Under Born approximation, in non-relativistic quantum mechanics, the scattering amplitude A ( | p ⟩ ⟩ → → | p ′ ⟩ ⟩ ) is: A ( | p ⟩ ⟩ → → | p ′ ⟩ ⟩ ) − − 1 = 2 π π δ δ ( E p − − E p ′ ) ( − − i ) ∫ ∫ d 3 r V ( r ) e − − i ( p − − p ′ ) r This is to be compared to the: ∫ ∫ d 3 k ( 2 π π ) 3 e i k r 0 ⟨ ⟨ p ′ , k | S | p , k ⟩ ⟩ where we look at the (connected) S-matrix entry for two electrons scattering off each other, treating one with "fixed" momentum as the source of the potential, and the other scattering off that potential.