$$\require{cancel}$$
A comparison between Equations ([e17.73]) and ([e17.75]) reveals that $\sigma_{\rm total} = \frac{4\pi}{k}\,{\rm Im}[f(0)] = \frac{4\pi}{k}\, {\rm Im}(f({\bf k},{\bf k})],$ because $$P_l(0)=1$$ . This result is known as the optical theorem , and is a consequence of the fact that the very existence of scattering requires scattering in the forward ($$\theta=0$$) direction, in order to interfere with the incident wave, and thereby reduce the probability current in that direction.