Scattering Theory: Difference between use of scattering amplitude and Born approximation?

[Shahzad Raza]

TL;DR

What is the difference between use of scattering amplitude and born approximation as calculation of both seem similar?

I am preparing for my final and studying scattering thoery and I was not able to connect the points and find difference between scattering amplitude of spinless particles and the born approximation both were looking alike from calculations and after spending a lot of time I have separated these main step to understand both. Below I have shared what I have understood from these steps please guide me if I am mistaken .

Scattering amplitude:
here we have taken the complete wave function representing the incident part and the scattered part .

here we have used green's theorem as scattered wave propagator and by simplifications we will get the equation 11.56

now by using born series we will get the solution of this equation for the n order approximation for the wave function, here only second order approximation is given.

here we have calculated the amplitude by comparing the equation 11.56 with the first equation which is the general formula for the scattering amplitude and then cross section with the help of amplitude

Born approximation:
And in case of born approximation we have calculated the cross section for the spherically symmetric potential .

So , in short at first we have calculated the general formula for scattering amplitude for all potentials and then in born approximation we specifically find formula for spherically symmetric potentials .

I have inserted images as I am learning LATEX yet .

 

[A. Neumaier]

TL;DR: What is the difference between use of scattering amplitude and Born approximation as calculation of both seem similar?

The latter is (as the name says) an approximation of the former, where only the first order term in the Born series is taken into account.