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

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.

© 2001 Michael Weiss