In this video and the next video, we will discuss angular momentum addition. So if we want to describe the spin won't have particle like electron in an oribtal motion. We should be able to simultaneously describe their internals degree of freedom spin and their spatial degree of freedom. Their orbital motion described typically by a wavefunction like spherical harmonics. Mathematically what that means is that we need to expand our ket space to the direct product space of the two dimensional cast space spin by the spin again, kets up and spin up and spin down kets. And the infinite dimensional cat space spent by the position eigen functions. So formally we can write like this, this is the direct product and the direct product between the position eigenket, the basis ket for the infinite dimensional space. And then here is the eigenkets of the spin space, you take the direct product and this is the basis ket in the combined expanded space. Now the rotation operator is still given by the same form. This J here is now the total angular momentum, the orbital plus spin. Now here we need to be careful because this orbital angular momentum operates on the position ket space, whereas spin angular momentum operates on the spin ket space. So to be precise, we should write our total angular momentum like this. So the L here is the orbital angular momentum, I is the identity operator in the spin space. So this is the orbital angular momentum operator operating on the extended stay extended ket space, but it doesn't do anything to the spin state, it only operates on the spatial part. Plus this part here, I here is the identity operator in the space spent by the position eigenket and here is a spin operator operating on the spin eigenkets. So this part here doesn't do anything to the spatial part of the extended ket space and only operates on the spin part. The general rotation operator therefore can be written as the direct product of the rotation operator defined by the orbital angular momentum. And the rotation operator defined by the spin angular momentum operator. And the finally the wave function is the, once again the defined by the inner product of the ket, alpha with the eigenket. This eigenket is now an extended eigenket which contains both the spatial part position eigenket and the spin eigenket. And we're going to write it down like this. So sigh is the usual way function defined by the inner product of the position eigenket with alpha and then the subscrete alpha and beta represents the spin state plus and minus spin up and spin down. The two spin component can be arranged in a two dimensional vector. So you can have the first element being sigh plus and then the second element being sigh minus corresponding to the spin down state. And the way function has the usual meaning. So if you take absolute value square, that represents the probability of finding a particle at position X. With spin plus or up or with spin down minus. Now, instead of the position eigenket we can of course use the familiar NLM as the basis function and here indexes the radial part, radial component of the wave function. And LMN specify the angular part, this is precisely what we did in the hydrogen atom problem. And the n, l, m are eigenkets of L square and LZ operators with Eigen values of l l plus 1h square and mh bar respectively. The spin part spin up and spin down are the eigenkets up, of course as square and see operators. And the eigen values are s times s plus one hpe are square and S is one half and therefore you get three H par score over 4. And then the M seven s. The eigen value for a sub Z is either plus one half h bar or minus1/2 h bar. So now the angular momentum state. Now we're ignoring the radial part here, so we're dropping the quantum number in and only collecting the angular part. We now have four quantum numbers, L and M describing the orbital part and S describing the spin part. So in the bracket notation, we just collect all these quantum numbers and dump it inside this ket vertical bar in the angle bracket on the right. And that's my ket describing these orbital and spin state specified by these four quantum numbers. Now, in this case we have chosen the eigenket to be the simultaneous eigenket of L square L of Z, S square and S of Z operators. Now, alternatively, of course, we can also choose the simultaneous eigenkets of J square, L square, S square and J sub Z because these guys are mutually commuting operators as well. And you can find simultaneous eigenkets of these four operators instead of these four operators and they give you equivalent description. Now let's take an example of spin one half system. So for the moment we're ignoring the orbital part when only considering the spin part but now we have two particles, two electrons, for example. So two spin one half particles, and we want to consider the total angular momentum of the two particle system, which is given by the summation of the vector sum of these two spin operators as S1 and S2. And of course, to be precise, these total spin operator S is the direct product of S1. This is a spin operator for particle one, I is the identity operator acting on particle two, and plus I identity operator acting on particle one. Direct product ID with spin operator operating on particle too, and of course the all the operators on the spin space for particle one would commute with all operators in spin space for particle too. So, whatever component X, Y Z, component or spin one should commute with the X Y Z components of spin to operator among themselves of course the spin operators X, Y Z component of spin 1 and X Y Z. Component of spin 2 should satisfy the usual commutation relation, That should be satisfied by any general angular momentum operators. So this should still be valid for S1, X Y Z components and X S2, X Y Z components, the commutation relation of the previous slides, that is the fact that the S1 and S2 components should commute. And among the components of S1 and among the components of S2, they should satisfy the usual angular momentum commutation relation should lead to this too. So this is a total and total spin angular momentum operator, and X Y Z component of the total spin angular momentum operators should satisfy the usual spin usual general angular momentum commutation relation. So now we can collect all these operators that we have defined, so S1 square should have the eigen value of S1, (S1 +1) h bar square, just like J square operator, same thing for S2 square. And S1 Z and S2 Z, should have these m1 and m2 times h bar eigen values, and then for S square which is defined as the sum of these S1 and S2, we would expect an eigen value of S times (S +1) h bar square. Following the general angular momentum properties, and the Z component of the total spin should give us an eigen value of m times h, bar, so once again we have two sets of mutually commuting operators. You