### D.65 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 a few times on the top of the lad­der . Every con­tri­bu­tion to the co­ef­fi­cient of a state comes from ap­ply­ing for times and for times, so all con­tri­bu­tions have com­pat­i­ble square roots. merely adds an 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 and 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:

where is the Kro­necker delta and

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.