In this video, we will derive a very general definition of angular momentum. So far we have discussed spin 1.5 system and orbital angular momentum. Now, we're going to generalize our discussion to a very general definition of angular momentum. Now, recall the angular momentum was defined as a generator of rotation, which led to this commutator bracket. This really is the fundamental properties of the angular momentum. Now define this J squared operator as this just the dot product of J vector with itself. We see that J squared commute with the individual component, whereas the individual components themselves, they don't commute with one another. What we can do is we can try to find a simultaneous Eigenket of J squared and one of the components of these XYZ component of J, and we choose J_z. We write down the simultaneous Eigenket of J squared and J_z as ket a, b. A here represents the eigenvalue of J squared operator and b is the eigenvalue for J_z operator. Now we define a ladder operator, J plus and J minus as J_x plus or minus I times J_y. Then you can show with a straightforward algebra that the commutator bracket between J plus and J minus results in 2h bar J_z. Also the commutator bracket between J_z and J plus minus give you this. But of course, J squared and J plus or J minus they commute. Why? Because J squared commute with J_x and J_y both. Now in order to see the physical meaning of this ladder operator, we apply J plus and J minus to the Eigen ket a, b and then we apply J_z operator to that. From the definition of the commutator bracket, we can re-express this J_ z times J plus minus as their commutator plus J plus minus times J_z. But from the commutator relation we have this. This commutator bracket can be replaced by plus minus h bar times J plus minus. That's this term here. Then this we apply J_z to this Eigen ket first and that simply result gives you eigenvalue b, and then J plus minus operator remaining there. That's this term here. This operation results in this, which means that this quantity J plus minus operating on Eigenket a, b is also an Eigen ket of J_.z With an eigenvalue b plus h bar for J plus b minus h bar for J minus. J plus and J minus operator raises and lowers the eigenvalue for J_z by h bar. On the other hand, J squared and J plus minus operator commute. This J plus minus H acting on a ket a, b. Afterwards, we operate J squared on it. It simply gives you these eigenvalue a for J squared times the J plus minus operator acting on a,b. We conclude that J plus minus operator acting on the Eigen ket a, b are also simultaneous Eigen ket sub J squared and J_z eigenvalue for J square is the same as before a. But eigenvalue for J_z is raised and lowered by h bar, which means that you operate J plus or J minus operator on ket a,b. Is a constant multiple to this eigenket a and b plus or minus h bar. This constant is to be determined by the normalization condition and this development you should notice is very similar to the creation and annihilation operators that we have seen in the harmonic oscillator problem. Now, because we have this latter operator, J plus and J minus, if you know one eigenket, then from that we can generate all eigenkets of J_z simply by applying J plus or J minus successfully. It turns out that the eigenvalue b of the operator J_z has an upper limit. We can see this by considering this operator here, J square minus J_z square, which simply is J_x square plus J_y squared. Using the definitions of J plus and J minus operators, that operator can be written like this, and then J minus operator is an Hermitian conjugate of J plus and vice versa. Therefore, we can write this equation in terms of J plus and J plus dagger Hermitian adjoint. Now we consider the expectation value of this operator for a state a and b. Simply by substituting this expression for J square minus J_z square operator, we can write it like this and if you look at this term here, that would simply be this ket here, defined by operating J plus dagger to state a, b. This part here is dual conjugate, so this here simply gives you the inner product of this ket with itself and gives you this absolute value square of this ket defined by operating J plus dagger operator onto a, b ket. Similarly, the second term is the absolute value squared of J plus operator acting on the ket a, b. These guys, because they are modulus squared, are obviously positive definite. It's greater than or equal to 0. It is zero only when these kets are null kets. What that means is that the a, the eigenvalue of J square minus b squared, b being the eigenvalue of J_z. That whole thing is greater than or equal to 0, which means that the value of b has a limited range. It has a maximum value and it will have a minimum value. The maximum value of b can be defined by this equation. By recognizing the fact that you can't increase the value of b anymore and therefore, when your b, it has its maximum value. Then applying J plus operator should give you a null ket. From this, if you have a b maximum, J plus operation results in zero or a null ket. Now, take this equation and operate J minus. The right-hand side isn't already a null ket, so whatever you do, you still get a null ket. Using the fact that these J minus times J plus from the straightforward algebra using their definitions, you can rewrite it like this, J squared minus J_z squared minus h bar times J_z. Substitute this for this here, then you get this equation here because a, b max is an eigenket of all of these three operator J squared, J_z square, and J_z. The operator here simply is replaced by their eigenvalues, a b max square and h bar times b max, that's zero. Therefore, this here should be 0, and that leads to a is equal to b max times b max plus h bar. Now, we make a very similar argument, and insist that there must be a minimum value for b as well. When b is a minimum value, you can lower this aeigenvalue anymore. When you apply j minus, then you should get on the whole kit and going through the same procedure, so this time you rewrite j plus times j minus as this, you get this results this time. Going through the same process as above. A is equal to b minimum times b minimum minus h bar. Now combine this equation with this equation up there. Another equation for a in terms of b-max, you will see that b max is equal to negative b min, and b max should be defined positive because it's supposed to be a maximum value. Your b value, the quantum number for j z should be between these negative b max and positive b max. Now we have determined the bounds of the eigen value b for operator j sub z. We should be able to obtain a b max by successfully applying j plus 2ab min, and because we can only apply this operator j plus on an integer number of times, this implies that the difference between b max and b minus should be an integer multiple of h bar. Since b max is equal to negative b min, you can substitute negative b max here for b min and get b max is equal to n h bar divided by 2, from which we get a is equal to h bar square times n over 2 times n over 2 plus 1. For a more convenient notation, we define j to be n over 2, and n once again is being an integer. j here is either an integer or a half integer. Now using this j, a can be written as h bar squared times J times j plus 1. Also, define m such that b is equal to m times h bar, and from this condition above here, we can see that m runs from negative j to positive j with a step of one. There are total of two j plus 1 allow the values of m, each value separately to each other from its nearest neighbor by 1. In summary, we can write these eigenvalue equations for j square and j z as this. Now instead of a and b, we're using j and m to specify the eigen kit, and j is related to the eigenvalue 4 j square operator, m is related to the eigenvalue of j operator, and once again, j is an integer or a half integer, and m runs from negative j to j. Now, this is the very general results for an angular momentum. We have not used the classical definition of angular momentum, and the only thing that we have used is the fact that the angular momentum is the generator of rotation. A generator over rotation should satisfy that fundamental commutation relation among themselves, among the x y z components of themselves, that was in the first slide of this lecture. That leads to this eigenvalue equations for j squared and j z operators, and this is the result that applies to any angular momentum operators in general peace be it spin, orbital, or any general combination of these two angular momentum.
