointer.  They are measurable quantities for a magnet.  You could decide to measure the x component of the magnetic moment.  You could decide to measure the y component,  or you could decide to measure the z component.  You might think, classically, you'd be right,  that you could measure all three of them simultaneously.  You could do an experiment to measure all three of the components  of the magnetic moment simultaneously,  and in that way figure out exactly where the magnetic moment is pointing.  Let's save that question whether you can measure all of them simultaneously  for an electron or not, but the answer is no.  But you can measure any one of them, the x component,  the y component, or the z component.  How do you do it?  Suppose I wanted to measure the x component.  The x is this way.  I put it in a big magnetic field,  and I check whether or not it emits a photon.  If it emits a photon, then it had one component or the other component.  And the only possible answers are minus one and plus one.  In other words, you'll either see a photon or you won't see a photon.  So the components of the electron spin have only two possible values,  the x component, the y component, and the z component.  And what's more, we're going to see in a minute  that the component in any old direction, if you measure it,  will only have two possible values.  Now, that's a little hard to think about,  but let me tell you right now what sigma one, sigma two, and sigma three are  is they represent the observable values of the components of the electron spin  along the three axes of space, the three axes of ordinary space.  I'll show you how that works and how we can construct the component  along any direction in a moment.  But notice that they do have sort of very similar properties, same eigenvalues.  So if you measure the possible values that you can get in an experiment,  for sigma one, you get one minus one.  For sigma three, you get one and minus one.  For sigma two, you get one and minus one.  That's all you can ever get when you actually measure them.  And we're going to see the same is true along any axis.  But before we do, let me tell you the last,  I think it's the last postulate of quantum mechanics.  And here is the probability interpretation.  Suppose you prepare a system in a state, a particular state,  and let's see, let's call that state B.  Somebody gave you an electron or whatever it happens to be in a particular quantum state B.  Now, there is some observable.  Let's call it M, some observable, M that has an eigenvalue lambda A,  which means that if you measure M, you can get the value lambda A,  or you may not get the value, you may get some other lambda.  But if you measure M, you get one of its eigenvalues.  And the associated eigenvector is A.  And you can see that there's a difference between them just to indicate.  M is an observable, meaning to say it's a Hermitian operator.  Lambda A is one of the possible values that you can measure.  And the associated eigenvector is lambda A.  B is just any old state, any old state, whatever,  not particularly an eigenstate of M, it could be or not.  It's just any old arbitrary state.  Then we could ask the question, if the system is prepared in B and we measure M,  what's the probability that the answer will be lambda A?  Lambda A are the possible values that you can measure.  M is the thing you're measuring.  And A is the eigenvector of M with eigenvalue lambda A.  The answer is that the probability is given by the expression,  the inner product of A with B times its complex conjugate.  One way, times its complex conjugate,  the easy way to write the complex conjugate is just to write it at time B A.  Which is just the same as multiplying this by its complex conjugate.  Remember that when you multiply a number by its complex conjugate,  you always get a real positive number.  If A and B are normalized, it's an easy theorem to prove that this is less than one,  less than or equal to one.  If A and B are unit length, then the inner product between them has a magnitude less than one.  So this number is always less than or equal to one, and it is the probability,  it's the probability, again I'll state what it is,  it's the probability that if you start with an arbitrary state A,  a system prepared in an arbitrary state A,  and you measure the measurable quantity M,  this is the probability that you get lambda A.  I'm prepared to say it.  I'm prepared to say it wrong.  Sorry, what did I say? Did I say it wrong?  You said it was prepared to state A.  Oh, sorry. Let me say it again.  I thought I said it the right way, but I probably said it wrong.  It's prepared in the state B,  arbitrarily prepared in the state B,  and the probability that you measure lambda A will be the inner product between B and A squared,  times its complex conjugate,  a real positive number given by the square,  or I call it the square, but I mean times the complex conjugate,  of A with B.  