Quantum Mechanics for Engineers 

© Leon van Dommelen 

12.11 General spin matrices
The arguments that produced the Pauli spin matrices for a system with
spin work equally well for systems with larger square
angular momentum.
In particular, from the definition of the ladder operators
it follows by taking the sum, respectively difference, that

(12.17) 
Therefore, the effect of either or 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 is real symmetric while is purely imaginary, equal
to times a real skewsymmetric matrix. Be sure to tell all you
friends that you heard it here first. Do watch out for the
wellinformed 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.