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] 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.

Unitary condition edit

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]

Partial wave expansion edit

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


where f is the partial scattering amplitude and P are the Legendre polynomials. The partial amplitude can be expressed via the partial wave S-matrix element S ( ) and the scattering phase shift δ as


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]


X-rays edit

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

Neutrons edit

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

Quantum mechanical formalism edit

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

Measurement edit

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

See also edit

References edit

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