Symplectic Geometry and Quantum Mechanics

1. Auflage 2006
ISBN: 978-3-7643-7575-1
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 368 Seiten

This book offers a complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Contents;6
2;Preface;12
3;Notation;16
4;I Symplectic Geometry;20
4.1;1 Symplectic Spaces and Lagrangian Planes;21
4.1.1;1.1 Symplectic Vector Spaces;21
4.1.2;1.2 Skew-Orthogonality;29
4.1.3;1.3 The Lagrangian Grassmannian;33
4.1.4;1.4 The Signature of a Triple of Lagrangian Planes;37
4.2;2 The Symplectic Group;45
4.2.1;2.1 The Standard Symplectic Group;45
4.2.2;2.2 Factorization Results in;56
4.2.3;2.3 Hamiltonian Mechanics;68
4.3;3 Multi-Oriented Symplectic Geometry;82
4.3.1;3.1 Souriau Mapping and Maslov Index;83
4.3.2;3.2 The Arnol’d–Leray–Maslov Index;91
4.3.3;3.3;101
4.3.4;Symplectic Geometry;101
4.4;4 Intersection Indices in Lag(n) and Sp(n);112
4.4.1;4.1 Lagrangian Paths;112
4.4.2;4.2 Symplectic Intersection Indices;117
4.4.3;4.3 The Conley–Zehnder Index;121
5;II Heisenberg Group, Weyl Calculus, and Metaplectic Representation;137
5.1;5 Lagrangian Manifolds and Quantization;138
5.1.1;5.1 Lagrangian Manifolds and Phase;138
5.1.2;5.2 Hamiltonian Motions and Phase;145
5.1.3;5.3 Integrable Systems and Lagrangian Tori;154
5.1.4;5.4 Quantization of Lagrangian Manifolds;160
5.1.5;5.5 Heisenberg–Weyl and Grossmann–Royer Operators;167
5.2;6 Heisenberg Group and Weyl Operators;173
5.2.1;6.1 Heisenberg Group and Schr¨ odinger Representation;174
5.2.2;6.2 Weyl Operators;180
5.2.3;6.3 Continuity and Composition;188
5.2.4;6.4 The Wigner–Moyal Transform;199
5.3;7 The Metaplectic Group;208
5.3.1;7.1 De.nition and Properties of;209
5.3.2;7.2 The Metaplectic Algebra;221
5.3.3;7.3 Maslov Indices on;227
5.3.4;7.4 The Weyl Symbol of a Metaplectic Operator;235
6;III Quantum Mechanics in Phase Space;247
6.1;8 The Uncertainty Principle;248
6.1.1;8.1 States and Observables;249
6.1.2;8.2 The Quantum Mechanical Covariance Matrix;250
6.1.3;8.3 Symplectic Spectrum and Williamson’s Theorem;255
6.1.4;8.4 Wigner Ellipsoids;264
6.1.5;8.5 Gaussian States;273
6.2;9 The Density Operator;281
6.2.1;9.1 Trace-Class and Hilbert–Schmidt Operators;282
6.2.2;9.2 Integral Operators;292
6.2.3;9.3 The Density Operator of a Quantum State;301
6.3;10 A Phase Space Weyl Calculus;313
6.3.1;10.1 Introduction and Discussion;314
6.3.2;10.2 The Wigner Wave-Packet Transform;320
6.3.3;10.3 Phase-Space Weyl Operators;327
6.3.4;10.4 Schrödinger Equation in Phase Space;334
6.3.5;10.5 Conclusion;341
7;A Classical Lie Groups;343
7.1;A.1 General Properties;343
7.2;A.2 The Baker–Campbell–Hausdor. Formula;345
7.3;A.3 One-parameter Subgroups of;345
8;B Covering Spaces and Groups;348
9;C Pseudo-Di.erential Operators;350
9.1;C.1 The Classes;351
9.2;C.2 Composition and Adjoint;351
10;D Basics of Probability Theory;353
10.1;D.1 Elementary Concepts;353
10.2;D.2 Gaussian Densities;355
11;Solutions to Selected Exercises;357
12;Bibliography;363
13;Index;372