# 2-particle scattering phase shifts

1. Apr 15, 2014

### Einj

Hi everyone. Consider the scattering amplitude and phase shift formalism for the study of scattering theory. One usually develops this techniques in the situation where we have in incoming plane wave being scattered by a fixed potential V. In particular one usually writes:
$$\psi_{\vec p}^{in}(\vec r)=e^{i\vec p\cdot\vec r/\hbar}+\frac{e^{ipr/\hbar}}{r}f(\theta,\phi).$$
Where $f(\theta,\phi)$ is the scattering amplitude and contains the potential V. Such amplitude can also be expressed in terms of the so-called phase shifts as:
$$f(\theta,\phi)=\sum_{\ell=0}^\infty \frac{1}{k}(2\ell+1)e^{i\delta_{\ell}}\sin\delta_\ell P_\ell(\cos\theta).$$

I was wondering if this formalism can be applied also to the case where there are two particles colliding against each other and how one can eventually express the cross section in this case.

Thanks