# 5.5. Calculation of scattering cross sections with partial waves¶

In this section we derive Eqs. 5.19 and 5.20. At a large distance from the scattering
centre we can make an *Ansatz* for the wave function. This consists of
the incoming beam and a scattered wave:
is the angle between the incoming beam and the
line passing through and the scattering centre. does not
depend on the azimuthal angle because the incoming wave has
azimuthal symmetry, and the spherically symmetric potential will not
generate contributions to the scattered wave. is
called the scattering amplitude. From the *Ansatz* it follows that the
differential cross section is given directly by the square of this
amplitude:
with
the appropriate normalisation.

Beyond , the solution can also be written in the form leaving out all contributions because of the azimuthal symmetry: where we have used the fact that is proportional to . Because the potential vanishes in the region , the solution is given by the linear combination of the regular and irregular spherical Bessel functions, and as we have seen this reduces for large to We want to derive the scattering amplitude by equating the expressions \eqref{eq_psi} and \eqref{eq_psi3} for the wave function. For large we obtain, using \eqref{bess_approx}: We write the right hand side of this equation as an expansion similar to that in the left hand side, using the following expression for a plane wave can also be written as an expansion in Legendre polynomials: so that we obtain: If we substitute the asymptotic form of in the right hand side, we find: Both the left and the right hand side of \eqref{equat} contain in- and outgoing spherical waves (the occurrence of incoming spherical waves does not violate causality: they arise from the incoming plane wave). For each , the prefactors of the incoming and outgoing waves should both be equal on both sides in \eqref{equat}. This condition leads to and

Using \eqref{eq_sigma}, \eqref{f_exp}, and \eqref{f_delta}, we can write down an expression for the differential cross section in terms of the phase shifts : For the total cross section we find, using the orthonormality relations of the Legendre polynomials: