In this video, we will discuss the measurement theory using the algebraic basis that we have established so far. In a Stern-Gerlach experiment, we stated that when we make a measurement, we collapse the quantum state into one of the allowed states. Now, using the linear algebra language, now, we can say more specifically that when we make a measurement, then the quantum state of the system collapses into one of the eigenstates of the operator representing the physical variable that we're measuring. Suppose a system was in a quantum state described by a ket Alpha, and we measure a physical variable represented by an operator A. The eigenstates or eigenkets of the A are represented here as these ket with a sub n symbols here. When we make a measurement, what happens, initially, it was in some quantum state represented by ket Alpha. We make measurement of A momentum, spin energy, whatever it might be, we are collapsing this state into one of the eigenstates of A. That's what's happening. If the initial state happens to be one of the eigenstates of operator A, then obviously, it is already in the eigenstate and therefore, it doesn't change. If the initial state is a sub k sum eigenstate of the operator A, we preserve the state. When the initial state is in the eigenstate of the measured variable, then we do not alter the quantum state. However, if the initial state Alpha is not one of the eigenstates, then it will collapse into one of the eigenstates. Into which one? We don't know. We can only talk about the probability of measuring certain values. The probability of measuring a sub n, the eigenvalue corresponding to this eigenstate, represented by this ket a sub n is given by this quantity here, the inner product between the eigenket a sub n and the initial state vector Alpha. Then it will give you some complex number, this inner product will, and we take absolute value square. That is the probability of measuring a sub n, as long as all these vectors are normalized that is. Pictorially we can say that initial state Alpha, we make measurements on the state of A, it collapses into one of the eigenstates, and which eigenstate will we measure or which eigenvalue will we measure or which eigenstate will the quantum state collapse to? We don't know, but we do know the probability of collapsing into this eigenstate a sub n is given by this quantity here. Even though we don't know precisely what value will we measure, we can predict the average value we will be getting by calculating the expectation value. Expectation value is given by this. Alpha here is the initial state and A is the operator representing the observable that we're measuring. You can take this equation here and between Alpha and A here, you insert identity operator and insert on other identity operator here. That's what we're doing here. Outer product between a sub n and a sub m summed over all m, that's one identity operator, a sub n, a sub n outer products summed over n. That's the second identity operator. Identity operator, multiply to operator or a vector doesn't do anything, so we can do this. Now, we look at this matrix element of operator A using its own eigenvectors as basis. We already saw that this gives you a diagonal matrix, where the diagonal elements are the eigenvalues. In mathematical terms, this can be represented by the Kronecker delta. Because delta mn represents a quantity that is zero for the case when m and n are not the same, and when m and n are the same, then this Delta is 1. It gets rid of one of these two summations and yields this expression here. What is this? This here, we just told you, is the probability of measuring a. A_n is the value that we will be getting and this is a probability associated with that value if you can sum over all this, that is, the definition of expectation value in probability theory. Our quantum mechanical definition of expectation value is consistent with the expectation value definition in probability theory. That's what I just mentioned and let's revisit Stern-Gerlach experiments again. Specifically, we will be looking at this second spin measurements along z with the electron coming in. Incidentally, electrons all have spin up along x direction. Now, the initial state is x plus. In other to distinguish the direction of z versus x, we added x here. X plus represents the spin up along x. That's the initial state here coming into the second, Stern-Gerlach experiments along z here. Now, the eigenstates of the spin along z, we inserted z here just to distinguish them once again from these spin up along x. Z plus is spin upstate along z direction. Z minus is a spin downstate along z direction. Now, the probability of measuring spin up and spin down along z, according to the probability definition that we just talked about, is given by this. It's the inner product between the initial state, which is x plus, and the eigenstate, which is z plus, we take absolute value square of the inner product, we get the probability of measuring spin up. Probability