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Okay, welcome to Lecture 15. Today we'll talk about the total spin of a composite system.

In most books you find what I do today under the heading of addition of angular momenta.

We call it total spin because the world addition is not entirely wrong, but it's not entirely correct either,

so we'd rather call it the total spin. So the physical question is what is the total spin,

or if you want to think classically or in special cases what is the total angular momentum of a bipartite system.

Bipartite is just a learned name for a system which has two parts, well, which system doesn't,

but it's a system which we think of having two parts. If we know the spin of the individual systems,

if we know the spin of each constituent system, this is a very natural question.

So if you have an elephant that makes pirouettes on the surface of the earth, then the elephant has an angular momentum, classically speaking,

and the earth has an angular momentum because you may have heard about it that the earth turns around in axis,

and so you have these two angular momenta classically, and you can ask if you look from outside,

you take the total system, what's the total angular momentum system of the elephant making the pirouettes and the earth turning.

It's a very reasonable physical question if you know the angular momentum of the elephant and the angular momentum of the earth.

Now this is classical thinking, and quantum mechanical thinking we know well, what is this angular momentum bit in the first place.

So let's be more precise in the context of quantum mechanics, more precisely in quantum mechanics, consider system A,

system A whatever, which has angular momentum, well system 1 has a Hilbert space which we call HA instead of 1, I don't know,

and it has on that operate three operators A1, A2, and A3, which you remember our convention they're all operators on HA,

or some dense subset thereof, and they satisfy this algebra Ai with Aj equals I times epsilon Ijk Ak, did I introduce the epsilon or not, I did not right?

Well the epsilon is just a short form of saying 0 minus 1 or plus 1, and if you have epsilon 1, 2, 3 you meant plus 1,

if you have any odd permutation of 1, 2, 3 you mean minus 1, if you have any even permutation of 1, 2, 3 you mean plus 1,

and if you have any non permutation of 1, 2, 3 like for instance 1, 1, 1, then you mean to say 0, and there's a summation convention here, sum over repeated index k.

This is the short form for writing down the three commutation relations for these guys, this is supposed to say this is an angular momentum algebra,

like we wrote them down, so we assume we have a system where you have an angular momentum algebra on it,

and we'll assume more, we'll assume that this is already a spin-JA system, spin-JA system, and we discover, we define what a spin-JA system is,

okay, so we have a system of a definite spin, remember that the orbital angular momentum of a particle, there are many spins in there,

so you have to look at a subspace thereof if you want to use this building block, but that can be done, so we have one spin-J system,

maybe I can then just delete this number 1 because the JA system is the first one, and then you have a spin-JB system, the A is just a label here,

so you have a second system, you have a JB system which has B, which has a Hilbert space HB, and on that you have operators B1, B2, B3,

self-adjoint operators that also satisfy this angular momentum algebra, so these are the two systems we have, a spin-JA and a spin-JB system,

okay, one system and another system, and the question is what is the angular momentum of the combined system, aha, so the combined system, the composite system,

we learned in the last lecture, the composite system is the system that has as its Hilbert space HA tensor HB, so I should have maybe said two distinguishable systems,

but well they already have different angular momentum, so HA tensor HB is the Hilbert space for the combined system, elephant plus earth,

but it's a quantum elephant and a quantum earth, well if that sounds funny, I ask you what is a classical elephant and a classical earth,

because everything is quantum isn't it, so of course the elephant is a quantum object and the earth is a quantum object, okay,

so the funny laugh at the people who say a classical elephant, so anyway, so you have the quantum elephant and the quantum earth,

you take them together and now you say well although I have that, I still want to be able to measure the angular momentum of the first part of the system,

and you know how that goes, you can do this by looking at the operators AI tensor, the identity on the second space, right,

if you, we talked about these guys, if you use these on those, they will act on this tensor product as operators,

so they go from HA, sorry HB, go from HA tensor HA to HA tensor HB, to HA tensor HB, right, the linear operators there by definition,

and that will just be the lift of this operator A1, A2, A3 to this composite system, so you can think, look at them together,

but then you only look at the earth or you only look at the elephant, that's this one, you also have the BI's,

they're the identity on HA with the BI's is similar, you can look at, you can look at these guys, okay,

and so the first thing you then check is that they still satisfy the angular momentum algebra, so if you now take commutator AI tensor identity on HB,

I will start writing ones for that because this gets a little tiring, okay, but let's write like this,

comma AJ tensor HB, then you will still get I epsilon IJK AK tensor HB,

so in other words, these guys still satisfy the angular momentum algebra and similarly for these guys, ID HA tensor BI's,

they also satisfy angular momentum algebra, so the lifted operators to the total system still are angular momentum operators, like we called them,

okay, so we have all of that, okay, and because we said that this is a spin JA and this is a spin JB system,

we already know from last time that we actually have a common eigen basis for those and so on.