Calculating Scattering Amplitude in Born Approx.

[Unkraut]

Homework Statement

I'm supposed to calculate the scattering amplitudes of some spherically symmetric potentials in the Born approximation and just trying to figure out how that works in general and what a scattering amplitude is actually.

Homework Equations

1+1=2

The Attempt at a Solution

Reading this page http://electron6.phys.utk.edu/QM2/modules/m7/born.htm, I understand the following:
With an incoming plane wave $$\psi_0(\vec r)=e^{ikz}$$ and a potential $$V(r)=\frac{\hbar^2}{2m}U(r)$$ the solution can be written as $$\psi(\vec r)=\psi_0(\vec r)+\int G(\vec r-\vec r')U(r')\psi(r')d^3r$$ where the Green's function G takes the form $$G(\vec r)=G(r)=-\frac{1}{4\pi}\frac{e^{ikr}}{r}$$ and for big r we have $$G(\vec r-\vec r')=-\frac{1}{4\pi}\frac{e^{ikr}}{r}e^{-ik\vec r \cdot \vec r'}$$.
Now $$\psi(\vec r)=e^{ikz}-\frac{1}{4\pi}\frac{e^{ikr}}{r}\int e^{-ik\vec r \cdot \vec r'}U(r')\phi(\vec r')d^3r'=e^{ikz}+f(\theta, \phi)e^{ikr}{r}$$ ?
That means $$f(\theta, \phi)=-\frac{1}{4\pi}\int e^{-ik\vec r \cdot \vec r'}U(r')\phi(\vec r')d^3r'$$. But this is dependent on r, isn't it? In other sources I find rougly the same, with no explanation why this is considered independent of r. In my eyes that looks just wrong. Physics is strange.

 

[Unkraut]

Ah, I see. It's not $$\vec r$$ but $$\hat r$$, i.e. the unit vector in radial direction. I misinterpreted the notation. Right?