In this video, we'll continue our discussion on angular momentum addition and and developed a general theory of angular momentum addition. Not specific to any spin or or real angular momentum, so consider any two angular Momentum, J1 and J2. It could be spin, it could be orbital angular momentum it's we're just keeping it general. They satisfy this commutation relations, so among themselves the X Y Z component of J1. Satisfy the usual angular momentum commutation relations, X, Y Z component of J2 the same, the X Y Z component of J1 and J2 commute with one another. And then we construct the total angular momentum operator, J, which is the sum of the direct product of J1 and the identity operator in the space of J2. And then plus the identity operator in the space of J1 direct producted with J2 operator. So we're going to simply, from now on, we're going to simply write this as just J is equal to J1 plus J2, implying that precisely rigorously speaking, we mean this. The rotation operator as before is described by this usual form exponential function of this J operator here. Total J and that is again related to the individual component J1 and J2. Through this direct product operation, the J is an angular momentum and a generator of rotation in the combined cat space, and therefore we should expect the same angular momentum commutation relation for the X Y Z component of this total angular momentum vector J. So, from the commutation relation, we find that there are four mutually commuting operators. J1 square, J2square, J square, and Jz or J1 sq J2 sq J1z and J2z now let us confirm the commutation relation between J square and J one square, J square can be written like this. So there is J1 square and J2 square and there is a cross term and the cross term can be written, here is the cross term for the Z component and the cross town for the X Y component can be expressed in terms of the latter operator, lowering and raising operators. And since J square commute with all components of J1, J 1x, J2y and J1 z. We can easily see from this expansion that yes, they do commute similarly J square commute with J2 square as well. However, J square operator, the total angular momentum operator square does not commute with the z component of one J 1 nor does it commute with the Z component of J2 however, it does commute with its own Jz component. So, you cannot add, you cannot add J square to the second set of commuting operators here because J square doesn't commute with J1z and J2z. Likewise, you cannot add J1z Or J2z to the first set of commuting operators. So this is it the four commuting operators are the maximum number of mutually commuting operators that you can have in this problem of angular momentum addition. And you have two choices either this, J1 square, J2 square, J1z, J1, J2z, or J1 square J2 square J square and Jz. So the igenkets simultaneous eigenkets for this choice can be written in terms of their eigen values or quantum numbers specifying their eigen values. J1, J2, M1 M2 and the dragon value equations follow the usual angular momentum eigen value equation as shown here. Or you can use the other set of mutual for mutually commuting operators J1 sq, J2 sq, J square and Jz. And then specify the simultaneous eigenket In terms of J1, J 2, J M and the I M value equation once again follows the usual eigen value equation for general angular momentum operators. Now these two basis sets are related to each other through a unitary transformation, and the unitary transformation can be found by simply multiplying to this, I can catch J1, J2, eigenket and multiply to that identity operator because J1, J 2M on M 2. This eigenket is complete or to normal eigenket the outer product and summed over all these M1M2 will give you identity operator. So you can multiply identity operator to this and it should be the same as the original eigenket. Now this equation gives you a transformation equation between this eigenket and this eigenket through this matrix element, this matrix element, the inner product between the M1 and M2. Eigenket and the J M eigenket is called clebsch-Gordon coefficient and chlebsch-Gordon coefficient have many important properties and very useful properties. And first, let's consider this J sub Z total angular momentum Z component obviously is the summation of the individual angular momentum Z component. So if you take J sub Z minus J1z minus J2z this is zero. So whatever ket that you're operating on to this, you will get zero null ket. Now multiply J1 J2 M1 M 2 from the left then because this J1, J2, M1, M2, R D. I get a text of J1z and J2z, so you can operate these operator to the bra here to get this and then Jz can operate on the ket here to give you M and that is zero. So we get this equation here Clebsch- Gordon coefficient vanish this the inner product is zero unless this is