That's the probability.  Now notice it's sort of symmetric in A and B.  It is symmetric in A and B,  but for the moment think of it asymmetrically.  You prepare the system in A,  you prepare the system in B,  and you measure M and ask what the probability for lambda A is.  That's the basic postulate of quantum mechanics,  always assuming that all of your vectors are normalized.  Always assuming that your vectors are normalized,  that your state B is normalized,  and that the eigenvector A is normalized.  Is that the cosine squared of the phase angle between them?  If these were real vectors in a real vector space,  a two-dimensional vector space,  it would be the cosine of an angle.  Since they're both complex vectors,  they have an angle between them.  Is that equal to the cosine squared of that angle?  What do you mean they have a cosine angle between them?  No such concept.  What is true is that if these are just ordinary three-dimensional vectors,  real vectors, and ordinary three-dimensions,  that the dot product of two vectors,  two unit vectors, is equal to the cosine of the angle between them.  So it's kind of like the cosine of the angle,  but it's not a real thing, not real.  In general, this won't be real.  That's why we multiply it by its complex conjugate.  Yeah.  By the theorem, it just proves that B is the eigenvector  of the different eigenvalues that are part of the equation.  Good, good, good, good, good, excellent.  If B happens to be one of the eigenvectors of M  with a different eigenvalue,  then A and B are orthogonal.  And so what it says is that if you prepare the system  in one eigenvector corresponding to a different eigenvalue,  then the probability that you get A is zero.  Good, perfect.  So if you prepare it in one eigenvector  and you measure the probability that you get a different eigenvalue is zero.  Good.  So that's part of the probabilistic interpretation here.  And all it says is if a quantity is definitely equal to one thing,  it's surely not equal to the other thing,  but expressed in terms of inner products.  A little more complicated, but yes.  Yeah, we're going to do many, many such things.  But this is the simplest of the things we could do.  And for the moment, this is our basic quantum postulative probabilities.  Let's give an example.  What's the probability?  Let's take the case where sigma three is equal to one.  Let's see, let's say it this way.  Here we prepare B is equal to one zero,  which means, by the way, that sigma three is equal to plus one.  But now we're not going to measure sigma three.  We're going to measure sigma one.  We're going to measure sigma one.  And we're going to ask what's the probability that when we measure sigma one  that we get plus one or minus one, as the case may be.  So let's ask what's the probability that if we prepare, here's the experiment,  we prepare the system with the spin up  by putting it in a big magnetic field pointing upward.  And then we measure sigma one, which is the component of the spin along the x-axis.  What's the probability that we get plus one?  What's the probability that we get minus one?  You can guess, but we'll work it out.  So what is A? A is an eigenvector of sigma one with eigenvalue, let's say, plus one.  So A is equal to one over root two, one over root two.  What's the inner product between these?  Quick, quick, quick, quick.  One over square root of two, right?  One over square root of two, what's the square of it?  One half.  So A, B is one over square root of two, and times its complex conjugate,  that's usually written this way, times its complex conjugate,  its absolute value squared, which is the same as multiplied by its complex conjugate,  is one half.  So what this tells us is if we prepare an electron in the upstate  and we measure the x component of spin,  we have a probability one half for getting plus,  and it will also be true that we will have probability one half for getting minus.  To see that, what's the eigenvector that's associated with minus one for sigma one?  We put a minus in here.  It doesn't change the answer.  Still one half.  So there's an example, and this is a characteristic example.  You prepare a system, in this case the system is prepared with the spin vertically up,  and it's measured in the horizontal direction.  Let's do the opposite.  We're going to get the same answer.  Let's do, we start B now.  This time we first prepare the electron horizontally,  and we measure the vertical component.  It's going to be the same in a product.  The horizontal, this time B,  just all we've done is interchange B and A.  I forgot which was which now, B and A.  We've prepared the system with its spin pointing,  with its x component positive,  and we've measured the z component.  