Advances in Partial Differential Equations

Symplectic Geometry and Quantum Mechanics

Authors: de Gosson, Maurice A.

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About this book

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.

Reviews

From the reviews:

"De Gosson’s book is an exhaustive and clear description of almost all the more recent results obtained in connected areas of research like symplectiv geometry, the combinatorial theory of the Maslov index, the theory of the metaplectic group and so on. It fills an important niche in the literature." -Mircea Crâsmareanu, Analele Stiintifice

"This book concerns certain aspects of symplectic geometry and their application to quantum mechanics. … This book seems best suited to someone who already has a solid background in quantum theory and wants to learn more about the symplectic geometric techniques used in quantization. … the book contains useful information about various important topics." (Brian C. Hall, Mathematical Reviews, Issue 2007 e)

“This book covers … symplectic geometry and their applications in quantum mechanics with an emphasis on phase space methods. … The exposition is very detailed and complete proofs are given. … the book takes a particularly fresh point of view on some of the topics and contains a lot of useful information for readers with some background in quantum theory and an interest in the use of symplectic techniques.” (R. Steinbauer, Monatshefte für Mathematik, Vol. 155 (1), September, 2008)


Table of contents (10 chapters)

Buy this book

eBook $159.00
price for USA (gross)
  • ISBN 978-3-7643-7575-1
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $209.00
price for USA
  • ISBN 978-3-7643-7574-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Symplectic Geometry and Quantum Mechanics
Authors
Series Title
Advances in Partial Differential Equations
Series Volume
166
Copyright
2006
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Basel
eBook ISBN
978-3-7643-7575-1
DOI
10.1007/3-7643-7575-2
Hardcover ISBN
978-3-7643-7574-4
Edition Number
1
Number of Pages
XX, 368
Topics