Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S18 / 1 / 5

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- Solution 5

Solution

The first Born approximation substitutes the incident wave for the exact wave inside the scattering integral. Thus

R_{B} = \frac{m ^{2}}{k ^{2} ℏ ^{4}} \int e^{2 ik x} U (x) d x^{2} .

For the attractive delta this is

β^{2} / k^{2}

, the leading small-

β / k

expansion of the exact result. Physically one neglects the distortion of the incident state by repeated interaction with the potential. At low

k

the Born result can exceed one, while the exact reflection remains bounded and approaches one; it is not valid in that limit.

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