In a scattering problem, the main objective is solving the Schrödinger equation

where H is the total Hamiltonian of the scattering system in the center of momentum, K is the kinetic energy and V is the potential energy. We seek for a solution ,

The solution can be decomposed

The solution of can be solve by Runge-Kutta method on the pdf

where and .

For , the solution of is

where and is the Riccati-Bessel function. The free wave function is

where is the Legendre polynomial.

Note that, if we have Coulomb potential, we need to use the Coulomb wave instead of free wave, because the range of coulomb force is infinity.

For , the solution of can be found by Runge-Kutta method, where R is a sufficiency large that the potential is effectively equal to 0. The solution of is shifted

where is the scattering matrix element, it is obtained by solving the boundary condition at . The scattered wave function is

put the scattered wave function and the free wave function back to the seeking solution, we have the

and the differential cross section

.

In this very brief introduction, we can see

- How the scattering matrix is obtained
- How the scattering amplitude relates to the scattering matrix

But what is scattering matrix? Although the page did not explained very well, especially how to use it.