L22.2 First Born Approximation. Calculation of the scattering amplitude 

The added momentum due to peturbation is very small compared to initial momentum. So only direction changes not magnitude can be assumed.

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Scattered wave has only a phase shift but still in the same state with the incident wave (we've derived this result in previous lectures - by considering a far away observation point aka observation point r >> r' while r' is where the potential exists) thus its momentum (kscattered) is equal to the incident wave's momentum (kincident) so the length (norm of k: |k|) is the same for both.