It's just exactly the same thing, one half.  Let's do a more complicated case.  I've told you what the eigenvectors of sigma two are.  So let's ask the most complicated question we can ask.  What's the probability  that if we prepare the state with sigma one equal positive one?  That means B is equal to one over square root of two,  one over square root of two.  And this time we measure sigma two,  not sigma three, but sigma two.  So this time B is one of the two eigenvectors of sigma two.  Let's take this one, one over square root of two,  I over square root of two.  Sorry, this is A now, this is A.  The eigenvector of sigma two.  So we start with the spin pointing along the x-axis,  and we measure it along the y-axis.  We start with the pointing along the x-axis.  That's an eigenvector of sigma one or along the one axis.  And now we measure it along the two axis.  Okay, can somebody compute the inner product of these two?  I think I can.  The inner product of these two,  we first of all have to complex conjugate.  So don't forget to complex conjugate.  But it's one over the square root of two  times one over the square root of two is one half.  And then plus, but we have to complex conjugate,  so that means minus I over two.  One over square root of two times one over square root of two  is one half with an I.  This is the inner product of A with B.  I don't know if it's A with B or B with A,  but it doesn't matter.  What's the complex conjugate of this with itself?  Let's work it out.  We have to multiply it by one over two plus I over two.  Quick answer?  One.  Not one.  What was that?  Mm-hmm.  Okay, the imaginary parts cancel.  We have a one half times a one half with an I,  so we have a minus one half times a one half with a minus I.  So the imaginary parts cancel.  The real parts are one half times one half is one quarter  plus one quarter, one half.  Okay?  So we see if we, the way to say this now  is if we line up the spin in any direction  by a strong magnetic field,  and then we measure the spin in a perpendicular direction,  we have a half problem in any, in fact, in any perpendicular  direction, we have a probability of a half that it's this way  and a half that it's this way.  So any polar, it's called the spin polarization,  any spin state, if we start it out with a strong magnetic field  and freeze it into some direction,  if we then remove the magnetic field  and measure an orthogonal component to spin,  it will have equal probability of being up and down.  What about the components of spin in some arbitrary direction?  Let's discuss the components of spin along some other axis.  We, maybe it's a good time to stop now and,  well, let's, let's do it.  And I promise to do this again, yeah.  Regarding orthogonal and this case, this case versus classical,  if you have two vectors that are 90 degrees apart,  you would say that...  Are we talking about vectors or pointers?  Pointers, there are 90 degrees apart,  we would often say this orthogonal,  and the projection of one and the other equals zero.  Yeah, okay, good, good, good, good, good.  So let's not, so right, so we have to,  distinguish vectors from pointers.  Right, so here we have a situation where we have,  where we have a pointer which has been forced to point in some direction  and we measure the pointiness in a,  what's called perpendicular direction, all right?  That's not, all right, good.  Or let's just say, yeah, okay, good, good, or better yet,  let's look at the eigenvector which corresponds to the pointiness plus one in the x direction  and compare it with the eigenvector that corresponds with the pointiness in another direction.  They're not orthogonal eigenvectors,  they're not orthogonal vectors in the, in this sense.  Yeah, yeah, right, comes out to one half instead of zero, right, good.  What is the thing which has zero probability of pointing up like this?  Yeah, the one that's pointing down.  The one that's pointing down in this sense is orthogonal to the one pointing up.  So we have to, so let's use the word perpendicular to mean ordinary space  and orthogonal, okay, here's our vocabulary.  Pointers instead of vectors,  vectors mean states, pointers mean pointers.  Orthogonality means states being orthogonal,  being distinctly different from each other,  measurably different, and perpendicular means perpendicular at 90 degrees in ordinary space, okay, good.  Independent means something a little different.  Independent doesn't mean a perpendicular.  So let's stick with the word.  No, independent simply means it's not pointing in the same direction.  And the set of vectors, there's a concept of independence, linear independence with,  which is not, which is different than orthogonal, mutual orthogonality.  So independence and orthogonality are quite different for linear independence anyway.  