The Transition to Chaos: Conservative Classical and Quantum Systems

Acknowledgements
Contents
1 Overview
	1.1 Introduction
	1.2 Historical Overview
	1.3 Plan of the Book
	References
2 Fundamental Concepts
	2.1 Introduction
	2.2 Conventional Perturbation Theory
	2.3 Integrable Systems
		2.3.1 Noether's Theorem
		2.3.2 Hidden Symmetries
	2.4 Poincaré Surface of Section
		2.4.1 Henon-Heiles System
		2.4.2 The HOCl Molecule and Birkhoff Coordinates
		2.4.3 Lattice Surfaces of Section
	2.5 Nonlinear Resonance and Chaos
		2.5.1 Single-Resonance Hamiltonians
			2.5.1.1 (2,2) Resonance
			2.5.1.2 (2,3) Resonance
		2.5.2 Two-Resonance Hamiltonian
	2.6 KAM Theory
		2.6.1 The KAM theorem (for N=2)
	2.7 The Definition of Chaos
		2.7.1 Lyapounov Exponent
		2.7.2 KS Metric Entropy and K-Flows
	2.8 Conclusions
	References
3 Area-Preserving Maps
	3.1 Introduction
	3.2 Twist Maps
		3.2.1 Derivation of a Twist Map from a Torus
		3.2.2 Generating Functions
		3.2.3 Birkhoff Fixed Point Theorem
		3.2.4 The Tangent Map
		3.2.5 Homoclinic and Heteroclinic Points
	3.3 The Standard Map
	3.4 Scaling Behavior of Noble KAM Tori
	3.5 Renormalization in Twist Maps
		3.5.1 Integrable Twist Map
		3.5.2 Nonintegrable Twist Map
		3.5.3 The Universal Map
	3.6 Bifurcation of M-Cycles
		3.6.1 Some General Properties
		3.6.2 The Quadratic Map
		3.6.3 Scaling in the Quadratic de Vogelaere Map
			3.6.3.1 Scaling Behavior of the Bifurcation Sequence
	3.7 Cantori
	3.8 Renormalization Map
		3.8.1 Expression for the Renormalization Map
		3.8.2 Fixed Points of the Renormalization Map
	3.9 Conclusions
	References
4 Chaotic Scattering
	4.1 Introduction
	4.2 The Complete Ternary Horseshoe
		4.2.1 Double Gaussian Potential Energy Peaks
		4.2.2 Delta-Kicked System
			4.2.2.1 Delay Time
			4.2.2.2 Symbolic Dynamics
			4.2.2.3 Incomplete Ternary Horseshoes
	4.3 Scattering Chaos in a Magnetic Dipole
	4.4 Model of Chlorine Ion in a Radiation Field
		4.4.1 Scattering Map
		4.4.2 Delay Time
		4.4.3 Symbolic Dynamics
	4.5 Chaos in the HOCl Molecular System
		4.5.1 Homoclinic Tangles
		4.5.2 Scattering Dynamics
	4.6 Conclusions
	References
5 Arnol'd Diffusion
	5.1 Introduction
	5.2 The Arnol'd Web
	5.3 Arnol'd Diffusion and Nekhoroshev Time
	5.4 Graphical Evolution of the Arnol'd Web
	5.5 Arnol'd Diffusion in an Optical Lattice
		5.5.1 Arnol'd Web
		5.5.2 Arnol'd Diffusion
	5.6 Stability of the Solar System
	5.7 Colliding Beam Synchrotron Particle Accelerator
	5.8 Conclusions
	References
6 Quantum Dynamics and Random Matrix Theory
	6.1 Introduction
	6.2 Invariant Measure for the GOE
	6.3 Probability Density that Extremizes Information
		6.3.1 Polar Form of Probability Density
		6.3.2 Cluster Expansion of the Probability Density
	6.4 Eigenvalue Statistics: Gaussian Orthogonal Ensemble
		6.4.1 Eigenvalue Number Density
		6.4.2 Eigenvalue Two-Body Correlations: 3-Statistic
		6.4.3 Eigenvalue Nearest Neighbor Spacing Distribution (GOE)
	6.5 Eigenvector Statistics: Gaussian Orthogonal Ensemble
	6.6 Thermalization of Quantum Systems
	6.7 Conclusions
	References
7 Bounded Quantum Systems
	7.1 Introduction
	7.2 Quantum Integrability
	7.3 Symmetries and Degeneracy
	7.4 The Quantized Baker's Map
	7.5 Quantum Billiards
		7.5.1 The Stadium Billiard
		7.5.2 The Sinai Billiard
		7.5.3 The Ripple Billiard
	7.6 Peres Test for Quantum Integrability
		7.6.1 Theory
		7.6.2 The D-Billiard
	7.7 Quantum Spin Models
		7.7.1 The XY Models with Anisotropy
		7.7.2 The XY Model with an Applied Magnetic Field
	7.8 Coupled Morse Oscillators
	7.9 Signatures of Chaos in a Soft Sinai Lattice
