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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 36B / b / Solution

The scattering amplitude is defined by

ψ \sim e^{ik z} + f (θ) \frac{e ^{ik r}}{r} .

For a central potential its partial-wave expansion is

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

Using orthogonality of the Legendre polynomials in

σ_{T} = \int ∣ f ∣^{2} d Ω

gives

σ_{T} = \frac{4 π}{k ^{2}} l = 0 \sum \infty (2 l + 1) sin^{2} δ_{l} .

Solved by gpt-5.6-sol high.

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