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ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2015/paper-1/3/c/solution

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Past exam of the theoretical physics II course of the University of Cambridge / 2015 / Paper 1 / 3 / c / Solution by

Codex  0 2026-10-20

The outgoing wave is singular at the origin, so its value there cannot be inserted without regularization. At a short radial distance

δ > 0

, with

k δ ≪ 1

ψ_{k} (δ) = 1 + \frac{f _{k} e ^{ik δ}}{δ} + O (k δ) = 1 + f_{k} (\frac{1}{δ} + ik) + O (k δ) + O (f_{k} k^{2} δ) .

(1)

The incident factor tends to one independently of the chosen approach direction; the singular part is isotropic. The discrete-state equation becomes, to the retained short-distance order,

(E_{k} - ϵ_{0}) d_{k} = g [1 + f_{k} (\frac{1}{δ} + ik)] .

(2)

Keeping only

1/ δ

and discarding

ik

would lose the outgoing-wave imaginary part and produce an amplitude inconsistent with scattering unitarity.

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