In
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, q ...
, 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 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, t ...
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 ma ...
:
where is the position vector; ; 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; is the outgoing spherical wave; is the scattering angle; and 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
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 Inte ...
.
The scattering amplitude is a 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 ...
; 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,
:
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 in the low-energy regime.
See also
* Veneziano amplitude
In theoretical physics, the Veneziano amplitude refers to the discovery made in 1968 by Italian theoretical physicist Gabriele Veneziano that the Euler beta function, when interpreted as a scattering amplitude, has many of the features needed to ...
* Plane wave expansion
References
Neutron
X-rays
Electron
Scattering
Diffraction
{{quantum-stub