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14.4: Optical Theorem

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  • 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.

    Contributors and Attributions

    • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)