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Eigenvectors of
for Spin

To find the eigenvectors of the operator
we follow precisely the same
procedure as we did for
(see previous example for details). The steps
are:
1. Write the eigenvalue equation

2. Solve the characteristic equation for the eigenvalues

3. Substitute the eigenvalues back into the original equation

4. Solve this equation for the eigenvectors

Here we go! The operator
, so that, in matrix notation the eigenvalue equation becomes

The characteristic equation is
, or

These are the same eigenvalues we found for
(no surprise!) Plugging
back into the equation, we obtain

Writing this out in components gives the pair of equations

which are both equivalent to
. Repeating the process for
, we find that
. Hence the two eigenvalues and
their corresponding normalized eigenvectors are

Jim Branson
2013-04-22