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Table of Contents
Intro; Preface; Contents; 1 Introduction; References; 2 Time-Dependent SchrÃœdinger Equation and Gaussian Wave Packets; 2.1 Dynamics of Mean Values and Uncertainties; 2.2 Direct Solution of the Riccati Equation; 2.3 Alternative Treatment via the Ermakov Equation Â#x83;; 2.3.1 Position and Momentum Uncertainties in Terms of Ermakov and Riccati Variables; 2.3.2 Consequences of the Wave Packet Spreading for the Probability Current; 2.4 Linearization of the Complex Riccati Equation; 2.5 Time-Dependent Green Function or Feynman Kernel.
2.5.1 Riccati Equations from the Green Function and Trigonometric Considerations2.6 Lagrange
Hamilton Formalism for Quantum Uncertainties; 2.7 Momentum Space Representation; 2.8 Wigner Function and Ermakov Invariant; 2.9 Representation of Canonical Transformations in Quantum Mechanics; 2.10 Algebraic Derivation of the Ermakov Invariant; 2.11 Generalized Creation and Annihilation Operators and Coherent States; 2.12 Application of the Ermakov Invariant to Transform Â#x83;; 2.13 Interrelations Between the Different Treatments Â#x83;; References; 3 Time-Independent SchrÃœdinger and Riccati Equations.
3.1 On Supersymmetry and Riccati Equations3.2 Nonlinear Version of Time-Independent Quantum Mechanics; 3.3 Complex Hamiltonians with Real Spectra; 3.4 Comparison of Time-Dependent and Time-Independent Systems; References; 4 Dissipative Systems with Irreversible Dynamics; 4.1 Different Approaches for Treating Open Dissipative Systems; 4.2 System-Plus-Reservoir Approaches; 4.2.1 Caldeira
Leggett Model and Kossakowski
Lindblad Generators; 4.2.2 Bateman Hamiltonian; 4.3 Effective Models Within the Canonical Formalism; 4.3.1 Caldirola
Kanai Hamiltonian; 4.3.2 Expanding Coordinate System.
4.4 Effective Models Using Nonlinear Modifications of the SchrÃœdinger Equation4.4.1 Models Based on Ehrenfest's Theorem and the Langevin Equation; 4.4.2 Models Based on Non-unitary Time-Evolution; 4.4.3 Models Based on a Smoluchowski Equation for the Probability Density; 4.5 Non-unitary Connections Between the Canonical and Nonlinear Approaches; References; 5 Irreversible Dynamics and Dissipative Energetics of Gaussian Wave Packet Solutions; 5.1 Direct Solution of the Riccati Equation, Ermakov Equation and Corresponding Invariant.
5.2 Position and Momentum Uncertainties in Terms of Ermakov Â#x83;5.3 Linearization of the Riccati Equation and Dissipative Lagrange
Hamilton Â#x83;; 5.4 New Qualitative Quantum Effects Induced by a Dissipative Environment; 5.4.1 Increase of Ground State Energy Due to Interaction with an Environment; 5.4.2 Bifurcation and Non-diverging Uncertainty Product; 5.4.3 Modified Plane Waves and Nonlinear Superposition; 5.4.4 Environmentally-Induced Tunnelling Currents and Resonant Energy Back-Transfer; 5.5 Time-Dependent Green Function for the Dissipative Case.
2.5.1 Riccati Equations from the Green Function and Trigonometric Considerations2.6 Lagrange
Hamilton Formalism for Quantum Uncertainties; 2.7 Momentum Space Representation; 2.8 Wigner Function and Ermakov Invariant; 2.9 Representation of Canonical Transformations in Quantum Mechanics; 2.10 Algebraic Derivation of the Ermakov Invariant; 2.11 Generalized Creation and Annihilation Operators and Coherent States; 2.12 Application of the Ermakov Invariant to Transform Â#x83;; 2.13 Interrelations Between the Different Treatments Â#x83;; References; 3 Time-Independent SchrÃœdinger and Riccati Equations.
3.1 On Supersymmetry and Riccati Equations3.2 Nonlinear Version of Time-Independent Quantum Mechanics; 3.3 Complex Hamiltonians with Real Spectra; 3.4 Comparison of Time-Dependent and Time-Independent Systems; References; 4 Dissipative Systems with Irreversible Dynamics; 4.1 Different Approaches for Treating Open Dissipative Systems; 4.2 System-Plus-Reservoir Approaches; 4.2.1 Caldeira
Leggett Model and Kossakowski
Lindblad Generators; 4.2.2 Bateman Hamiltonian; 4.3 Effective Models Within the Canonical Formalism; 4.3.1 Caldirola
Kanai Hamiltonian; 4.3.2 Expanding Coordinate System.
4.4 Effective Models Using Nonlinear Modifications of the SchrÃœdinger Equation4.4.1 Models Based on Ehrenfest's Theorem and the Langevin Equation; 4.4.2 Models Based on Non-unitary Time-Evolution; 4.4.3 Models Based on a Smoluchowski Equation for the Probability Density; 4.5 Non-unitary Connections Between the Canonical and Nonlinear Approaches; References; 5 Irreversible Dynamics and Dissipative Energetics of Gaussian Wave Packet Solutions; 5.1 Direct Solution of the Riccati Equation, Ermakov Equation and Corresponding Invariant.
5.2 Position and Momentum Uncertainties in Terms of Ermakov Â#x83;5.3 Linearization of the Riccati Equation and Dissipative Lagrange
Hamilton Â#x83;; 5.4 New Qualitative Quantum Effects Induced by a Dissipative Environment; 5.4.1 Increase of Ground State Energy Due to Interaction with an Environment; 5.4.2 Bifurcation and Non-diverging Uncertainty Product; 5.4.3 Modified Plane Waves and Nonlinear Superposition; 5.4.4 Environmentally-Induced Tunnelling Currents and Resonant Energy Back-Transfer; 5.5 Time-Dependent Green Function for the Dissipative Case.