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So welcome to lecture number 13 of the course. Today we'll discuss spin. Well, spin is often

this rather mystic quantity of which you mostly say that in classical physics it doesn't exist

and so on and so forth. Today I try to present one avenue towards it that like

maybe a few others try to demystify it by particularly spelling out what the

logic behind it is and by maybe using terminology which is more or less the

standard terminology however in one or two places deviates from it in order not

to give the wrong impressions which then mystifies the subject rather than

demystify it. Well it starts all very innocently namely by noting that in the

previous lecture at the end we used Stone's theorem to define angular

momentum. So we defined angular momentum and here I would like to give it a more

precise name as it's also often called in literature we defined orbital angular

momentum. So maybe last time I should have said we define the orbital angular

momentum operator but anyway that's what we did and well so there's the first

thing a bit of terminology here so momentum is not momenta so that's

singular but one usually means by it a collection of three operators namely

these li's that we had last time so if you have li from i 1 2 3 you say you

have the angular momentum operator of course the idea is that you say this is

actually a vector consisting of l1 l2 l3 well it is not a vector it's also not an

operator valued vector however this terminology stems from there that you

say have one angular momentum okay so I will use it in that form if I say angular

momentum operator I mean these three operators it's the first little

terminological hurdle okay so fine that's not too bad and Stone's theorem

gave us the domains d li stone such that these angular momentum operators over l2

are self adjoint okay self adjoint so they are indeed quantum mechanical

observables all this was quite well and they were given by these three

expressions l1 equals minus I x2 del 3 psi okay if they act on the side minus

x3 del 2 psi l2 as minus I x3 del 1 psi minus x1 del 3 psi and finally l3 psi

is minus 1 x1 del 2 psi minus x2 del 1 psi there they are so this we got from

Stone's theorem from this construction of this unitary now we also saw that by

this corollary of Stone's theorem we had, we may compromise self-adjointness, but

keeping essential self-adjointness, but keeping essential self-adjointness, to

gain a domain equals a range, to gain domain equals range, meaning that the LI's

go from, well I use, I think I use the Latin S, that you start in Schwarz space

and you go to Schwarz space, so that the range is also Schwarz space, okay, and you

see the advantage of this is once you do that you may formulate

commutation relations, then one can calculate L1 comma L2, if I restrict

attention to size that come from Schwarz space, for psi in Schwarz. So what was

this commutator? This is the definition of the commutator, this is L1 after L2

minus L2 after L1, and you see, you see the reason why I needed to restrict to

this domain of essential self-adjointness, because only then do I

know that the first, if this one acts on the psi, it will yield something that

this guy can eat, okay. So we have this, and to make it blatantly obvious, well

you know that this is always, is an operator acts on this, but that usually

here we save brackets by not always writing the argument in brackets, but

this is of course L1 of L2 psi minus L2 of L1 psi, like this, right, so it's the

definition of the composition, and we may now simply calculate this, so we plug

this in, okay. So what is this? The L1, let's calculate this for L1, L2, this is

minus i bracket L1 is x2 del 3 minus x3 del 2 acting on, so it acts on this L2

psi, but L2 psi is minus i times x3 del 1 psi minus x1 del 3 psi, that's this, and