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Atlantis studies in mathematical physics

Lecture 10: Scattering Theory. Time-Dependent Formalism. Wave Operators

Gianfausto Dell’Antonio

Published Jan 1, 2016• 0 citations
Publisher

Abstract

Scattering theory, in Quantum as in Classical Mechanics, describes those effects of the interaction of a system of N particles which can be measured when the components of the system have become spatially separated so that they can be uniquely identified and the mutual interactions have become negligible. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.2991/978-94-6239-115-4_10

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Scattering theory, in Quantum as in Classical Mechanics, describes those effects of the interaction of a system of N particles which can be measured when the components of the system have become spatially separated so that they can be uniquely identified and the mutual interactions have become negligible. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Aims and problems of quantum scattering theory Scattering theory is the branch of physics concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world. In quantum mechanics, a typical scattering process involves a beam of particles prepared with a given energy and with a more or less defined linear momentum. One then studies either the collision with a similar wave packet, or the collision of the given wave packet against a fixed target. In particular, for a two-body elastic scattering problem with conservative forces, we work in the centre of mass frame, hence reducing the original problem to the analysis of a particle in an external field of forces. Indeed, in Section 7.2 we already used such a technique in the investigation of the hydrogen atom. The crucial difference with respect to Section 7.2 is that, in scattering problems, the continuous spectrum and its perturbations are studied, whereas the Balmer formula has to do with bound states. In the following section we outline the basic aspects of what is called time-dependent scattering. Time-dependent scattering The physical situation we would like to study is as follows: a beam of particles is prepared in a state that is approximately free and allowed to evolve towards the scatterer. After the collision we detect the scattered beam far away from the interaction region.
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.
In this lecture we give more details of an alternative approach to quantum scattering theory, initiated by V. Enss, This approach is based on a geometric analysis of the behavior for t→±∞t \rightarrow \pm \infty t→±∞ of the solutions of Schroedinger’s equation for initial data in the subspace of absolute continuity for the hamiltonian H . These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.