Scattering length

As an example on how to compute the s-wave (i.e. angular momentum ${\displaystyle l=0}$) scattering length for a given potential we look at the infinitely repulsive spherical potential well of radius ${\displaystyle r_{0}}$ in 3 dimensions. The radial Schrödinger equation (${\displaystyle l=0}$) outside of the well is just the same as for a free particle:

${\displaystyle -{\frac {\hbar ^{2}}{2m}}u''(r)=Eu(r),}$

where the hard core potential requires that the wave function ${\displaystyle u(r)}$ vanishes at ${\displaystyle r=r_{0}}$, ${\displaystyle u(r_{0})=0}$. The solution is readily found:

${\displaystyle u(r)=A\sin(kr+\delta _{s})}$.

Here ${\displaystyle k={\sqrt {2mE}}/\hbar }$ and ${\displaystyle \delta _{s}=-k\cdot r_{0}}$ is the s-wave phase shift (the phase difference between incoming and outgoing wave), which is fixed by the boundary condition ${\displaystyle u(r_{0})=0}$; ${\displaystyle A}$ is an arbitrary normalization constant.

One can show that in general ${\displaystyle \delta _{s}(k)\approx -k\cdot a_{s}+O(k^{2})}$ for small ${\displaystyle k}$ (i.e. low energy scattering). The parameter ${\displaystyle a_{s}}$ of dimension length is defined as the scattering length. For our potential we have therefore ${\displaystyle a=r_{0}}$, 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 ${\displaystyle a_{s}}$ has the same low energy scattering properties as a hard sphere of radius ${\displaystyle a_{s}}$.) To relate the scattering length to physical observables that can be measured in a scattering experiment we need to compute the cross section ${\displaystyle \sigma }$. 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 ${\displaystyle z}$-axis):

${\displaystyle \psi (r,\theta )=e^{ikz}+f(\theta ){\frac {e^{ikr}}{r}}}$

where ${\displaystyle f}$ is the scattering amplitude. According to the probability interpretation of quantum mechanics the differential cross section is given by ${\displaystyle d\sigma /d\Omega =|f(\theta )|^{2}}$ (the probability per unit time to scatter into the direction ${\displaystyle \mathbf {k} }$). If we consider only s-wave scattering the differential cross section does not depend on the angle ${\displaystyle \theta }$, and the total scattering cross section is just ${\displaystyle \sigma =4\pi |f|^{2}}$. The s-wave part of the wavefunction ${\displaystyle \psi (r,\theta )}$ is projected out by using the standard expansion of a plane wave in terms of spherical waves and Legendre polynomials ${\displaystyle P_{l}(\cos \theta )}$:

${\displaystyle e^{ikz}\approx {\frac {1}{2ikr}}\sum _{l=0}^{\infty }(2l+1)P_{l}(\cos \theta )\left[(-1)^{l+1}e^{-ikr}+e^{ikr}\right]}$

By matching the ${\displaystyle l=0}$ component of ${\displaystyle \psi (r,\theta )}$ to the s-wave solution ${\displaystyle \psi (r)=A\sin(kr+\delta _{s})/r}$ (where we normalize ${\displaystyle A}$ such that the incoming wave ${\displaystyle e^{ikz}}$ has a prefactor of unity) one has:

${\displaystyle f={\frac {1}{2ik}}(e^{2i\delta _{s}}-1)\approx \delta _{s}/k\approx -a_{s}}$

This gives:

${\displaystyle \sigma ={\frac {4\pi }{k^{2}}}\sin ^{2}\delta _{s}=4\pi a_{s}^{2}}$

• Fermi pseudopotential
• Neutron scattering length

• Landau, L. D.; Lifshitz, E. M. (2003). Quantum Mechanics: Non-relativistic Theory. Amsterdam: Butterworth-Heinemann. ISBN 0-7506-3539-8.