Okay, now let's come to classical pointers.  I'm interested in something, I'm interested now in the component of the pointer  along an arbitrary direction, along an arbitrary direction in real space.  What direction?  Let's pick a direction.  How do we pick a direction?  We pick a direction by picking a set of components of a unit vector.  We pick a unit vector and pick a set of components in,  let's call it a unit vector, so unit pointer, my goodness.  A unit pointer, we take a unit pointer.  Now this is in real three-dimensional space.  One, two, three, one is the same as X.  Two is the same as Y.  Three is the same as Z.  Same thing, X, Y, Z.  And now I pick some arbitrary pointer.  So some arbitrary pointer is pointing along that direction.  And let's take it to be a unit vector, which means it's one unit in length.  It's described by a set of three components.  The sums of the squares of the components have to add up to one.  But here are the three components.  Let's call this vector N.  And a unit vector is usually described by putting a little hat over it like that.  N hat means a unit vector.  This is a standard notation for a unit vector.  A unit pointer, my goodness, please correct me, don't allow me to do that.  A unit pointer is indicated by a little hat.  And it has a set of components.  The components are in X and Y and in Z, or in one and two and in three.  Those are the components.  And they're basically this component, this component, and this component.  You add them up and you get the pointer that's pointing in some arbitrary direction.  These are the components of the pointer.  Now what about the component of the spin along the direction N?  How do we calculate that?  That the component of the spin, or the component of the spin pointer,  along the arbitrary direction N is given, let's call it the spin.  Now we're thinking classically.  We're thinking about a classical magnetic moment, a classical spin.  The dot product, sigma dot N, is the component of sigma along the direction of N.  That's what a dot product is.  It's the component of one vector along the axis of another vector.  Sorry, pointer.  What is sigma dot N?  It's the first component, or the X component, called sigma one times N one plus sigma two times N two plus sigma three times N three.  Well this gives us an interesting candidate in the quantum mechanics for the operator, which is the component of spin along a direction N.  Namely, we take the operator sigma one, multiply it by N one, add two with the operator sigma two times N two,  add two with the operator sigma three, and multiply it by N three.  That's a candidate, and it is, it is the correct candidate, which we can call sigma dot N, but it's an operator.  Why is it an operator?  The N's are numbers.  They're just numerical numbers that multiply here.  Sigma's are matrices.  If we multiply a matrix by a number, we just multiply all the entries by the same number, and then we add them up.  We can write this down actually, given three N one N two and N three.  We can add them up.  Let's do it.  N one times sigma one, what's sigma one?  Sigma one is one one, and if I multiply it by N one, it looks like this.  That's sigma one times N one.  What about sigma two times N two?  Well, sigma two is minus i, i, and now we multiply it by N two.  Sigma three times N three.  We take N three, which is one minus one, and we multiply it by N three.  So that's just N three and three.  Zero zero.  Now we add them up.  What do we get?  On the diagonal, these have no diagonal elements.  This has diagonal.  So we get N three, N three, minus N three.  We get N one minus i N two, and N one plus i N two.  There's three components, N one N two and N three.  The sums of the squares should be equal to one because it's a unit vector.  And here is the operator that corresponds to measuring the components of spin along the direction N.  So what's the experiment?  The experiment is we take an electron and we put it in the magnetic field pointing along the N direction,  and we see what we get, up or down.  This is the operator that corresponds to that.  N three on the diagonal, minus N three, N one minus i N two.  Is this her mission?  Yes, it's her mission.  Real, real, and this one's the complex conjugate of this.  So it's her mission.  Is it square equal to one?  What's that?  Any unit vector.  Well, I don't expect you to be able to see it offhand.  So what I'm going to do is I'm going to show you an important property and let you check it yourself.  An important property of these sigma matrices, one that you can check,  but I'll derive it's importance and then allow you to check it by yourself.  Let's take this sigma one plus sigma two, sigma one N one plus sigma two N two plus sigma three N three.  Leave it in this form, not in this form, leave it in this form and square it.  