Scattering theory
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In physics, scattering theory is a framework for studying and understanding scattering events. Two types of scattering can be considered, wave scattering and particle scattering. Wave scattering corresponds to the collision or scattering of a wave with some material object (for instance sunlight scattered by rain drops to form a rainbow) while particle scattering is associated to the behaviour of many particles hitting some obstacle (for instance alpha particles scattered by gold atoms in the historical Rutherford scattering experiment). In quantum mechanics, both phenomena are linked via the wave-particle duality. Scattering theory is the name usually given to describe both type of events in a same mathematical frame. In mathematics, scattering theory is this particular mathematical frame that is, the study of the particular type of differential equations with specific boundary conditions that are used in this context.
In acoustics, the differential equation is the wave equation, and scattering studies how its solutions, the sound waves, scatter from solid objects or propagate through non-uniform media (such as sound waves, in sea water, coming from a submarine). In the case of classical electrodynamics, the differential equation is again the wave equation, and the scattering of light or radio waves is studied. In atomic and molecular physics, the scattered particle may correspond to atoms and molecules, governed by the Schroedinger equation and interacting via the potential energy surfaces or to electrons scattered by an atomic or molecular target via Coulomb interaction. In the latter case, such electron-molecule collisions often induce a strong modification of the molecular potential surfaces and cause vibrational excitation of or dissociative electron attachment to the target. In particle physics, the equations are those of QED, QCD and the Standard Model, the solutions of which correspond to fundamental particles.
The example of quantum chemistry is particularly interesting, as the theory is reasonably complex while still being "visible" and reasonably tractable. Of particular note is that the objects being scattered themselves correspond to the bound state solutions of some differential equation. Thus, for example, the hydrogen atom corresponds to a solution to the Schroedinger equation with an inverse-square law central potential. The scattering of two hydrogen atoms will disturb the state of each atom, resulting in one or both becoming excited, or even ionized. Thus, collisions can be either elastic (the internal quantum states of the particles are not changed) or inelastic (the internal quantum states of the particles are changed). From the experimental viewpoint the observable quantity is the cross section. From the theoretical viewpoint the key quantity is the S matrix.
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.
