quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...

, the scattering amplitude is the probability amplitude 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, th ...

in a stationary-state scattering process.''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 mad ...

\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 In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...

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 Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

of the scattering amplitude is

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

. The scattering amplitude is a probability amplitude; the differential

cross-section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Ab ...

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 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 ...

or classical electron radius, ₀.

Neutrons

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

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.

References

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

X-rays

Neutrons

Quantum mechanical formalism

Measurement

See also

References