	7.10 Conclusions
	References
8 Manifestations of Chaos in Quantum Scattering Processes
	8.1 Introduction
	8.2 Random Matrix Theory and Nuclear Scattering Processes
	8.3 Scattering Theory
		8.3.1 Hamiltonian
		8.3.2 Energy Eigenstates
		8.3.3 The Reaction Matrix
		8.3.4 The Scattering Matrix
	8.4 Wigner–Smith and Partial Delay Times
		8.4.1 Delay Time of a Wave Packet
		8.4.2 Delay Times for Multichannel Scattering
		8.4.3 Delay Times and Complex Poles
	8.5 Scattering in the Ripple Waveguide
		8.5.1 Scattering Resonances in a Ripple Waveguide
		8.5.2 Wigner–Smith Delay Times for a Chaotic Scattering System
			8.5.2.1 Basis States for the Reaction Region
			8.5.2.2 Signatures of Chaos in Waveguide Scattering
	8.6 COE and GOE
		8.6.1 Circular Orthogonal Ensembles
		8.6.2 Lorentzian Orthogonal Ensembles
		8.6.3 The Relation Between COE and OE
			8.6.3.1 Equivalence of COE and OE When N→∞
		8.6.4 When Does a GOE Hamiltonian Yield a COE S-Matrix?
	8.7 Experimental Observation of RMT Predictions
		8.7.1 Experimental Nuclear Spectral Statistics
		8.7.2 Experimental Molecular Spectral Statistics
	8.8 Conclusions
	References
9 Semiclassical Theory: Path Integrals
	9.1 Introduction
	9.2 Green's Function and Density of States
	9.3 The Path Integral
	9.4 Semiclassical Approximation
		9.4.1 Method of Stationary Phase
		9.4.2 The Semiclassical Green's Function
		9.4.3 Conjugate Points
	9.5 Energy Green's Function
		9.5.1 General Expression
		9.5.2 Density of States
	9.6 Gutzwiller Trace Formula
		9.6.1 Stationary Phase Approximation
			9.6.1.1 Monodromy Matrix
			9.6.1.2 Response Function: Two Degrees of Freedom
	9.7 Anisotropic Kepler System
	9.8 Diamagnetic Hydrogen
		9.8.1 The Model
		9.8.2 Absorption Cross Section
		9.8.3 Experiment
		9.8.4 Semiclassical Cross Section
	9.9 Conclusions
	References
10 Time-Periodic Quantum Systems
	10.1 Introduction
	10.2 Floquet Theory
		10.2.1 Floquet Matrix
		10.2.2 Floquet Hamiltonian
	10.3 Quantum Nonlinear Resonances
		10.3.1 Two Primary Resonance Model
		10.3.2 Floquet Eigenstates
		10.3.3 Quantum Resonance Overlap
		10.3.4 Floquet Eigenvalue Nearest Neighbor Spacing
		10.3.5 Quantum Renormalization
	10.4 Dynamics of a Driven Bounded Particle
		10.4.1 Driven Particle in Infinite Square Well
		10.4.2 Avoided Crossings and High Harmonic Radiation
	10.5 Dynamical Tunneling in Atom Optics Experiments
		10.5.1 Hamiltonian for Atomic Center-of-Mass
		10.5.2 Average Momentum of Cesium Atoms
		10.5.3 Floquet Analysis of Tunneling Oscillations
	10.6 Quantum Delta-Kicked Rotor
		10.6.1 The Schrödinger Equation for the Delta-Kicked Rotor
		10.6.2 KAM-Like Behavior of the Quantum Delta-Kicked Rotor
		10.6.3 The Floquet Map
		10.6.4 Dynamic Anderson Localization
	10.7 Microwave-Driven Hydrogen
		10.7.1 Experimental Apparatus
		10.7.2 One-Dimensional Approximation
			10.7.2.1 Nonlinear Resonances
	10.8 Arnol'd Diffusion in Quantum Systems
		10.8.1 Arnol'd Diffusion in the Driven Optical Lattice
			10.8.1.1 Floquet States
			10.8.1.2 Behavior of Quantum States
	10.9 Quantum Control
		10.9.1 The Model (Classical Dynamics)
		10.9.2 The Model (Quantum Dynamics)
		10.9.3 Floquet States
		10.9.4 STIRAP
		10.9.5 Avoided Crossings
	10.10 Conclusions
	References
A Classical Mechanics Concepts
	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
	References
B Simple Models
	B.1 The Pendulum
		B.1.1 Libration—Trapped Orbits (E0g)
	B.2 Double-Well Potential
		B.2.1 Below the Barrier—(Eo<0)
		B.2.2 Above the Barrier—(Eo>0)
	B.3 Infinite Square-Well Potential
	B.4 One-Dimensional Hydrogen
		B.4.1 Zero Stark Field
