Scattering amplitude

In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.[1]

The latter is described by the wavefunction

${\displaystyle \psi (\mathbf {r} )=e^{ikz}+f(\theta ){\frac {e^{ikr}}{r}}\;,}$

where ${\displaystyle \mathbf {r} \equiv (x,y,z)}$ is the position vector; ${\displaystyle r\equiv |\mathbf {r} |}$; ${\displaystyle e^{ikz}}$ is the incoming plane wave with the wavenumber k along the z axis; ${\displaystyle e^{ikr}/r}$ is the outgoing spherical wave; θ is the scattering angle; and ${\displaystyle f(\theta )}$ is the scattering amplitude. The dimension of the scattering amplitude is length.

The scattering amplitude is a probability amplitude; the differential cross-section as a function of scattering angle is given as its modulus squared,

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

Partial wave expansion

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,[2]

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

where f_ℓ is the partial scattering amplitude and P_ℓ are the Legendre polynomials.

The partial amplitude can be expressed via the partial wave S-matrix element S_ℓ (${\displaystyle =e^{2i\delta _{\ell }}}$) and the scattering phase shift δ_ℓ as

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

Then the differential cross section is given by[3]

${\displaystyle {\frac {d\sigma }{d\Omega }}=|f(\theta )|^{2}={\frac {1}{k^{2}}}\left|\sum _{\ell =0}^{\infty }(2\ell +1)e^{i\delta _{\ell }}\sin \delta _{\ell }P_{\ell }(\cos \theta )\right|^{2}}$,

and the total elastic cross section becomes

${\displaystyle \sigma =2\pi \int _{0}^{\pi }{\frac {d\sigma }{d\Omega }}\sin \theta \,d\theta ={\frac {4\pi }{k}}\operatorname {Im} f(0)}$,

where Im f(0) is the imaginary part of f(0).

X-rays

The scattering length for X-rays is the Thomson scattering length or classical electron radius, r₀.

Neutrons

The nuclear neutron scattering process involves the coherent neutron scattering length, often described by b.

Quantum mechanical formalism

A quantum mechanical approach is given by the S matrix formalism.

Measurement

The scattering amplitude can be determined by the scattering length in the low-energy regime.

See also

• Veneziano amplitude
• Plane wave expansion

References

1. Quantum Mechanics: Concepts and Applications Archived 2010-11-10 at the Wayback Machine By Nouredine Zettili, 2nd edition, page 623. ISBN 978-0-470-02679-3 Paperback 688 pages January 2009
2. Michael Fowler/ 1/17/08 Plane Waves and Partial Waves
3. Schiff, Leonard I. (1968). Quantum Mechanics. New York: McGraw Hill. pp. 119–120.