can choose these four S1 square, S2 square and S square, the combined spin square and the combined spin Z component or you can choose S1 square S2 square and the Z component of S1 and S2 operators separately. So those two, those two sets form mutually commuting operators and therefore we can choose as our basis set, the simultaneous Saigon cats of either this set, first set of operators or the second set of operators. So the simultaneous Saigon cats of the first set of operators here can be denoted by these index, specifying the eigen values of these eigenkets values for these four operators. So S1, S2 S and M and the precise equation for the I am values are given in the previous slide and we're just using these numbers to denote the eigenket values of these four operators. Because S1 and S2 are common to both, we can simplify this notation to be just S and M by specifying S and M for the second set of again, mutually commuting operators. The simultaneous eigenket, are specified by these eigen value, once againS1, S2, m1, m2, once again S1, S2 are common to both. And so we're just specifying eigen 2 to denote these simultaneous eigenket, for these eigenket 2 representation corresponding to the second set of operators. The possible values of m1, m2 are simply + and -1/2 we already know that, and therefore we can spell out these 4 different possibilities for m1 m2 eigenkets. So +, + 1/2 + 1/2 -1/2+1 /2+ 1/2 -1/2 -1/2 -1/2, so these are the 4 possibilities for the S, m representation corresponding to the first set of commuting operators, S can be either 1 or 0. Why, because this this is the vector sum, the total spin, S is a vector sum of S1 individual spin, S1,S2, so it will have the maximum value when S1, S2 are aligned parallel to each other. And then in that case the magnitude of the some would be simply the sum of the magnitude of these individual vectors, so 1/2 + 1/2 =.1. And the minimum value of the vector sum will correspond to the case where these two vectors, S1 and S2 are anti parallel and therefore cancel each other out. And then in that case you get 0, now from the general properties of the general angular momentum, for a given value of S, m, the Z component value is between -S + 1/2, I'm sorry,- S+ -S2 + S with an imager interval. So for S =1 and can be either 1, 0 or -1, for the case when S = 0, the only possibility is M = 0, so these are the 4 possible states for S, m basis set, so now we have 3 eigenkets with S=1 total spin of 1. And we call that we call those 3 eigenekets has spin triple it, and then this 1 eigenket, 1 state corresponding to x = 0, and we call that spin singlet state, these 2 bases set, these S, m basis set 4 eigenkets of the S, m representation. And the 4 eigenkets that we derive in the previous slide corresponding to m1 m2 representation, they are equivalent, these are 4 basis sets that can represent a general spin state of this to spin system. You can use either one and the perfect analogy would be a two dimensional space, can be represented by this X Y unit factor shown here. And you can rotate this X Y unit factor to an arbitrary orientation here to make X prime and Y prime and you can represent all vectors in the two dimensional space equally well using either this basis set or that basis set. And also just like the two Cartesian coordinate system are related to each other by rotation matrix. The two basis sets that we have derived for these two spin systems sm representation and m1 and m2 representation. Those two basis sets are related to each other by unitary transformation. So to find the precise relation, we first note that s equals one and m equals one. This is the case when the spin is all maximum, the z value is maximum. And the maximum z value is possible only when both spins are aligned. Both spins are plus one half. And then we apply a lowering operator S minus to this plus plus. To this sorry to this s equals one m equals one state in order to obtain s equals one m equals zero state. Now we use the usual definition of s minus. Now the s minus is summation of s one minus plus s two minus, because the s operator is the sum of s one and s two. So s minus operator is also the sum of s one minus and s two minus. Use the usual definition of the lowering operator for each spin s one and s two. And we know that this is the general angular momentum equation for a lowering operator for general angular momentum operator J. So the spin s minus s one minus s two minus. All should satisfy this equation here. So first we apply s minus two. s equals one and m equals one state. And use the equation in the previous slide, we obtain this because s equals one and m equals one. That turns out to be simply square root of two h bar and then we obtain s equals one and m equals zero state. That's the left hand side. Now right hand side, we apply to the plus plus. This is the Eigen ket in the m one and m two representation on that one, we apply s one minus plus s two minus which is equal to s minus here. So we're doing the same thing but in the m one and m two representation. And so apply s one, s one minus to this to get minus plus. So we're lowering s one only here and then s two minus on the second m two. Okay, to obtain plus minus and the co options are defined by the usual same equation has shown here. And then you obtain this h bar the right hand side is h bar minus plus plus plus minus. So this is the equation, s one s equals one, m equals zero state in sm representation is equal to one over square two plus quantity plus minus plus minus plus in the m one m two representation. Finally, s equals one m equals negative one can be obtained by applying a lowering operator one more time to this equation. Or simply by recognizing okay, in order to achieve m equals negative one. The only possibility is both m one and m two are negative one half. Finally, the expression for s equals zero and m equals zero can be obtained by constructing a ket that is orthogonal to all of the three previous kets that we have obtained for s equals one state. And then we will see that that's the only possibility is s equal zero. Only possibility for s equals zero m equals zero is one of our square root of two times plus minus minus minus plus state. So in summary, we can write it like this. So s equals one m equals one is plus plus. That's the only possibility, both spin up. Likewise, s equals one, m equals negative one. This is minus minus both spins down. And then s equals one m equals zero is a symmetric combination of spin up, spin down and spin down spin up. s equals zero equals zero is the anti symmetry combination of up down down up state. The coefficients linking the two basis sets are called the Clebsch-Gordan coefficients. This represents a transformation from one basis set to another, and it is a unitary transformation. And Clebsch-Gordan coefficients are very very important in the theory of atomic structure.
