Scattering theory



  In mathematics and physics, scattering theory is a framework for studying and understanding the nuclei. More precisely, scattering consists of the study of how solutions of partial differential equations, propagating freely "in the distant past", come together and interact with one another or with a boundary condition, and then propagate away "to the distant future".

The direct scattering problem is the problem determining the distribution of scattered radiation/particle flux basing on the characteristics of the scatterer.

The inverse scattering problem is the problem of determining the characteristics of an object (its shape, internal constitution, etc.) from measurement data of radiation or particles scattered from the object.

Since its early statement for radiolocation, the problem has found vast number of applications, such as echolocation, geophysical survey, medical imaging and quantum field theory, to name just a few.

In theoretical physics

In mathematical physics, scattering theory is a framework for studying and understanding the interaction or molecules, governed by the Schrödinger equation.

Elastic and inelastic scattering

The example of scattering in S matrix.

Topics in physics

According to the optics classification of the Optical Society of America this field consists of the following topics:

  • Long-wave scattering
  • Mie theory
  • Multiple scattering
  • Scattering measurements
  • Brillouin scattering
  • Molecular scattering
  • Particle scattering
  • Raman scattering

The mathematical framework

In mathematics, scattering theory deals with a more abstract formulation of the same set of concepts. For example, if a differential equation is known to have some simple, localized solutions, and the solutions are a function of a single parameter, that parameter can take the conceptual role of time. One then asks what might happen if two such solutions are set up far away from each other, in the "distant past", and are made to move towards each other, interact (under the constraint of the differential equation) and then move apart in the "future". The scattering matrix then pairs solutions in the "distant past" to those in the "distant future".

Solutions to differential equations are often posed on manifolds. Frequently, the means to the solution requires the study of the spectrum of an operator on the manifold. As a result, the solutions often have a spectrum that can be identified with a Hilbert space, and scattering is described by a certain map, the S matrix, on Hilbert spaces. Spaces with a discrete spectrum correspond to bound states in quantum mechanics, while a continuous spectrum is associated with scattering states. The study of inelastic scattering then asks how discrete and continuous spectra are mixed together.

An important, notable development is the inverse scattering transform, central to the solution of many exactly solvable models.

References

  • Lectures of the European school on theoretical methods for electron and positron induced chemistry, Prague, Feb. 2005
 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Scattering_theory". A list of authors is available in Wikipedia.