Quantum Mechanics for Mathematicians

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American Mathematical Soc., 1 janv. 2008 - 387 pages
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This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. It addition to traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin, and it introduces the reader to functional methods in quantummechanics. This includes the Feynman path integral approach to quantum mechanics, integration in functional spaces, the relation between Feynman and Wiener integrals, Gaussian integration and regularized determinants of differential operators, fermion systems and integration over anticommuting (Grassmann)variables, supersymmetry and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer formula for the index of the Dirac operator. Prior to this book, mathematicians could find these topics only in physics textbooks and in specialized literature.This book is written in a concise style with careful attention to precise mathematics formulation of methods and results. Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory.Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis.
  

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Table des matières

Classical Mechanics
3
Basic Principles of Quantum Mechanics
63
4 Notes and references
146
2 Onedimensional Schrodinger equation
155
3 Angular momentum and SO3
180
5 Hydrogen atom and SO4
193
6 Semiclassical asymptotics I
205
Spin and Identical Particles
217
Path Integral Formulation of Quantum Mechanics
239
Integration in Functional Spaces
289
3 Gaussian Wiener integrals
301
Fermion Systems
307
Supersymmetry
343
Bibliography
373
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