The Phase-Amplitude (PhA) representation of a wave function

The PhA representation of an oscillatory wave function is $\psi (r)=y\ \sin (\phi )$, where $y(r)$ is the amplitude and $\phi (r)$ the phase. Since these quantities depend on distance $r$ slowly and generally monotonically, they can be calculated numerically out to large distances with a relatively small number of mesh-points. A linear equation for $y^{2}$ exists that has been overlooked in the past. The advantage of this equation is that it avoids the non-linearity difficulties encountered with the equation for $y$ given In 1930 W. E. Milne. This equation will be shown and a solution method will be described, that uses expansions into Laguerre polynomials. A numerical example for the Coulomb potential will be presented, including the region of turning points.