As and example of another problem with spherical symmetry, we solve the
3D symmetric harmonic oscillator
We have already solved this problem in Cartesian coordinates.
Now we use spherical coordinates and angular momentum eigenfunctions.
The eigen-energies are
is the number of nodes in the radial wave function and
total angular momentum quantum number.
This gives exactly the same set of eigen-energies as we got in the Cartesian solution
but the eigenstates are now states of definite total angular momentum and z component of angular momentum.