		B.4.2 Nonzero Stark Field
	References
C Symmetries and the Hamiltonian Matrix
	C.1 Space-Time Symmetries
		C.1.1 Continuous Symmetries
			C.1.1.1 Time Translation
			C.1.1.2 Space Translation
			C.1.1.3 Rotation
		C.1.2 Discrete Symmetries
			C.1.2.1 Parity
			C.1.2.2 Time Reversal
	C.2 Structure of the Hamiltonian Matrix
		C.2.1 Space-Time Homogeneity and Isotropy
			C.2.1.1 Time Translation Invariance
			C.2.1.2 Space Translation Invariance
			C.2.1.3 Rotational Invariance
		C.2.2 Time Reversal Invariance
			C.2.2.1 Integer Spin
			C.2.2.2 Half-Integer Spin
	References
D Invariant Measures
	D.1 General Definition of Invariant Measure
		D.1.1 Invariant Metric (Length)
		D.1.2 Invariant Measure (Volume)
	D.2 Hermitian Matrices
		D.2.1 Real Symmetric Matrix
			D.2.1.1 Polar Form of Measure for Real Symmetric Matrix
		D.2.2 Complex Hermitian Matrices
			D.2.2.1 Polar Form of Measure for Complex Hermitian Matrix
		D.2.3 Quaternion Real Matrices
			D.2.3.1 Polar Form of Invariant Measure for NN Quaternion Real Matrix
		D.2.4 General Formula for Invariant Measure of Hermitian Matrices
	D.3 Unitary Matrices
		D.3.1 Symmetric Unitary Matrices
		D.3.2 General Unitary Matrices
		D.3.3 Symplectic Unitary Matrices
		D.3.4 General Formula for Invariant Measure of Unitary Matrices
		D.3.5 Orthogonal Matrices
	D.4 Volume of Invariant Measure for Unitary Matrices
	References
E Quaternions
	E.1 General Properties of Quaternions
	E.2 Quaternions in the GOE
	References
F Gaussian Ensembles
	F.1 Gaussian Orthogonal Ensemble (GOE)
		F.1.1 Probability Density and Quaternion Matrices for GOE
		F.1.2 Cluster Functions for GOE
	F.2 Gaussian Unitary Ensemble (GUE)
		F.2.1 Complex Hermitian Matrices and Invariant Measure
		F.2.2 Polar Form of Measure for Complex Hermitian Matrix
		F.2.3 Probability Density for GUE
		F.2.4 Cluster Functions for GUE
		F.2.5 Eigenvalue Number Density for GUE
		F.2.6 3-Statistics for GUE
		F.2.7 Eigenvalue Nearest Neighbor Spacing Distribution for GUE (N=2)
	F.3 Gaussian Symplectic Ensemble (GSE)
		F.3.1 Quaternion Real Matrices: GSE
		F.3.2 Polar Form of Invariant Measure for NN Quaternion Real Matrix: GSE
		F.3.3 Probability Density for GSE
		F.3.4 Cluster Functions for GSE
	References
G Circular Ensembles
	G.1 Vandermonde Determinant
	G.2 Circular Unitary Ensemble (CUE)
	G.3 Circular Orthogonal Ensemble (COE)
	G.4 Circular Symplectic Ensemble (COE)
	G.5 Cluster Functions for Circular Ensembles
		G.5.1 Circular Unitary Ensemble (CUE)
		G.5.2 Circular Orthogonal Ensemble (COE)
		G.5.3 Circular Symplectic Ensemble (CSE)
	G.6 3-Statistics for Circular Ensembles
		G.6.1 Circular Unitary Ensemble
		G.6.2 Circular Orthogonal Ensemble
		G.6.3 Circular Symplectic Ensemble
	References
H Maxwell's Equations for 2-d Billiards
	H.1 Transverse Magnetic Modes
	H.2 Transverse Electric Modes
	Reference
I Moyal Bracket and Wigner Function
	I.1 The Wigner Function
	I.2 Ordering of Operators
	I.3 Moyal Bracket
	References
J Gaussian, S.I., and Atomic Units
	J.1 Maxwell's Equations in Gaussian Units
		J.1.1 Physical Quantities Expressed in Gaussian Units
		J.1.2 Values of Fundamental Constants Expressed in Gaussian Units
		J.1.3 The Conversion Between Gaussian and SI Units
		J.1.4 The Conversion Between Gaussian and Atomic Units
	J.2 Maxwell's Equations in S.I. Units
		J.2.1 Conversion Between S.I. Units and Atomic Units
	References
K Hydrogen in a Constant Electric Field
	K.1 The Schrödinger Equation
		K.1.1 Equation for Relative Motion
		K.1.2 Solution for λ0=0
	K.2 One-Dimensional Hydrogen
	References
Author Index
Subject Index