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Symplectic Geometry and Quantum Mechanics

By: Gosson, Maurice de [author.].
Material type: materialTypeLabelBookSeries: Operator Theory: Advances and Applications ; 1660.Publisher: Basel : Birkh�user Basel, 2006. Description: XX, 368 p. online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9783764375751.DDC classification: 512.55 | 512.482
Contents:
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.
Summary: 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 Schr�dinger equation in phase space whose solutions are related to those of the usual Schr�dinger 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. 0
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PK Kelkar Library, IIT Kanpur
Available EBKS0007970
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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 Schr�dinger equation in phase space whose solutions are related to those of the usual Schr�dinger 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. 0

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