12.11 General spin matrices

The arguments that produced the Pauli spin matrices for a system with spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em
/\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$ work equally well for systems with larger square angular momentum.

In particular, from the definition of the ladder operators

\begin{displaymath}
{\widehat J}^+ \equiv {\widehat J}_x + {\rm i}{\widehat J}...
... {\widehat J}^- \equiv {\widehat J}_x - {\rm i}{\widehat J}_y
\end{displaymath}

it follows by taking the sum, respectively difference, that
\begin{displaymath}
{\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.