Editing Scattering amplitude (section)

== Partial wave expansion ==

{{Main article|Partial wave analysis}}

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,<ref>[http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/Scattering_II.htm Michael Fowler/ 1/17/08 Plane Waves and Partial Waves]</ref>
:<math>f=\sum_{\ell=0}^\infty (2\ell+1) f_\ell P_\ell(\cos \theta)</math>,
where {{math|''f<sub>ℓ</sub>''}} is the partial scattering amplitude and {{math|''P<sub>ℓ</sub>''}} are the [[Legendre polynomial]]s. The partial amplitude can be expressed via the partial wave [[S-matrix]] element {{math|''S<sub>ℓ</sub>''}} (<math>=e^{2i\delta_\ell}</math>) and the '''scattering phase shift''' {{math|''δ<sub>ℓ</sub>''}} as
:<math>f_\ell = \frac{S_\ell-1}{2ik} = \frac{e^{2i\delta_\ell}-1}{2ik} = \frac{e^{i\delta_\ell} \sin\delta_\ell}{k} = \frac{1}{k\cot\delta_\ell-ik} \;.</math>

Then the total cross section<ref name=Schiff1968>{{cite book|last=Schiff|first=Leonard I.|title=Quantum Mechanics|url=https://archive.org/details/quantummechanics00schi_086|url-access=limited|date=1968|publisher=McGraw Hill|location=New York|pages=[https://archive.org/details/quantummechanics00schi_086/page/n138 119]–120}}</ref>
:<math>\sigma = \int |f(\theta)|^2d\Omega </math>,
can be expanded as{{r|landau}}
:<math>\sigma = \sum_{l=0}^\infty \sigma_l, \quad \text{where} \quad \sigma_l = 4\pi(2l+1)|f_l|^2=\frac{4\pi}{k^2}(2l+1)\sin^2\delta_l</math>
is the partial cross section. The total cross section is also equal to <math>\sigma=(4\pi/k)\,\mathrm{Im} f(0)</math> due to [[optical theorem]].

For <math>\theta\neq 0</math>, we can write{{r|landau}}
:<math>f=\frac{1}{2ik}\sum_{\ell=0}^\infty (2\ell+1) e^{2i\delta_l} P_\ell(\cos \theta).</math>