Quantum Mechanics for Engineers 

© Leon van Dommelen 

D.76 Harmonic oscillator revisited
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
The difference from {D.15} is that a harmonic
oscillator potential has replaced the Coulomb
potential. A suitable rescaling is now
, which produces
where is the energy in half quanta.
Split off the expected asymptotic behavior for large
by defining
Then satisfies
Plug in a power series , then the
coefficients must satisfy:
From that it is seen that the lowest power in the series is
, not being
acceptable. Also the series must terminate, or blow up will occur.
That requires that . So the
energy must be with
an integer no smaller than , so at least zero.
Therefore, numbering the energy levels from 1 like for the hydrogen
level gives the energy levels as
That are the same energy levels as derived in Cartesian coordinates,
as they should be. However, the eigenfunctions are different. They
are of the form
where is some polynomial of degree , whose lowest
power of is . The value of the azimuthal quantum
number must run up to like for the hydrogen atom. However,
in this case must be odd or even depending on whether is odd
or even, or the power series will not terminate.
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.