Scattering length

The scattering length in quantum mechanics describes low-energy scattering. For potentials that decay faster than ${\displaystyle 1/r^{3}}$ as ${\displaystyle r\to \infty }$, it is defined as the following low-energy limit:

${\displaystyle \lim _{k\to 0}k\cot \delta (k)=-{\frac {1}{a}}\;,}$

where ${\displaystyle a}$ is the scattering length, ${\displaystyle k}$ is the wave number, and ${\displaystyle \delta (k)}$ is the phase shift of the outgoing spherical wave. The elastic cross section, ${\displaystyle \sigma _{e}}$, at low energies is determined solely by the scattering length:

${\displaystyle \lim _{k\to 0}\sigma _{e}=4\pi a^{2}\;.}$