Angular momentum operators and their algebra
PROFESSOR: So angular momentum, we need to deal with angular momentum, and the inspiration for it is classical. We have L is r cross p. Classically. So let's try to just use that information and write the various operators. And in fact, we're lucky in this case. The operators that we would write inspired by the classical definition are good operators and will do the job. So what do we have? Lx, if you remember the cross-product rule, that would be y Pz minus z Py. Now you can think of this thing as a cyclic, like a circle when you have x and Px, y and Py, and z and Pz. Things are cyclically symmetric. There's no real difference between this core. And so you can go cyclically here. So you say, let's go cyclical on this index. Ly is equal to the next cyclic to y in that direction is z Px minus x Pz. And Lz is equal to x Py minus y Px. And these things, I'll think of them as the operators. Let's put hats to everything. The first thing I can wonder wit...