Symplectic Geometry and Quantum Mechanics

  • Maurice de Gosson
Part of the Operator Theory: Advances and Applications book series (OT, volume 166)

Table of contents

  1. Front Matter
    Pages i-xx
  2. Symplectic Geometry

  3. Heisenberg Group, Weyl Calculus, and Metaplectic Representation

    1. Front Matter
      Pages 121-121
    2. Pages 195-233
  4. Quantum Mechanics in Phase Space

    1. Front Matter
      Pages 235-235
    2. Pages 271-302
  5. Back Matter
    Pages 333-368

About this book

Introduction

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.

Keywords

Heisenberg group Lie group Weyl calculus phase space quantum mechanics symplectic geometry

Authors and affiliations

  • Maurice de Gosson
    • 1
  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany

Bibliographic information

  • DOI https://doi.org/10.1007/3-7643-7575-2
  • Copyright Information Birkhäuser Verlag 2006
  • Publisher Name Birkhäuser Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-7643-7574-4
  • Online ISBN 978-3-7643-7575-1