Symplectic Geometry and Quantum Mechanics [electronic resource] / by Maurice Gosson.
Tipo de material: TextoSeries Operator Theory: Advances and Applications ; 166 | Operator Theory: Advances and Applications ; 166Editor: Basel : Birkhuser Basel, 2006Descripción: XX, 368 p. online resourceTipo de contenido:- text
- computer
- online resource
- 9783764375751
- SpringerLink (Online service)
- Mathematics
- Topological Groups
- Integral Transforms
- Operator theory
- Differential equations, partial
- Quantum theory
- Mathematical physics
- Mathematics
- Topological Groups, Lie Groups
- Mathematical Methods in Physics
- Partial Differential Equations
- Integral Transforms, Operational Calculus
- Operator Theory
- Quantum Physics
- 512.55 23
- 512.482 23
- QA252.3
- QA387
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.
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 Schrȵdinger equation in phase space whose solutions are related to those of the usual Schrȵdinger 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.
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