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Memoirs of the American Mathematical Society

Mathematical foundations of quantum scattering theory for multiparticle systems

Israel Michael Sigal

Published Jan 1, 1978• 23 citations
Publisher
DOI
10.1090/memo/0209

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The most important physical aspects of quantum mechanics are interactions in bound systems and predictions relevant to particle scatterings. A special accent is placed upon the foundation of quantum scattering theory from the first principles of physics, without resorting to any free parameters. Such a complete theory, which is autonomously and consistently devised on a rigorous mathematical basis, represents a genuine modern candidate for a powerful predictor of real events in experiments. For modern scattering theory, the following two principal questions emerge: the existence and completeness of the Moller wave operators. They directly imply the unitarity of the S-matrix and the correctness of the boundary conditions. Such a general and versatile methodological concept largely surpasses the frames of scattering theory.

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Elements of Quantum Scattering Theory

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Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
The most important physical aspects of quantum mechanics are interactions in bound systems and predictions relevant to particle scatterings. A special accent is placed upon the foundation of quantum scattering theory from the first principles of physics, without resorting to any free parameters. Such a complete theory, which is autonomously and consistently devised on a rigorous mathematical basis, represents a genuine modern candidate for a powerful predictor of real events in experiments. For modern scattering theory, the following two principal questions emerge: the existence and completeness of the Moller wave operators. They directly imply the unitarity of the S-matrix and the correctness of the boundary conditions. Such a general and versatile methodological concept largely surpasses the frames of scattering theory.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.