12.11 Gen­eral spin ma­tri­ces

The ar­gu­ments that pro­duced the Pauli spin ma­tri­ces for a sys­tem with spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em
/\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$ work equally well for sys­tems with larger square an­gu­lar mo­men­tum.

In par­tic­u­lar, from the de­f­i­n­i­tion of the lad­der op­er­a­tors

{\widehat J}^+ \equiv {\widehat J}_x + {\rm i}{\widehat J}_...
...d {\widehat J}^- \equiv {\widehat J}_x - {\rm i}{\widehat J}_y

it fol­lows by tak­ing the sum, re­spec­tively dif­fer­ence, that
{\widehat J}_x = {\textstyle\frac{1}{2}} {\widehat J}^+ + {...
...{\widehat J}^+ + {\rm i}{\textstyle\frac{1}{2}} {\widehat J}^-
\end{displaymath} (12.17)

There­fore, the ef­fect of ei­ther ${\widehat J}_x$ or ${\widehat J}_y$ is to pro­duce mul­ti­ples of the states with the next higher and the next lower mag­netic quan­tum num­ber. The mul­ti­ples can be de­ter­mined us­ing (12.9) and (12.10).

If you put these mul­ti­ples again in ma­tri­ces, af­ter or­der­ing the states by mag­netic quan­tum num­ber, you get Her­mit­ian tridi­ag­o­nal ma­tri­ces with nonzero sub and su­per­diag­o­nals and zero main di­ag­o­nal, where ${\widehat J}_x$ is real sym­met­ric while ${\widehat J}_y$ is purely imag­i­nary, equal to ${\rm i}$ times a real skew-sym­met­ric ma­trix. Be sure to tell all you friends that you heard it here first. Do watch out for the well-in­formed friend who may be aware that form­ing such ma­tri­ces is bad news any­way since they are al­most all ze­ros. If you want to use canned ma­trix soft­ware, at least use the kind for tridi­ag­o­nal ma­tri­ces.