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strangerep
#60
Apr3-12, 08:57 PM
Sci Advisor
P: 1,919
Quote Quote by tom.stoer
I guess strangerep wants to discuss the states [...]
Not necessarily. The previous answers seemed to be missing the mark somehow,
so I wanted check Mr Vodka's prior knowledge of Lie groups, QM, and their use in
the classification of elementary particles.

Quote Quote by mr. vodka View Post
I'm familiar with the derivation yes. However, I
don't recall SO(3) being mentioned (but I'm quite familiar with group theory,
so no need to hold back). I took a quick look at Ballentine (happened to be my
lying on my desk) and I don't see SO(3) being mentioned in the derivation
either.
Ballentine covers the rotation group SO(3) much earlier, so he assumes the
reader already knows about how those noncommuting ##J_x,J_y,J_z## operators
form the Lie algebra of so(3) (or su(2), but let's skip that for the moment).
He also assumes that by section 7 the reader knows about the relationship between
a Lie group like SO(3) and the Lie algebra so(3) which "generates" the group.

From what you said above, I get the impression you're not yet clear on the intimate
connection between angular momentum and the rotation group SO(3) ?

It seems the derivation in Ballentine is similar to the one I saw in
my QM class: we basically define the angular momentum operator J as something
that satisfies the well-known commutation relations, and then it was stated
that Jē and J_z form a CSCO (that it's a SCO is clear, but we didn't see an
argument for the "Complete" part [which, I suppose, depends on the context of
the angular momentum]; maybe this is related to your SO(3) reference).
No, it's "complete" because there's only one "Casimir" invariant operator for
the SO(3). (A Casimir invariant is an operator which commutes with all the
group operators, other than the trivial identity operator.) Since this
derivation focuses only with angular momentum and nothing else, he
doesn't need to consider other stuff at this point.

One defines the classifying quantum numbers j and m such that [...]
With some similar arguments one can also argue that [itex]-j \leq m \leq j[/itex].
OK, that's close enough for present purposes.

But I don't think my confusion stems from me misunderstanding
(regular) spin (?).
The feeling I got reading through earlier parts of this thread is that you
were having trouble transferring the principles in the quantum theory of
angular momentum over to more general cases of classification of elementary
particles.

But... to ensure none of us waste our time..., maybe you should restate your
question(s) as clearly as you can now?