###

Rewriting
Using

We wish to use the
and
operators to help us solve the central potential problem.
If we can rewrite
in terms of these operators, and remove all the angular derivatives,
problems will be greatly simplified.
We will work in Cartesian coordinates for a while, where we know the commutators.
First, write out
.

Group the terms.

We expect to need to keep the radial derivatives
so lets identify those by dotting
into
.
This will also make the units match
.

By adding these two expressions, things simplify a lot.

We can now solve for
and we have something we can use in the Schrödinger equation.

The Schrödinger equation now can be written with only radial derivatives and
.

Jim Branson
2013-04-22