Optical theorem

In physics, the optical theorem is a general law of wave scattering theory, which relates the zero-angle scattering amplitude to the total cross section of the scatterer.^[1] It is usually written in the form

${\displaystyle \sigma ={\frac {4\pi }{k}}~\mathrm {Im} \,f(0),}$

where f(0) is the scattering amplitude with an angle of zero, that is the amplitude of the wave scattered to the center of a distant screen and k is the wave vector in the incident direction.

Because the optical theorem is derived using only conservation of energy, or in quantum mechanics from conservation of probability, the optical theorem is widely applicable and, in quantum mechanics, ${\displaystyle \sigma _{\mathrm {tot} }}$ includes both elastic and inelastic scattering.

The generalized optical theorem, first derived by Werner Heisenberg, follows from the unitary condition and is given by^[2]

${\displaystyle f(\mathbf {n} ,\mathbf {n} ')-f^{*}(\mathbf {n} ',\mathbf {n} )={\frac {ik}{2\pi }}\int f(\mathbf {n} ,\mathbf {n} '')f^{*}(\mathbf {n} ,\mathbf {n} '')\,d\Omega ''}$

where ${\displaystyle f(\mathbf {n} ,\mathbf {n} ')}$ is the scattering amplitude that depends on the direction ${\displaystyle \mathbf {n} }$ of the incident wave and the direction ${\displaystyle \mathbf {n} '}$of scattering and ${\displaystyle d\Omega }$ is the differential solid angle. When ${\displaystyle \mathbf {n} =\mathbf {n} '}$, the above relation yields the optical theorem since the left-hand side is just twice the imaginary part of ${\displaystyle f(\mathbf {n} ,\mathbf {n} )}$ and since ${\displaystyle \sigma =\int |f(\mathbf {n} ,\mathbf {n} '')|^{2}\,d\Omega ''}$. For scattering in a centrally symmetric field, ${\displaystyle f}$ depends only on the angle ${\displaystyle \theta }$ between ${\displaystyle \mathbf {n} }$ and ${\displaystyle \mathbf {n} '}$, in which case, the above relation reduces to

${\displaystyle \mathrm {Im} f(\theta )={\frac {k}{4\pi }}\int f(\gamma )f(\gamma ')\,d\Omega ''}$

where ${\displaystyle \gamma }$ and ${\displaystyle \gamma '}$ are the angles between ${\displaystyle \mathbf {n} }$ and ${\displaystyle \mathbf {n} '}$ and some direction ${\displaystyle \mathbf {n} ''}$.