The Transition to Chaos : Conservative Classical Systems and Quantum Manifestations

This book provides a thorough and comprehensive discussion of classical and quantum chaos theory for bounded systems and for scattering processes. Specific discussions include: • Noether’s theorem, integrability, KAM theory, and a definition of chaotic behavior. • Area-preserving maps, quantum billi...

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Main Author: Reichl, Linda
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: New York, NY Springer New York 2004, 2004
Edition:2nd ed. 2004
Series:Institute for Nonlinear Science
Subjects:
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 1 Overview
  • 2 Fundamental Concepts
  • 3 Area-Preserving Maps
  • 4 Global Properties
  • 5 Random Matrix Theory
  • 6 Bounded Quantum Systems
  • 7 Manifestations of Chaos in Quantum Scattering Processes
  • 8 Semiclassical Theory—Path Integrals
  • 9 Time-Periodic Systems
  • 10 Stochastic Manifestations of Chaos
  • A Classical Mechanics
  • A.1 Newton’s Equations
  • A.2 Lagrange’s Equations
  • A.3 Hamilton’s Equations
  • A.4 The Poisson Bracket
  • A.5 Phase Space Volume Conservation
  • A.6 Action-Angle Coordinates
  • A.7 Hamilton’s Principal Function
  • A.8 References
  • B Simple Models
  • B.1 The Pendulum
  • B.2 Double-Well Potential
  • B.3 Infinite Square-Well Potential
  • B.4 One-Dimensional Hydrogen
  • B.4.1 Zero Stark Field
  • B.4.2 Nonzero Stark Field
  • C Renormalization Integral
  • C.3 References
  • D Moyal Bracket
  • D.1 The Wigner Function
  • D.2 Ordering of Operators
  • D.3 Moyal Bracket
  • D.4 References
  • E Symmetries and the Hamiltonian Matrix
  • E.1 Space-Time Symmetries
  • E.1.1 Continuous Symmetries
  • E.1.2 Discrete Symmetries
  • E.2 Structure of the Hamiltonian Matrix
  • E.2.1 Space-Time Homogeneity and Isotropy
  • E.2.2 Time Reversal Invariance
  • E.3 References
  • F Invariant Measures
  • F.1 General Definition of Invariant Measure
  • F.1.1 Invariant Metric (Length)
  • F.1.2 Invariant Measure (Volume)
  • F.2 Hermitian Matrices
  • F.2.1 Real Symmetric Matrix
  • F.2.2 Complex Hermitian Matrices
  • F.2.3 Quaternion Real Matrices
  • F.2.4 General Formula for Invariant Measure of Hermitian Matrices
  • F.3 Unitary Matrices
  • F.3.1 Symmetric Unitary Matrices
  • F.3.2 General Unitary Matrices
  • F.3.3 Symplectic Unitary Matrices
  • F.3.4 General Formula for Invariant Measure of Unitary Matrices
  • F.3.5 Orthogonal Matrices
  • F.4 References
  • G Quaternions
  • G.1 References
  • H Gaussian Ensembles
  • H.1 Vandermonde Determinant
  • H.2 Gaussian Unitary Ensemble (GUE)
  • H.3 Gaussian Orthogonal Ensemble (GOE)
  • M.6 Expectation Value of the Generating Function (Part 2)
  • M.7 Average Response Function Density
  • M.7.1 Saddle Points for the Integration over a
  • M.7.2 Saddle Points for the Integration over ?
  • M.7.4 Wigner Semicircle Law
  • M.8 References
  • N Average S-Matrix (GOE)
  • N.1 S-Matrix Generating Function
  • N.2 Average S-Matrix Generating Function
  • N.3 Saddle Point Approximation
  • N.4 Integration over Grassmann Variables
  • N.5 References
  • O Maxwell’s Equations for 2-d Billiards
  • O.1 References
  • P Lloyd’s Model
  • P.1 Localization Length
  • P.2 References
  • Q Hydrogen in a Constant Electric Field
  • Q.1 The Schrödinger Equation
  • Q.1.1 Equation for Relative Motion
  • Q.2 One-Dimensional Hydrogen
  • Q.3 References
  • Author Index
  • H.4 Gaussian Symplectic Ensemble (GSE)
  • H.5 References
  • I Circular Ensembles
  • 1.1 Vandermonde Determinant
  • 1.2 Circular Unitary Ensemble (CUE)
  • 1.3 Circular Orthogonal Ensemble (COE)
  • 1.4 Circular Symplectic Ensemble (COE)
  • 1.5 References
  • J Volume of Invariant Measure for Unitary Matrices
  • J.1 References
  • K Lorentzian Ensembles
  • K.1 Normalization of AOE
  • K.2 Relation Between COE and AOE
  • K.4 Invariance of AOE under Inversion
  • K.4.1 Robustness of AOE under Integration
  • K.5 References
  • L Grassmann Variables and Supermatrices
  • L.1 Grassmann Variables
  • L.2 Supermatrices
  • L.2.1 Transpose of a Supermatrix
  • L.2.2 Hermitian Adjoint of a Supermatrix
  • L.2.3 Supertrace of a Supermatrix
  • L.2.4 Determinant of a Supermatrix
  • L.3 References
  • M Average Response Function (GOE)
  • M.3 Gaussian Integral for Response Function Generating Function
  • M.4 Expectation Value of the Generating Function (Part 1)
  • M.5 The Hubbard-Stratonovitch Transformation -