What do we get?  Well, let's write it out.  Sigma one N one plus sigma two N two plus sigma three N three times sigma one N one plus sigma two N two plus sigma three N three.  Let's multiply.  We're going to get a whole slew of terms.  The first set of terms are going to be things like sigma one squared times N one squared.  When I multiply this out, I'll get sigma, what's sigma one squared?  One.  So from the N one squared sigma one squared terms, we'll get N one squared.  What about from the sigma two times sigma two term plus N two squared?  What about from the sigma three N three term plus N three squared?  But what's that?  That's one.  All right, now let's look at the thing which multiplies N one times N two.  Here's the thing which is going to multiply N one times N two, N one N two.  And what's it going to contain?  It's going to contain sigma one sigma two.  But then there's another term like that which is sigma two sigma one.  All right, you've got one term which is sigma one on the left, sigma two on the right.  And the other term has sigma two on the left, sigma one on the right.  Why do I bother writing sigma one, sigma two, and sigma two, and sigma one instead of just calling them both sigma one sigma two?  Because when you multiply matrices, the order counts.  The order counts when you multiply matrices.  Matrices in general have different products when you multiply them in different orders.  The technical term is that they don't commute.  So it's important that these things all, when you multiply them by themselves, they give you one.  Yeah, these are, each of the square of these gives the identity.  I know, but you just kind of prove that for the, for the sigma dot N.  We're going to prove that.  All right, then we have N two and three times sigma two, sigma three plus sigma three, sigma two.  And then another term similar, which is N one times N three.  All right.  Well, this adds up to one, which means the unit matrix.  This is the unit matrix one.  What's this doing here?  This is bad.  We don't want that there.  We want the square of the component of the spin in any direction to be the same as in every other direction.  There's nothing special about any of these directions.  Well, try it out.  Multiply, let's multiply, let's multiply sigma, this you go home and do yourself.  Prove to yourself that sigma one times sigma two is the opposite, is the negative of sigma two times sigma one.  They're anti-commute.  They're anti-commute.  This is a special property.  They anti-commute, which is to say that when you change the order, it changes their sign.  This is something to check.  You sit down and do it numerically.  Sigma one sigma two plus sigma two sigma one is zero.  Sigma two sigma three plus sigma three sigma two is zero.  We have five minutes.  We have three minutes.  Let's do one case.  One case for fun, let's say.  Let's do, I've avoided imaginary numbers by doing sigma one and sigma three.  And one times sigma three, what's that?  That's equal to one one zero zero times one zero zero minus one.  And that's equal to, let's see, this time this is zero.  This time this is minus one.  This time this is one.  And this time this is zero.  Did I get that right?  I think I did.  Now let's do it the other way.  One zero zero minus one zero one one zero.  All right?  And the first element one times zero plus zero times one that's zero again.  But now let's go to one times one is one.  And now we go down to this corner at zero times zero minus one zero.  So you notice when you multiply them in opposite order,  you get exactly the same thing except for a minus sign.  When you add these, sigma one sigma three and sigma three sigma one,  add them together you get zero.  Same is true for sigma one sigma two and sigma two sigma three.  So these are zero.  So what do we learn?  We've learned that with this particular choice of matrices,  the square of any component of sigma along any axis, any axis whatever,  is one.  That means the eigenvalues of such a matrix are plus and minus one.  So any matrix where did I write it?  I have it written down.  Here it is.  The eigenvalues of any matrix of this form are plus and minus one.  This we'll check next time.  But what it says is the possible measurable values of the spin along any axis of plus or minus one.  This is a very weird thing in quantum mechanics.  You can only get plus or minus one if you measure it along any axis.  But then take another axis.  You can also only get plus or minus one and plus or minus one.  It's the probabilities which reflect whether two ends are close.  We'll come to it next time.  That's enough for this time.  I've probably saturated you.  I've certainly saturated myself.  The preceding program is copyrighted by Stanford University. Please visit us at stanford.edu.
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