In this video, we will discuss some properties of rotation operators. First, let's look at the commutation of rotation operation in three-dimensional space. First, we should notice that the rotations about two different axis in general do not commute. To illustrate the point, we consider two successive rotations about x and z axis. Let's consider these rectangular block. We first rotate this block about z-axis by 90 degree, like this, and then rotate about x-axis by 90 degree once again, in this orientation. If you reverse the operation and do the x rotate on first to get this and then z, you get an orientation of the block that's clearly different from the original, the other case. This shows that in general, two successive rotation by two different axis do not commute. Now, this is a key distinguishing factor or property of rotation operation from say, the translation operation for which translation by two different directions in general commute. Now, before we move on, we define active and passive rotation. You can rotate an object as we have just done in the previous slide, instead of coordinate system. This rotation of the object or vector itself is usually called active rotation. Now, the angle for these active rotation is taken to be positive for a counterclockwise rotation and negative for a clockwise rotation. By this convention, the rotation matrix for a rotation about z -axis can be written like this. Now, instead of rotating the vector or object itself, you can equivalently rotate the coordinate system in the opposite sense. We simply switch the sign of the rotational angle and get the same result. This is called a passive rotation. You can use either one, but once you have chosen one or the other, then you have to be consistent. In this class, we will use active rotation. Now, we define the infinitesimal rotation by just making the rotation angle very small. Then we expand these cosine and sine function in an infinite series. For small angle, we can ignore all the higher-order terms and retain only the angle or only the terms with second-order in the rotation angle and you get this. You can do the same exercise for rotation matrices for the rotation about x-axis and rotation about y-axis. Now we have these matrix representation for a infinitesimal rotation. Now, we can directly calculate the commutation relation between these matrices. You calculate the matrix multiplication of R_x and R_y and then you calculate the matrix multiplication of R_y and R_x in reverse order. Then you take the difference between these two matrices to calculate the commutation relation and you get this. You will see, comparing this matrix with the R sub z matrix, then you will see that the R sub z matrix for a rotation by Epsilon squared, and those are the two angles here. Then subtract the identity matrix, so 1, 1, 1 here, you take that off. Then this matrix will give you just this. You can represent this right-hand side as rotation about z-axis by an angle Epsilon squared, minus a rotation by zero angle by about any axis. This is the commutation relations for a rotation matrices for three-dimensional space. Now, the rotation matrix for a vector in three-dimensional space in general could be different from the rotation operators for quantum states, state kets. But here we postulate that for every three by three rotation matrix R, which we just have dealt with in three-dimensional space, there is a corresponding rotation operator D in the appropriate ket space. This operator makes the same rotation as this three by three rotation matrix to the wave function or state ket. This is a postulate. Based on this postulate, then the algebraic properties of these operators D should be the same as the algebraic properties of these three-by-three rotation matrices. What that means is that there is an identity matrix. There should be an identity operator which doesn't really do anything, it's just multiply on identity. It's the same. There is a closure. If you multiply two matrices, then it gives you another matrix which represents another rotation in the three-dimensional space. If you make two successive rotation operators, then the product results in another rotation operation in the same ket space, of course. Then there is an inverse for every matrix, there is an inverse when you multiply them together, you get an identity. Just like that, for any rotation operator in the ket space, there's an inverse. When you multiply together, you get an identity operator. Then the successive multiplication of matrices are associative and successive operation of rotational operation is also associative. With this, we now can translate the commutation relations we obtained for the matrices, three-by-three matrices to the commutation relation among the rotation operators D. We write out like this. The rotation, infinitesimal rotation operation about x-axis by an angle Epsilon times the infinitesimal rotation about y-axis minus in the reverse order product of the reverse order should be equal to the rotation operation about z-axis by Epsilon square minus the identity. Now, we use the definition of these infinitesimal rotation operation in the using these angular momentum operator. Now we call the operator here is represented by an exponential function. Here we retain up to the Epsilon square order for each exponential functions. If you do that, then you can work out this equation and you will see that all the other terms cancel out and we can only collect the Epsilon squared terms and that gives us these commutation relation between angular momentum components x and y components of the angular momentum is related to the z-component of the angular momentum like this, they do not commute, they result in commutator bracket, results in z-component of the angular momentum. More generally, you can write it like this, x, i, j, k are the xyz components of the angular momentum. Epsilon ijk is the Levi-Civita symbol, so when ijk are cyclic combinations, it's one, it's an anti cyclic, then it's negative 1, otherwise it's zero. It's a commutation relation that gives you a fundamental property of angular momentum. Now, contrast this angle of the commutation relation among the different components of the angular momentum with the commutation relation among the different components of the linear momentum. They all commute as long as i and j are different, they commute, but angular momentum don't. The fundamental difference really comes from the fact that the two successive rotation about two different axis do not commute as we have seen in the first slide in this lecture, whereas the translations along to different direction, they commute. This commutation relation really once again represents all the relevant properties of rotation operations into adventure really stem from this commutation relation among the different components of angular momentum vector.
