 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-2/36b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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) .

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} .

 New to topics? Read the docs here!