In quantum physics, the scattering amplitude is the

probability amplitude In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density. Probability amplitudes provide a relationship between the qu ...

of the outgoing

spherical wave The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seis ...

relative to the incoming

plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, ...

in a stationary-state

scattering process Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...

.''Quantum Mechanics: Concepts and Applications''
By Nouredine Zettili, 2nd edition, page 623. Paperback 688 pages January 2009 The plane wave is described by the

wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...

\psi(\mathbf) = e^ + f(\theta)\frac \;,

where

\mathbf\equiv(x,y,z)

is the position vector;

r\equiv, \mathbf,

;

e^

is the incoming plane wave with the wavenumber along the axis;

e^/r

is the outgoing spherical wave; is the scattering angle; and

f(\theta)

is the scattering amplitude. The

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

of the scattering amplitude is length. The scattering amplitude is a

; the differential cross-section as a function of scattering angle is given as its

modulus squared In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square ...

, :

\frac = , f(\theta), ^2 \;.

X-rays

The scattering length for X-rays is the Thomson scattering length or

classical electron radius The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic self-interaction energ ...

, ₀.

Neutrons

The nuclear

neutron scattering Neutron scattering, the irregular dispersal of free neutrons by matter, can refer to either the naturally occurring physical process itself or to the man-made experimental techniques that use the natural process for investigating materials. Th ...

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

Quantum mechanical formalism

A quantum mechanical approach is given by the

S matrix In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More forma ...

formalism.

Measurement

The scattering amplitude can be determined by the

scattering length The scattering length in quantum mechanics describes low-energy scattering. For potentials that decay faster than 1/r^3 as r\to \infty, it is defined as the following low-energy limit (mathematics), limit: : \lim_ k\cot\delta(k) =- \frac\;, wher ...

in the low-energy regime.

References

Neutron X-rays Electron Scattering Diffraction {{quantum-stub

X-rays

Neutrons

Quantum mechanical formalism

Measurement

See also

References