3D Problems Separable in Cartesian Coordinates
We will now look at the case of potentials that separate in Cartesian coordinates.
These will be of the form.
In this case, we can solve the problem by separation of variables.
The left hand side of this equation depends only on
, while the right side
In order for the two sides to be equal everywhere, they
must both be equal to a constantwhich we call
part of the solution satisfies the equation
Treating the other components similarly we get
and the total energy is
There are only a few problems which can be worked this way but they are important.