I found that the derivation of most “google result” is not clear enough. So here is my derivation. Before process, people may need to review the pervious post.

Most people start from,

that annoying me because it is somehow not “natural”. Why there is a “minus” sign? Why the is the first term? For my self, a more natural way is,

where are complex numbers, but that is still not so natural, because in numerical calculation, for simplicity, there is no complex number, we only have,

The first term is alway there as it is the free solution and bounded at . the second term is caused by the potential.

The goal is to find a solution take the form

where the first term is free wave and the second term is scattered wave. The solution for elastics scattering is

we used the substitution,

.

The radial function can be solved using Rungu-Kutta method on the equation,

and the solution of at far away is,

.

the arrow means . So, the problem is how to rewrite the solution. In the way, we will see how the phase shift or the S-matrix was found.

The free solution is the spherical wave,

The spherical Bessel function cna be express as Heankel function

The sign is outgoing wave.

Since the should be normalized, we can se and .

We put back

By setting

,

we have the first term is the free wave function. In the second term, . Notice that

That cancel the term in . And we have

some people will write the as and the S-matrix,

where

.

Remember that this is the S-matrix for elastics scattering.