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
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
Additional note: There is also a direct expression for the Clebsch-Gordan coefficients:
There are also resources on the web to compute these coefficients. See {N.13} for additional information.