In this video, we will discuss rotation operation. Now we first discuss infinitesimal translation operation by defining this operator T of dx, which represents an infinitesimal translation by this vector dx. What that means is that if you apply these translation operator to a position eigenket, with an eigenvalue x prime. It changes the ket into a position eigenket, with an eigenvalue x prime plus dx prime. This addition here of course is a vector sum. Now you apply this translation operator to an arbitrary ket Alpha, and then you express this arbitrary ket in the position basis. If you recall, this is the position eigenket, our basis vector, and this is the coefficients. But this is a continuously variable case. You change the summation into an integral, and then the coefficients here represents the wave function in position representation. So this is a function of position. Now, you apply the translation operator to the position ket. Now this is a number. This is an integral, so you only operate these operator onto this ket here, and it changes from this equation. It changes x prime into x prime plus dx prime. Now, but this, you can simply change, this is an integral over x. You can simply change the variable of integration and then turn it into this, and this here is the new wave function after translation. If you look at the changes in wave function before and after the operation of these translation operator, it changes this original wave function to this wave function. That's represented here. If you just explicitly write out this dot product or inner product represents wave functions, you can write it like this. Now, the translation operator must be unitary. Otherwise it's not going to preserve these inner product. There to preserve the inner product before and after, no after and before the translation operation. These product of T dagger T should be identity, and that's the condition for unitary operation. Also, we can infer other properties of translational operator and from the nature of the translation operation. The successive application of translation operation is equal to a single translation operation by a vector sum of these two translation vectors. The inverse operation simply is equal to translation in the opposite direction by the same amount, and if you take the limit of translation vector going to 0, this is a typo, I'm sorry, this should be 0. Then in the limit of this translation vector going to 0, you're not doing anything to your ket. It should be equal to the identity operator, and so if you write out these infinitesimal translation operator in this functional form, then you will be able to convince yourself that this operator satisfy all the properties in the previous slide, and it is a proper representation of this infinitesimal translation operation. Here K is a vector and an operator. So K is a vector in the sense that it has three components, and individual components are all operators. Hermitian operator, that is. Now let's consider the commutation relation between the position operator x and the translation infinitesimal translation operator T. Then you can see that x and xT, product. You apply translation first to a ket and then position operator next. If you do that, then the translation operation changes your position eigenket to x plus dx, and the applying the position operator to this simply pulls out these eigenvalue. So x prime plus dx here is the vector which is an eigenvalue of this eigenket for this positional operator x. If you switch the order. You do the x operation first, then you pull out these eigenvalue first, and then you operate your T. But because this x prime is just a constant, it goes out, and then these operator T operates only on the ket, and then it turns into x plus d plus x, x plus dx prime ket. If you calculate the commutator bracket between position operator x and the translation operator T of dx, and acting on a position [inaudible] x prime. Then from these two equation, you will see that that is equal to this. Now, we're talking about infinitesimal translation. What do we mean by that? Its dx prime vector is very, very small, so that is approximately equal to this x prime ket. From this, we can derive the commutation relation, commutator bracket between position operator and a translation operator, infinitesimal translation operator, they don't commute, and the commutator gives you this dx prime vector. If you use this functional form for the operator T, then this commutator relationship can be turned into this commutator relationship between the x position operator and the K operator in this definition here. They don't commute if the x, i and j, so the subscript i and j are equal, they don't commute, the commutator bracket gives you i. If i and j are not the same, then they do commute. What does that mean? I, j here subscript represents the XYZ component of the position in this K vector. If we are talking about the same components, same x component, same y component, and same z component, they don't commute. But for the other components between x and y, x component of the position and y component of K vector, then they do commute. Now from these commutator relation, we identify these k vector to be position operator vector divided by h bar of course. Then we can write our translation operator like this, and the commutator bracket between x and k in the previous slide then yields this correct commutation relation between position and momentum operators, which