Scattering amplitude

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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 .


The scattering amplitude is defined by the S operator :

Are there

  • the initial state and the final state with defined momentum , i.e. eigenstates of the momentum operator ,
  • the impulses of the states
  • the energy of the states
  • the mass (physics) of the states and
  • 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 :

If a plane wave parallel to the z-axis is assumed for the incoming wave , this results in:

Cross section

The differential cross section is given by:

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

with the wave number and the imaginary part of the scattering amplitude for the scattering angle zero.

Partial wave development

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

in which

  • the partial scattering amplitude
  • the Legendre polynomial
  • is the index for angular momentum.

The partial scattering amplitude can be expressed by the S-matrix element and the scattering phase:

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 .

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

The scatter length can be defined with the help of the partial scatter amplitude:

Usually, however, only the scattering length of the s waves is referred to as the scattering length.


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