 Partial-wave expansion of a scattering amplitude

ID: partial-wave-expansion-of-a-scattering-amplitude

For a central potential and the convention

S_{l} = 1 + 2 i f_{l} = e^{2 i δ_{l}}

f (θ) = \frac{1}{k} \sum_{l = 0}^{\infty} (2 l + 1) f_{l} P_{l} (cos θ), f_{l} = e^{i δ_{l}} sin δ_{l} .

The identity

∣ S_{l} ∣ = 1

is equivalent to

Im f_{l} = ∣ f_{l} ∣^{2}

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