The time-independent Schrödinger equation is
Using the Laplacian in spherical coordinate. and Set
The angular part,
The radial part,
To simplify the first term,
A more easy form of the radial function is,
The effective potential
We can use Rungu-Kutta method to numerically solve the equation.
The initial condition of has to be 0. (home work)
I used excel to calculate a scattered state of L = 0 of energy 30 MeV. The potential is a Wood-Saxon of depth 50 MeV, radius 3.5 fm, diffusiveness 0.8 fm.
Another example if bound state of L = 0. I have to search for the energy, so that the wavefunction is flat at large distance. The outermost eigen energy is -7.27 MeV. From the radial function, we know it is a 2s orbit.