Symmetry Trick

A general trick to find eigenvectors and eigenvalues for operators: symmetry. Imagine two operators $S$ and $T$, where eigenvectors for $S$ are easier to find, and their eigenvalues are all distinct. Moreover, suppose $S$ and $T$ commute: $ST=TS$. This arises in many physical situations, where $S$ represents a symmetry of the system. We can find eigenvectors for $S$ by reasoning instead of by computation.

Well, with all this set up, if $v$ is an eigenvector for $S$ then $Sv=\beta v$. But observe that $Tv$ is also an eigenvector for $S$:

And, since the eigenvalues of $S$ are all different, we have necessarily that $v$ and $Tv$ must be proportional:

and thus $v$ is an eigenvector for $T$!
The associated eigenvalue is different, of course, but can be computed easily by computing $Tv$.