When a slow particle scatters off a short ranged scatterer (e.g. an impurity in a solid or a heavy particle) it cannot resolve the structure of the object since its de Broglie wavelength is very long. The idea is that then it should not be important what precise potential V(r) one scatters off, but only how the potential looks at long length scales. The formal way to solve this problem is to do a partial wave expansion (somewhat analogous to the multipole expansion in classical electrodynamics), where one expands in the angular momentum components of the outgoing wave. At very low energy the incoming particle does not see any structure, therefore to lowest order one has only a spherical symmetric outgoing wave, the so called s-wave scattering (angular momentum l = 0). At higher energies one also needs to consider p and d-wave (l = 1,2) scattering and so on. The concept behind describing low energy properties in terms of a few parameters and symmetries is the idea of renormalization.
As an example on how to compute the s-wave (i.e. angular momentum l = 0) scattering length for a given potential we look at the infinitely repulsive spherical potential well of radius r0 in 3 dimensions. The radial Schrödinger equation (l = 0) outside of the well is just the same as for a free particle:
where the hard core potential requires that the wave function u(r) vanishes at r = r0, u(r0) = 0. The solution is readily found:
Here ; is the s-wave phase shift (the phase difference between incoming and outgoing wave), which is fixed by the boundary condition u(r0) = 0; A is an arbitrary normalization constant.
One can show that in general for small k (i.e. low energy scattering). The parameter as of dimension length is defined as the scattering length. For our potential we have therefore a = r0, in other words the scattering length for a hard sphere is just the radius. (Alternatively one could say that an arbitrary potential with s-wave scattering length as has the same low energy scattering properties as a hard sphere of radius as). To relate the scattering length to physical observables that can be measured in a scattering experiment we need to compute the cross section σ. In scattering theory one writes the asymptotic wavefunction as (we assume there is a finite ranged scatterer at the origin and there is an incoming plane wave along the z-axis)
where f is the scattering amplitude. According to the probability interpretation of quantum mechanics the differential cross section is given by dσ / dΩ = | f(θ) | 2 (the probability per unit time to scatter into the direction ). If we consider only s-wave scattering the differential cross section does not depend on the angle θ, and the total scattering cross section is just σ = 4π | f | 2. The s-wave part of the wavefunction ψ(r,θ) is projected out by using the standard expansion of a plane wave in terms of spherical waves and Legendre polynomials Pl(cosθ)
By matching the l = 0 component of ψ(r,θ) to the s-wave solution ψ(r) = Asin(kr + δs) / r (where we normalize A such that the incoming wave eikz has a prefactor of unity) one has