zero, unless M is equal to M1 plus M2. This simply corresponds to the conservation of the Z component of angular momentum. But it follows from this purely algebraic operations dealing with the transformation of clebsch-Gordon coefficient. Second, the clebsch-Gordon coefficient vanished unless your J total J is between the difference between J1 and J2 and the sum of J1 and J2. This really follows naturally from the factor some, so when J1 and J2 are aligned parallel to each other, then you get this. When there are anti parallel, you get this and your J the total J has to be in between. This condition also follows from the consideration of dimensionality of the two vector space that we're taking our direct product. So for a given value of J1 and J2, there are a total of 2J1 plus one times 2J 2 plus eigenket. Right because for a given value of J1 there are 2J1 plus one allowed values of Ms of one, same thing for J2. So that their total number of eigenkets and therefore the dimensionality of your ket space Is the product of the two. The vector space spent by these two are 2J1 plus one times, 2J2 plus one dimensional. Now in the J M representation there are 2J plus one eigenkets for each value of J. Now the total number of eigenkets should then be just summing these 2J plus one from the minimum value of J to the maximum value of J. If the minimum is the difference and the maximum is the sum, then we can make the sum and show simply that turns out to be the product 2J1 plus 1 and 2J2 plus one. So the dimensionality of the vector space is preserved if you restrict the values of J as in between the difference and the sum. Now the clebsch-Gordon coefficient, represents the transformation between two orthogonal metrices. The one set,one orthogonal metrices is J1, J2, M1 M2, that's one set, and then the other set is J1, J2 J M. Those are the two different sets and in general they should be unitary by convention. The coefficient are taken to be real, so the real unitary matrices are simply orthogonal matrices. So being orthogonal means that the inverse transformation is equal to itself. So being unitary would require that this is equal to the complex conjugate of this, but they're all real numbers, so they should simply be equal to each other. And the orthogonality condition simply gives you these two equations now, what does this mean? The two equations in the previous slide It follows from the orthogonality of the eigenkets and what do I mean by that? Okay, so the clebsch-Gordon cautions forms these matrix the columns are indexed by J and M and and these rows are indexed by M1 and M2. And this matrix is an orthogonal matrix, what that means is that all of the columns, all of the columns, column vectors. These all these different column vectors are orthogonal to one another, and all of the row vectors are orthogonal to one another. And those two conditions are represented generally by the two equations in the previous slide, these two equations here, so this represents the orthogonality among rows. This represents the orthogonality among columns Now consider a special case when j is equal to J prime, M is equal to M prime is equal to M1,M2. And that will give you this equal to and so the quantity in here simply falls out to be the absolute value square when you sum it over M1, M2, it gives you one just simply is the normalization conditions. So the all of the column and row vectors are normalized. Sometimes the Clebsch-Gordon coefficient are expressed in terms of the Wigners 3j symbol. And this is just a notation, so instead of having expressing the Clebsch- Gordon coefficient as this inner product between the M1 and M2 eigenket and J M eigenket. You can express this in terms of this, so this is Wagner's 3J symbol It's just a different notation. It has the same meaning as this now, lastly, let me summarize the meaning of Clebsch-Gordon coefficient, what are we doing here? We are taking a matrix in J1 M1 representation, and J2 M2 representation. So this is the matrix in the ket space corresponding to the first angular momentum. Eigen state and the second angular momentum eigen space, you take a direct product of the two to produce this matrix in the combined space. Now this taking a direct product will give you the representation in the M1 and M2 representation. You're using J1, J 2, M1, M 2 as the eigenket If you use the orthogonal matrix or unitary matrix consisting of the Clebsc-Gordan coefficient. You can transform this matrix into a block diagonal matrix in the J M representation and this individual blocks corresponds to the various allowed values of J between which runs between absolute value of J1 minus J2 to J1 plus J2. That those are different values of J and they are all in block diagonal form. And each block has a dimension of 2J plus one this is the algerian meaning of Clebsch-Gordon coefficient.
