D.66 How to make Clebsch-Gordan tables

The procedure of finding the Clebsch-Gordan coefficients for the combination of any two spin ladders is exactly the same as for electron ones, so it is simple enough to program.

To further simplify things, it turns out that the coefficients are all square roots of rational numbers (i.e. ratios of integers such as 102/38.) The step-up and step-down operators by themselves produce square roots of rational numbers, so at first glance it would appear that the individual Clebsch-Gordan coefficients would be sums of square roots. But the square roots of a given coefficient are all compatible and can be summed into one. To see why, consider the coefficients that result from applying the combined step down ladder ${\widehat J}^-_{ab}$ a few times on the top of the ladder ${\big\vert j\:j\big\rangle }_a{\big\vert j\:j\big\rangle }_b$. Every contribution to the coefficient of a state ${\big\vert j\:m\big\rangle }_a{\big\vert j\:m\big\rangle }_b$ comes from applying ${\widehat J}^-_a$ for $j_a-m_a$ times and ${\widehat J}^-_b$ for $j_b-m_b$ times, so all contributions have compatible square roots. ${\widehat J}^-_{ab}$ merely adds an $m_{ab}$ dependent normalization factor.

You might think this pattern would be broken when you start defining the tops of lower ladders, since that process uses the step up operators. But because ${\widehat J}^+{\widehat J}^-$ and ${\widehat J}^-{\widehat J}^+$ are rational numbers (not square roots), applying the up operators is within a rational number the same as applying the down ones, and the pattern turns out to remain.

Additional note: There is also a direct expression for the Clebsch-Gordan coefficients:

\begin{eqnarray*}
\lefteqn{\big\langle j\,m\big\vert j_1\,m_1\big\rangle \big\...
...\frac{(-1)^z}{\prod_{i=1}^3 (z-z_{{\rm l}i})!(z_{{\rm h}i}-z)!}
\end{eqnarray*}

where $\delta$ is the Kronecker delta and

\begin{eqnarray*}
& z_{{\rm l}1} = 0 \quad
z_{{\rm l}2} = j_1+m_2-j \quad
...
... \quad
z_{\rm h} = \min(z_{\rm h1},z_{\rm h2},z_{\rm h3}) \\
\end{eqnarray*}

Carefully coded, this one seems to be numerically superior at larger angular momenta. Either way, these coefficients will overflow pretty quickly.

There are also resources on the web to compute these coefficients. See {N.13} for additional information.