For the common
problem of central potentials
we use the obvious rotational symmetry to find that the angular momentum,
, operators commute with
but they do not commute with each other.
We want to find two mutually commuting operators
which commute with
, so we turn to
which does commute with each component of
We chose our two operators to be
Some computation reveals that we can write
With this the kinetic energy part of our equation will only have derivatives in
assuming that we have eigenstates of
The Schrödinger equation thus separates
into an angular part (the
term) and a radial part (the rest).
With this separation we expect (anticipating the angular solution a bit)
will be a solution.
will be eigenfunctions of
so the radial equation becomes
We must come back to this equation for each
which we want to solve.
We solve the angular part of the problem in general
using angular momentum operators.
We find that angular momentum is quantized.
integers satisfying the condition
The operators that raise and lower the component of angular momentum are
We derive the
functional form of the Spherical Harmonics
using the differential form of the angular momentum operators.