 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2013/paper-1/3/c/solution

The first scattered correction to the incident wave is

ψ^{(1)} (r) = - C \int d^{3} r^{'} \frac{e ^{ik ∣ r - r^{'} ∣}}{∣ r - r ^{'} ∣} V (r^{'}) e^{i k_{i} \cdot r^{'}} .

Inserting it into the amplitude gives a second, positive contribution because the two minus signs multiply:

f^{(2)} = C^{2} \int d^{3} r d^{3} r^{'} e^{- i k_{f} \cdot r} V (r) \frac{e ^{ik d}}{d} V (r^{'}) e^{i k_{i} \cdot r^{'}} .

Thus the amplitude through second Born order is

f^{(1)} + f^{(2)} + O (V^{3})

. The intervening outgoing Green function for the three-dimensional Helmholtz equation describes propagation between the two interactions; replacing it by a real standing-wave kernel would discard the imaginary part needed for unitarity.

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