Methods of Modern Mathematical Physics V3 (Scattering Theory) – Michael Reed, Barry Simon – 1st Edition

Description

Scattering theory is the study of an interacting on a scale of time and/or distance which is large compared to the scale of the interaction itself. As such, it is the most effective means, sometimes the only means, to study microscopic nature. To understand the importance of scattering theory, consider the variety of ways in which it arises.

First, there are various phenomena in nature (like the blue of the sky) which are the result of scattering. In order to understand the phenomenon (and to identify it as the result of scattering) one must understand the underlying and its scattering theory. Second, one often wants to use the scattering of or whose dynamics on knows to determine the structure and position of small or inaccessible objects.

For example, in x-ray crystallography (which led to the discovery of DNA), tomography, and the detection of underwater objects by sonar, the underlying dynamics is well understood. What one would like to construct are correspondences that link, via the dynamics, the position, shape, and internal structure of the object to the scattering data.

Ideally, the correspondence should be an explicit formula which allows one to reconstruct, at least approximately, the object from the scattering data. The main test of any proposed particle dynamics is whether one can construct for the dynamics a scattering theory that predicts the observed experimental data.

Scattering theory was not always so central the physics. Even thought the Coulomb cross section could have been computed by Newton, had he bothered to ask the right question, its calculation is generally attributed to Rutherford more than two hundred years later. Of course, Rutherford’s calculation was in connection with the first experiment in .

Table of Contents

Scattering theory

An overview of scattering phenomena
Classical particle scattering
The basic principles of scattering in Hilbert space


Quantum scattering I: Two-body case
Quantum scattering II: N-body case
Quantum scattering III: Eigenfunction expansions
Quantum scattering IV: dispersion relations
Quantum scattering V: central potentials
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