Scattering amplitude

The scattering amplitude is a quantity of scattering theory that describes the directional dependence of the scattering wave when a plane wave is scattered at a scattering center. It has the dimension length and connects the S matrix with the cross section . ${\displaystyle f}$

definition

The scattering amplitude is defined by the S operator : ${\displaystyle f(p\to p')}$ ${\displaystyle S}$

${\displaystyle \langle p'|S|p\rangle =\delta ^{(3)}({\vec {p}}'-{\vec {p}})+{\tfrac {\mathrm {i} }{2\pi m}}\delta (E'-E)f(p\to p')}$

Are there

• ${\displaystyle |p\rangle }$the initial state and the final state with defined momentum , i.e. eigenstates of the momentum operator , ${\displaystyle |p'\rangle }$
• ${\displaystyle {\vec {p}},{\vec {p}}'}$ the impulses of the states
• ${\displaystyle E,E'}$the energy of the states
• ${\displaystyle m}$the mass (physics) of the states and
• ${\displaystyle \delta }$the Dirac distribution .

Alternative definition

An alternative representation is presented below, which is also often used as a definition. In it, the scattering amplitude can be written as a function of the energy of the incoming state as well as the angle between and , since the S operator and thus also the scattering amplitude are invariant under rotations : ${\displaystyle \theta }$ ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {p}}'}$

${\displaystyle {\begin{aligned}\psi _{\mathrm {out} }({\vec {p}}')&=\langle p'|\psi _{\mathrm {out} }\rangle =\langle p'|S|\psi _{\mathrm {in} }\rangle =\int \mathrm {d} ^{3}{\vec {p}}\,\langle p'|S|p\rangle \langle p|\psi _{\mathrm {in} }\rangle =\int \mathrm {d} ^{3}{\vec {p}}\,\langle p'|S|p\rangle \,\psi _{\mathrm {in} }(p)\\&=\psi _{\mathrm {in} }({\vec {p}}')+{\frac {\mathrm {i} }{2\pi m}}\int \mathrm {d} ^{3}{\vec {p}}\,\delta (E'-E)f(p\to p')\psi _{\mathrm {in} }({\vec {p}})\\&=\psi _{\mathrm {in} }({\vec {p}}')+{\frac {\mathrm {i} }{2\pi m}}f(E',\theta )\int \mathrm {d} ^{3}{\vec {p}}\,\delta (E'-E)\;\psi _{\mathrm {in} }({\vec {p}})\end{aligned}}}$

If a plane wave parallel to the z-axis is assumed for the incoming wave , this results in: ${\displaystyle \psi _{\mathrm {in} }}$

${\displaystyle \psi _{\mathrm {out} }(p')=e^{\mathrm {i} p'z}+f(E',\theta )\;{\frac {e^{\mathrm {i} p'r}}{r}}}$

Cross section

The differential cross section is given by:

${\displaystyle {\frac {d\sigma }{d\Omega }}=|f(\vartheta )|^{2}\;.}$

For total cross section , there is a connection via the optical theorem :

${\displaystyle \sigma _{\mathrm {tot} }=\int _{4\pi }{\frac {d\sigma }{d\Omega }}\cdot d\Omega ={\frac {4\pi }{k}}~\mathrm {Im} \,f(0)}$

with the wave number and the imaginary part of the scattering amplitude for the scattering angle zero. ${\displaystyle k}$ ${\displaystyle \mathrm {Im} \,f(0)}$

Partial wave development

In the partial wave expansion, the scattering amplitude is expressed as a sum over partial waves :

${\displaystyle f(\vartheta )=\sum _{\ell =0}^{\infty }(2\ell +1)\;f_{\ell }(k)\;P_{\ell }(\cos \vartheta )}$

in which

• ${\displaystyle f_{\ell }(k)}$ the partial scattering amplitude
• ${\displaystyle P_{\ell }(\cos \vartheta )}$the Legendre polynomial
• ${\displaystyle \ell }$ is the index for angular momentum.

The partial scattering amplitude can be expressed by the S-matrix element and the scattering phase: ${\displaystyle S_{\ell }=e^{2i\delta _{\ell }}}$ ${\displaystyle \delta _{\ell }}$

${\displaystyle f_{\ell }={\frac {S_{\ell }-1}{2ik}}={\frac {e^{2i\delta _{\ell }}-1}{2ik}}={\frac {e^{i\delta _{\ell }}\sin \delta _{\ell }}{k}}={\frac {1}{k\cot \delta _{\ell }-ik}}\;.}$

It should be noted that the partial scattering amplitude , the S-matrix element and the scattering phase are implicit functions of the scattering energy or the momentum . ${\displaystyle f_{\ell }}$ ${\displaystyle S_{\ell }=e^{2i\delta _{\ell }}}$ ${\displaystyle \delta _{\ell }}$ ${\displaystyle k}$

The total scattering cross-section can thus be expressed as:

${\displaystyle \sigma _{\text{total}}={\frac {4\pi }{k^{2}}}\sum _{l=0}^{\infty }(2l+1)\sin ^{2}\delta _{l}\;.}$

The scatter length can be defined with the help of the partial scatter amplitude: ${\displaystyle a_{\ell }}$

${\displaystyle f_{\ell }(p){\xrightarrow[{p\rightarrow 0}]{}}-a_{\ell }\cdot p^{2\ell }}$

Usually, however, only the scattering length of the s waves is referred to as the scattering length. ${\displaystyle a_{0}}$ ${\displaystyle (\ell =0)}$

literature

• John R. Taylor: Scattering Theory - The Quantum Theory of Nonrelativistic Collisions, 1983.