Loose Ends

Why $j(j+1)$, instead of $j^2$? 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 $m^2$'' ($m$ being the projection on the $z$-axis, as usual) like so:

$$\frac{(-j)^2 + (-j+1)^2 +\ldots+(j-1)^2+j^2}{2j+1}$$

and verify that $j(j+1)/3$ results. Now argue that for any vector v, $\vert{\bf v}\vert^2 = v_x^2 + v_y^2 + v_z^2$, so the average value of $m^2$ 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 $j^2$? I suggested, loosely, that ``$SO(3)$ is classical, $SU(2)$ is quantum''. The eigenvalue $j(j+1)$ applies to representations of $SO(3)$ as well as $SU(2)$.

The culprit here is the finite dimensionality of the representation. To call $j$ the ``classical value'' for the angular momentum is misleading. Let us return to ``unnatural units'', where $\hbar$ is small. The ``classical'' magnitude for angular momentum is $j\hbar$, with large $j$. We get the ``classical limit'' by simultaneously letting $j \rightarrow \infty$ and $\hbar \rightarrow 0$, while keeping the product constant. In the classical limit, the quantum expression $\sqrt{j(j+1)}\hbar$ simplifies to $j\hbar$.

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 $l$ can grow as large as one wishes. The intrinsic spin, $s$, cannot - $s=\frac{1}{2}$ for an electron, $s=1$ for a photon. Even for a large atom like uranium, $s$ 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 $s$ would imply an enormous energy which would blow the nucleus apart.) In practice, the classical limit makes no sense for $s$, 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 $\sigma_1: SU(2) \rightarrow SU(H_1)$ and $\sigma_2: SU(2) \rightarrow SU(H_2)$. $H_1$ and $H_2$ are the Hilbert spaces for two separate physical systems. The Hilbert space of the combined system is the tensor product $H_1\otimes H_2$, and the representation $\sigma_1\otimes\sigma_2:SU(2)\rightarrow SU(H_1\otimes H_2)$ begets the angular momentum operator for the combined system. Even if $\sigma_1$ and $\sigma_2$ are irreducible representations, $\sigma_1\otimes\sigma_2$ generally won't be, but will decompose into a direct sum of irreducible representations:

$$\sigma_1\otimes\sigma_2: SU(2) \rightarrow SU(K_1)\oplus\ldots\oplus SU(K_r)$$

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

Enough said.

© 2001 Michael Weiss

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