The Transition to Chaos [E-Book] : Conservative Classical Systems and Quantum Manifestations / by Linda E. Reichl.
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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Full text |
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Personal Name(s): | Reichl, Linda E., author |
Edition: |
Second Edition. |
Imprint: |
New York, NY :
Springer,
2004
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Physical Description: |
XVIII, 675 p. 154 illus. online resource. |
Note: |
englisch |
ISBN: |
9781475743500 |
DOI: |
10.1007/978-1-4757-4350-0 |
Series Title: |
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Institute for Nonlinear Science
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Subject (LOC): |
- 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)
- 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
- 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.