Comparing with the QM scattering, we have to discard the ( 2 m ) 2 as they arise due to differing normalizations of momentum eigenstate in QFT compared to QM and obtain: ∫ ∫ V ( r ) e − − i ( p − − p ′ ) r d 3 r = e 2 | p − − p ′ | 2 − − i ε ε where Fourier transforming both sides, solving the integral and taking ε ε → → 0 at the end will yield V ( r ) = e 2 4 π π r as the Coulomb potential.