Symplectic Geometry and Quantum Mechanics

• Κύριος συγγραφέας: Gosson, Maurice de (Συγγραφέας)
• Συγγραφή απο Οργανισμό/Αρχή: SpringerLink (Online service)
• Μορφή: Ηλεκτρονική πηγή Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: Basel : Birkhäuser Basel, 2006.
• Σειρά: Operator Theory: Advances and Applications ; 166
• Θέματα: Mathematics.; Topological groups.; Lie groups.; Integral transforms.; Operational calculus.; Operator theory.; Partial differential equations.; Physics.; Quantum physics.; Topological Groups, Lie Groups.; Mathematical Methods in Physics.; Partial Differential Equations.; Integral Transforms, Operational Calculus.; Operator Theory.; Quantum Physics.

 This book is devoted to a rather complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a rigorous presentation of the basics of symplectic geometry and of its multiply-oriented extension. Further chapters concentrate on Lagrangian manifolds, Weyl operators and the Wigner-Moyal transform as well as on metaplectic groups and Maslov indices. Thus the keys for the mathematical description of quantum mechanics in phase space are discussed. They are followed by a rigorous geometrical treatment of the uncertainty principle. Then Hilbert-Schmidt and trace-class operators are exposed in order to treat density matrices. In the last chapter the Weyl pseudo-differential calculus is extended to phase space in order to derive a Schrödinger equation in phase space whose solutions are related to those of the usual Schrödinger equation by a wave-packet transform. The text is essentially self-contained and can be used as basis for graduate courses. Many topics are of genuine interest for pure mathematicians working in geometry and topology. 

 Symplectic Geometry -- Symplectic Spaces and Lagrangian Planes -- The Symplectic Group -- Multi-Oriented Symplectic Geometry -- Intersection Indices in Lag(n) and Sp(n) -- Heisenberg Group, Weyl Calculus, and Metaplectic Representation -- Lagrangian Manifolds and Quantization -- Heisenberg Group and Weyl Operators -- The Metaplectic Group -- Quantum Mechanics in Phase Space -- The Uncertainty Principle -- The Density Operator -- A Phase Space Weyl Calculus.