In this video we will construct matrix representation for rotation operators. And we do this by using the eigenkets of J squared and Jz operators, as our basis set. And the matrices for J square and Jz, using their own eigenkets as a basis set will obviously be a diagonal matrices. So the diagonal elements will be the eigenvalues of the operator J square here and here the diagonal elements are the eigenvalues of J sub z, operators. Now we then proceed to obtain matrix elements for the latter operators J plus and J minus, and to do this, we first consider this matrix element, so J plus dagger times, J plus. Now we recall the identity for using the definition of J plus operator, and then you can show that they're equal to this J square- Jz square- h bar Jz. And then because then now we have expressed this product operator in terms of J square and Jz operator for which our ket j, m or the eigenkets. So we can replace these operators with their respective eigenvalues, and get this here, but we also know that J plus operator raises the value for Jz by 1. So, this operating J plus on eigenkets Jm will result j and m plus 1 within of course, a multiplicatie constant. So our task at hand is to determine this constant here, which we call cj sub m, cjm +. Now to evaluate that we notice that the original matrix element that we try to evaluate here is by definition simply the absolute value square of J plus acting on jm ket. So this here is simply equal to this constant c sub j,m plus squared, and that is equal to this as we have shown that. So we obtain this equation down here for c sub jm plus, and then since we are free to choose a face factor, we choose c sub jm +. To be a real value and simply take a square root, and obtain this value here for c sub jm+, and write the equation j + acting on jm, results in this constant c sub jm + that we just obtained. Multiply to a ket jm + 1 and we go through the same process for j-, operator and obtain this similar equation. Obviously j minus lower the value of m by 1, and then the constant here is determined similarly to the case of c sub jm+. So now we can write down a general matrix element for these lateral operators, j + and j-, so here the bra is j prime and m prime and which may be different from j and m here. And using this equation, we can write down the results as this, and they are zero unless j prime and jay are the same. And also the matrix element is zero unless, m prime is equal to either m + 1 for the case of j plus operator, or m -1 for the case of j- operator. Now we're ready to construct the matrix representation for a rotation operator. So the general rotation operator, is expressed by this exponential of these angular momentum operator j vector, dot producted with these n unit vector, along the axis of rotation. And we specify these matrix element for the same j so j prime is j prime is supposed to be here and we simply replaced it with j, because we know that for the different values of j prime and j the matrix element is zero. And that's because the j square operator commute with all components of j, and therefore j square operator commute with this exponential operator, rotation operator itself. And therefore these j sub m, is an eigenket, for this j square operator, and also is an eigenket for this rotation operator, and therefore by the orthogonality of these eigenkets, different values of j will result in a null ket. For a given j now, we can construct (2j + 1) x (2j + 1) matrices for corresponding to different values of m prime and m here. And this (2j + 1) x (2j + 1) matrices, form a 2j + 1 dimensional irreducible representation, for the rotation group. What do we mean by matrix representation? Once again, if a group of matrices are homomorphic to the original group, by homomorphic we mean that they preserve the group multiplication. So, if you multiply these two operators and, find the matrix representation for that product rotation, that is equal to the matrix multiplication of the two matrices representing these individual operators separately. So, you either do the group multiplication for the operator first, and then find the matrix representation, that gives you the same matrix as the case. Where you find the matrix for each operator first, and perform matrix multiplication of these two matrices. Those to give you the same results if that's the case, these set of matrix form a representation of these original rotation operator group. And we're saying that this (2j + 1) x (2j + 1) matrix that we have obtained here forms a (2j + 1) dimensional representation. The rotation matrix that are not characterized by a specific value of j. Then you can then have an arbitrary dimensional matrix, but then you can always break down that large matrix in a black diagonal form as shown here. And by block diagonal form we mean that there are non non-zero matrix blocks, along the diagonal, and all the others are 0. And these individual blocks, will be a sub matrix of (2j + 1) x (2j + 1) dimension. Further these individual blocks of (2j + 1) x (2j + 1), cannot be further broken down into a smaller blocks, hence irreducible. So this is what we mean by irreducible representation, you cannot further break them down into block diagonal form irreducible, and they form a representation preserving the group multiplication. So these matrices, (2j + 1) x (2j + 1) matrices form irreducible representation of the group of rotation operators. And therefore whatever we do with the rotation operators, we can perform the same mathematical operations, using these matrices. So now we consider the most general form of a rotation operator expressed by Euler angles. And if you recall, the Euler rotation is represented by these triple product of rotation operations by an angle, alpha, beta and gamma about z, y and z axis successively. So the general matrix representation for this rotation operator is, by multiplying bra, jm prime, and then ket jm from the left and from the right respectively. And because this jm ket and bra, are eigenkets and again bras of j sub z operator, so this operator acting on jm, gives you e to the negative im gamma here. And this operator here acting on the bra, that gives you e to the negative m prime alpha, and then you're left with this matrix element for the rotation operator about y axis. And we're going to denote this as a little d (j) sub prime m. Now, for the case of j equal to one-half, we already found these rotation matrix using palace spin matrices from the previous lecture. Now we consider the next case, simplest case of j = 1, and to obtain matrix element for j = 1 case, we first expressed j sub y in terms of the latter operators. Then, we already know the matrix element for the latter operators that we have obtained a few slides earlier. And so it's straightforward to obtain these three by three matrix for j sub y, for the case of j = 1 here. And we arranged the matrix in such a way that m = 1 represents the first row and column, m = 1 represents the second row and column, and m = -1, is the last row and column. Of course this arrangement is arbitrary, and you can choose whichever arrangement you want, as long as you're consistent. And then we show that if you take a cubic power of this matrix, it becomes the same as or a cubic power of jy matrix divided by h bar, that is equal to the matrix is self divided by h bar. And using that, you can express the infinite sum of this exponential operator, you divide that into two even powered term and the odd powered term. And collecting all the even powered term for jy operator will give you a cosine functional beta, and collecting all the odd power term with respect to jy, will ll give you a sign of beta. And notice that this is true only for j = 1, it is not in general true for other j values. And finally using that, we can write these three by three matrix for this little d matrix, representing the rotation by y axis, about by an angle beta, for the case of j = 1. And this is the matrix, so multiplying to this matrix exponential negative I alpha, I and prime alpha, and exponential negative i. And gamma, will complete the full matrix representing a general rotation operator, for Euler angles, alpha, beta and gamma, for the ket space represented by j = 1 case.
