This note rederives the harmonic oscillator solution, but in spherical coordinates. The reason to do so is to obtain energy eigenfunctions that are also eigenfunctions of square angular momentum and of angular momentum in the -direction. The derivation is very similar to the one for the hydrogen atom given in derivation {D.15}, so the discussion will mainly focus on the differences.
The solutions are again in the form with the
the spherical harmonics. However, the radial functions
are different; the equation for them is now
Split off the expected asymptotic behavior for large
by defining
Plug in a power series , then the
coefficients must satisfy:
Therefore, numbering the energy levels from 1 like for the hydrogen
level gives the energy levels as
Note that for even , the power series proceed in even powers of . These eigenfunctions are said to have even parity: if you replace by , they are unchanged. Similarly, the eigenfunctions for odd expand in odd powers of . They are said to have odd parity; if you replace by , they change sign.