# 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.