leads to the uncertainty between position and momentum. Now, finite translation simply is obtained by successively applying these infinitesimal translation. Now consider a finite translation by a vector delta x prime, these delta x prime is not a small number, small vector. You can apply these infinitesimal translation many, many times. Take the limit of these N going to infinity, as N goes to infinity, this thing, it becomes very small. Delta x prime divided by N it becomes very small, so this quantity inside the parenthesis becomes this infinitesimal translation operator, and then you do it more many, many N times and take the limit of N going to infinity. This here simply gives you an exponential function, and of course, we are dealing with the operators here. The exponential function of an operator is as before, defined by this infinite summation consistent with the infinite series expression of an exponential function. Further, we notice that the translation along different directions commute. If you want to go from A to B, you can either go this along the x direction, say, and then next along the y direction and reach B. That is equal to the translation along the y direction first and then x, it gives you the same results. The two commute, and this is consistent with these momentum operator along different directions commute. The finite translation operator can now be written as a vector dot product between the position operator vector and a translation vector, both of these two are three-dimensional vectors in three-dimensional space. This is just a constant vector, p here is an operator. What we have done so far is we have derived an operator that translates a wave function or a state ket by a finite amount. This delta r prime is a three-dimensional vector. What operator is related to this translation operation? Linear momentum is fundamentally related to these translation operation, that's what we have derived here. Now we go through the same exercise for time evolution. Recall the time evolution operator is defined when we discuss this Schrodinger time-dependent equation, we define this time evolution operator as an exponential function of Hamiltonian operator. This operator translates the wave function at time t equals t_0 to time t. Now, we define an infinitesimal time evolution operator as this; analogously to the infinitesimal translation operation. Instead of linear momentum p, we have used the Hamiltonian operator to define this infinitesimal time evolution. Just as momentum generate translation, Hamiltonian generates time evolution. Now we're ready to talk about rotation operator, so we proceed to define the rotation operation in exactly the same way. We define this rotation operator acting on a ket, and you rotate the ket by a certain angle. This is the rotated ket. For rotation, we need to specify two things. One is the rotation axis about which you make the rotation and the angle by which you make the rotation. The rotation operator here should be specified with these two things. So n hat, here hat is not an operator, so this hat represents a unit vector. This n hat is a unit vector specifying the axis of rotation, and Phi here is the rotation angle. Now we define the infinitesimal rotational operator in just the same way as before. So 1 minus I and this quantity here multiply by these infinitesimal angle by which you are rotating. In the case of translation, this vector here was momentum. In the case of time evolution, this operator here was Hamiltonian, and you can guess for rotation, this vector here will be angular momentum. However, in this development of theory, we have not made any assumption about angular momentum, we have not used the classical definition of angular momentum at all. We're simply defining this factor as a generator of rotation operation, and you will see that this vector will satisfy all the properties of angular momentum. Here is the summary of the infinitesimal operators that we have generated. In general, infinitesimal operator is defined like this. This is the infinitesimal operator, this Epsilon here is the amount by which we're making the transformation. Epsilon goes here and the operator multiplied to these Epsilon with constant I is the operator that generate this transformation. For translation, generator is momentum divided by h bar, in three-dimension, it's here. For time evolution, we're dealing with a scalar. The generator is simply Hamiltonian operator divided by h bar, for rotation, we have this J vector, angular momentum vector that producted with the unit vector specifying the rotation axis divided by h bar. Finite rotation operation is obtained once again by successive operation of infinitesimal rotation operator. By the same logic that we have used for finite translation operation, we take the limit of N going to infinity, we simply get these exponential function again. Once again, we have not used the classical definition of angular momentum as the cross-product between position and momentum vector. The general definition that we have used here is therefore applicable to both orbital angular momentum and spin angular momentum. Recall, spin angular momentum does not depend on spatial coordinate at all, it is an intrinsic properties. Therefore, you cannot use this classical definition to define spin angular momentum. Spin angular momentum can only be defined properly using the development that we have described today as a generator of rotational motion.
