Parabolic coordinates simplify the wave equation of an azimuthally symmetric Coulomb potential. To get from Spherical to Parabolic Coordinates, make the following substitution.
The inverse of this substitution is:
Use the chain rule to convert derivatives from the spherical coordinates to parabolic coordinates. First convert the r derivatives.
Grunt work yields a cleaner form.
Now convert the θ derivatives.
Putting the two results together and doing a little more digging yields
The Laplacian Operator in parabolic coordinates is:
The form of the equations becomes very simple if one considers azimuthally independent solutions.
Now consider the Schrödinger Equation with a Coulomb potential in these coordinates.
Multiplying the whole equation by the sum of the parabolic coordinates, one finds.
Slightly further simplified:
For many situations in physics, the desired solution must be asymptotic to a plane wave. To search for such a solution, make the following substitution.
Solving for k to eliminate the last term from the Schrödinger Equation,
the remaining function satisfies a much simpler equation:
Now make a substitution to write this equation in terms of a dimensionless coordinate.
The result is in the form of the confluent hypergeometric equation.
In the second given form of η, β is the relative velocity in units of c. I prefer to write this equation with the terms proportional to the same power of ρ grouped.