of measuring spin down along z is given by this equation. Now, this probability we know are both 1/2. We can now, construct the ket x plus in terms of z plus and z minus. Now, the dot-product between x plus and z plus, and inner product between x plus and z minus, are the quotients of the z plus vector in the expansion with using z plus, and z minus. We know the coefficients absolute value squared is 1/2, 1/2. We can just choose 1 over square root of 2 for this and 1 over square root of 2, e to the i Delta 1, here for the second coefficient. Now, of course in general, we should have a phase factor in the first coefficients 2. However, the overall phase factor common to both of these two can be chosen arbitrarily according to the normalization condition. Therefore, only the relative phase factor between these two coefficients are physically meaningful and so we'd simply choose the phase factor for the first coefficient to be one and then just leave this e to the i Delta 1 as the phase factor defining the relative phase between these two coefficients. We can do the same for x minus the probabilities of measuring spin up and down along z in the case where the incident electron all have spin down along x will also be 1/2. We basically have the same condition as x plus. We can write down x minus in terms of z plus and z minus the same again, however, we do know that x minus is supposed to be orthogonal to x plus. We basically use the same expression with a negative sign so that these two vectors are orthogonal to each other. We have constructed the vector representation of these two ket vectors, x plus and x minus, using the basis of z plus and z minus. This coefficient together with this coefficient forms a two-dimensional column vector representing x plus. This coefficient and this coefficient together form a two-dimensional column vector representing ket vector x minus. We can then proceed to construct the matrix representation of operator S_x. The operator S_x can be represented by this summation of two outer products. This is the matrix representation of S_x using x plus and x minus as a basis. In this case, we already know that this matrix will be a diagonal matrix where the diagonal elements are the eigenvalues. In this case, plus h-bar over 2, for the second case, negative h-bar over 2. Now, once we construct this because we know x plus and x minus in terms of z plus minus, z plus minus here, plug it in. You can re-express this equation in terms of z plus and z minus. Now, this will give you the matrix representation S of x in the basis of z plus and z minus. You can do the same with S_y, the y-directional spin and get this expression. The y plus and y minus eigen kets of S_y is expressed in terms of z plus and z minus like this. There is another phase factor, Delta 2 here going in. To determine the phase factor Delta 1 and Delta 2, we should consider the sequential spin measurements along x and y. If you do this, you will still get equally split beams because the probability of measuring spin up and spin down along y using the eigenstates ups of S_x either spin up along x and spin down along x, they are also 1/2. Using these, you basically go through the same process that we have just gone through. You will be able to determine the dot-product absolute value square between the y eigenstates and x eigenstates are all one-half. This condition allows you to determine the relationship between Delta 1 and Delta 2. That relationship turns out to be this. That gives you that the difference between this two-phase factor, Delta 1 and Delta 2 should be Pi over 2, either plus Pi over 2 or minus Pi over 2. Now we choose the negative sign so that the x and y-direction corresponds to the x and y coordinate system in the right-handed coordinate system. Specifically, we choose Delta 1 to be 0 and Delta 2 to be Pi over 2. With that, we have this complete equation for the three spin operators: S of x, S of y, and S of z. If you put it in a matrix form, they look like this. Now, we note that the spin operators do not commute with one another. In other words, if you multiply the two spin operators S_x and S_y, for example. S_x times S_y is not equal to S_y times S_x. In quantum mechanics, we define a commutator bracket as this. A and B surrounded by this square bracket is defined as AB minus BA. If these two operators commute, then this commutator bracket will be 0. If they don't commute, then the commutator bracket will not be 0. For spin operators: S_x, S_y, and S_z, the commutator bracket can be written like this. Here Epsilon ijk is a mathematical symbol which is 1 when ijk is cyclic and negative 1 when it's anti-cyclic, and 0 otherwise. In general, this is not 0. Therefore, the spin operators do not commute. Instead, if you define S square operator as the summation of S_x square plus S_y square plus S_z square, then this S square operator commute with the individual component. Now this commutation relation has important ramifications as we will discuss in the next video.
