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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 35E / c / Solution

The Gaussian transform is

V (q) = (2 π)^{3/2} V_{0} r_{0}^{3} e^{- q^{2} r_{0}^{2} /2} .

Therefore

\frac{d σ}{d Ω} = \frac{2 π m ^{2} V _{0}^{2} r _{0}^{6}}{ℏ ^{4}} e^{- q^{2} r_{0}^{2}}, q = 2 k sin (ϑ /2) .

A sufficient weak-scattering regime is kinetic energy much larger than

∣ V_{0} ∣

(more intrinsically

m ∣ V_{0} ∣ r_{0}^{2} / ℏ^{2} ≪ 1

).

Solved by gpt-5.6-sol high.

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