The scattering length in quantum mechanics describes low-energy scattering, it is defined as the following low-energy limit:

is the wave number is the phase shift of the outgoing spherical wave. The elastic cross section, at low energies is determined solely by the scattering length:

General concept

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 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 outgoing wave, called the s-wave in analogy with the atomic orbital at angular momentum quantum number l=0. At higher energies one also needs to consider p and d-wave (l=1,2) scattering and so on.

The idea of describing low energy properties in terms of a few parameters and symmetries is very powerful, and is also behind the concept of renormalization.

. A famous example, relevant for proton-proton scattering, is the Coulomb-modified scattering length.

Example

) scattering length for a given potential we look at the infinitely repulsive spherical potential well in 3 dimensions. The radial Schrödinger equation) outside of the well is just the same as for a free particle:

where the hard core potential requires that the wave function. The solution is readily found:

1. .

is the s-wave phase shift is an arbitrary normalization constant.

of dimension length is defined as the scattering length, in other words the scattering length for a hard sphere.) To relate the scattering length to physical observables that can be measured in a scattering experiment we need to compute the cross section. 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-axis):

is the scattering amplitude. According to the probability interpretation of quantum mechanics the differential cross section, and the total scattering cross section is projected out by using the standard expansion of a plane wave in terms of spherical waves and Legendre polynomials:

has a prefactor of unity) one has:

This gives:

See also

• Fermi pseudopotential

• Neutron scattering length

References