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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]}
At large distances from the centrally symmetric scattering center, the plane wave is described by the wavefunction^{[2]}

where is the position vector; ; is the incoming plane wave with the wavenumber k along the z axis; is the outgoing spherical wave; θ is the scattering angle (angle between the incident and scattered direction); and 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,

The asymptotic form of the wave function in arbitrary external field takes the form^{[2]}

where is the direction of incidient particles and is the direction of scattered particles.

When conservation of number of particles holds true during scattering, it leads to a unitary condition for the scattering amplitude. In the general case, we have^{[2]}

Optical theorem follows from here by setting

In the centrally symmetric field, the unitary condition becomes

where and are the angles between and and some direction . This condition puts a constraint on the allowed form for , i.e., the real and imaginary part of the scattering amplitude are not independent in this case. For example, if in is known (say, from the measurement of the cross section), then can be determined such that is uniquely determined within the alternative .^{[2]}

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,^{[3]}

- ,

where *f _{ℓ}* is the partial scattering amplitude and

Then the total cross section^{[4]}

- ,

can be expanded as^{[2]}

is the partial cross section. The total cross section is also equal to due to optical theorem.

For , we can write^{[2]}

The scattering length for X-rays is the Thomson scattering length or classical electron radius, r_{0}.

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

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

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

**^***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- ^
^{a}^{b}^{c}^{d}^{e}^{f}Landau, L. D., & Lifshitz, E. M. (2013). Quantum mechanics: non-relativistic theory (Vol. 3). Elsevier. **^**Michael Fowler/ 1/17/08 Plane Waves and Partial Waves**^**Schiff, Leonard I. (1968).*Quantum Mechanics*. New York: McGraw Hill. pp. 119–120.