The remainder of this section goes into more detail on this calculation but is currently notationally challenged.
Recall the standard method of finding eigenvectors and eigenvalues:
For a matrix times a nonzero vector to give zero, the determinant of the
matrix must be zero.
This gives the ``characteristic
equation'' which for spin
systems will be a quadratic equation in
To find the eigenvectors, we simply replace (one at a time) each of the
eigenvalues above into the equation
Now specifically, for the operator
, the eigenvalue equation
The characteristic equation is
Similarly, we find for the eigenvalue
Jim Branson 2013-04-22