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#### Loose Ends

Why , instead of ? Well, group representation theory tells us so! - as already mentioned in the section titled Spin One-Half''. If you consult the computations in a textbook, you will find it all hinges on the commutators of the Lie algebra. You may also find it amusing to compute the average value of '' ( being the projection on the -axis, as usual) like so:

and verify that results. Now argue that for any vector v, , so the average value of should be one-third the magnitude of the angular momentum vector.

How do we reconcile the verdict of group representation theory with the classical value ? I suggested, loosely, that  is classical, is quantum''. The eigenvalue applies to representations of as well as .

The culprit here is the finite dimensionality of the representation. To call the classical value'' for the angular momentum is misleading. Let us return to unnatural units'', where is small. The classical'' magnitude for angular momentum is , with large . We get the classical limit'' by simultaneously letting and , while keeping the product constant. In the classical limit, the quantum expression simplifies to .

Bohr elevated this and similar limiting relations into a guiding principle in the old quantum theory. He termed it the Correspondence Principle. Sommerfeld called it a magic wand that only worked in Copenhagen, a back-handed compliment.

The orbital angular momentum number can grow as large as one wishes. The intrinsic spin, , cannot - for an electron, for a photon. Even for a large atom like uranium, would be at most a few hundred, assuming the spin of all the electrons, protons, and neutrons combined constructively. (But such a large value of would imply an enormous energy which would blow the nucleus apart.) In practice, the classical limit makes no sense for , another sense in which intrinsic spin is fundamentally non-classical.

The addition rules for angular momentum come from the following considerations: suppose we have two representations and . and are the Hilbert spaces for two separate physical systems. The Hilbert space of the combined system is the tensor product , and the representation begets the angular momentum operator for the combined system. Even if and are irreducible representations, generally won't be, but will decompose into a direct sum of irreducible representations:

The addition rules now all fall out from theorems of group representation theory.

Enough said.

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