Normal view MARC view ISBD view

Quantum Theory from a Nonlinear Perspective : Riccati Equations in Fundamental Physics.

By: Schuch, Dieter.
Material type: TextTextSeries: eBooks on Demand.Fundamental Theories of Physics Ser: Publisher: Cham : Springer, 2018Copyright date: ©2018Description: 1 online resource (261 pages).Content type: text Media type: computer Carrier type: online resourceISBN: 9783319655949.Subject(s): Quantum theory-Mathematics | Riccati equationGenre/Form: Electronic books.Additional physical formats: Print version:: Quantum Theory from a Nonlinear Perspective : Riccati Equations in Fundamental PhysicsDDC classification: 515.35 LOC classification: QC173.96-174.52QC174Online resources: Click here to view this ebook.
Contents:
Intro -- Preface -- Contents -- 1 Introduction -- References -- 2 Time-Dependent Schrödinger 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 ƒ -- 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 Considerations -- 2.6 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 ƒ -- 2.13 Interrelations Between the Different Treatments ƒ -- References -- 3 Time-Independent Schrödinger and Riccati Equations -- 3.1 On Supersymmetry and Riccati Equations -- 3.2 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 Schrödinger Equation -- 4.4.1 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 ƒ -- 5.3 Linearization of the Riccati Equation and Dissipative Lagrange--Hamilton ƒ -- 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 -- 5.6 Dissipative Schrödinger Equation in Momentum Space -- 5.6.1 Friction Term in Momentum Space -- 5.6.2 Wave Packet Solutions in Momentum Space -- 5.6.3 Time-Dependent Green Function in Momentum Space -- 5.7 Wigner Function and Ermakov Invariant for the Dissipative Case -- 5.8 Algebraic Derivation of the Dissipative Ermakov Invariant -- 5.9 Generalized Creation and Annihilation Operators and Coherent States ƒ -- References -- 6 Dissipative Version of Time-Independent Nonlinear Quantum Mechanics -- References -- 7 Nonlinear Riccati Equations in Other Fields of Physics -- 7.1 Riccati Equations in Statistical Thermodynamics -- 7.2 The Logistic or Verhulst Equation -- 7.3 Nonlinear Dynamics with Hopf Bifurcation.
7.4 Solitons and Riccati Equations -- 7.4.1 Burgers Equation -- 7.4.2 Korteweg--de Vries Equation -- 7.4.3 Connections Between the Soliton Equations -- 7.5 Complex Riccati Equation in Classical Optics -- 7.6 Ermakov Equation for Bose--Einstein Condensates -- 7.7 Ermakov Equation in Cosmology -- 7.8 Complex Riccati Equation and Pythagorean Triples -- References -- 8 Summary, Conclusions and Perspectives -- References -- Appendix A Method of Linear and Quadratic Invariants -- Appendix B Position and Momentum Uncertainties in the Dissipative Case -- Appendix C Classical Lagrange--Hamilton Formalism in Expanding Coordinates -- Appendix D On the Connection Between the Bateman Hamiltonian and the Hamiltonian in Expanding Coordinates -- D.1 The Case c = 0 -- D.2 The Case a = 0 -- D.3 The Case b = 0 -- Appendix E Logarithmic Nonlinear Schrödinger Equation via Complex Hydrodynamic Equation of Motion -- Index.
Tags from this library: No tags from this library for this title. Log in to add tags.
Item type Current location Call number URL Status Date due Barcode
Electronic Book UT Tyler Online
Online
QC173.96-174.52QC174 (Browse shelf) https://ebookcentral.proquest.com/lib/uttyler/detail.action?docID=5231551 Available EBC5231551

Intro -- Preface -- Contents -- 1 Introduction -- References -- 2 Time-Dependent Schrödinger 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 ƒ -- 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 Considerations -- 2.6 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 ƒ -- 2.13 Interrelations Between the Different Treatments ƒ -- References -- 3 Time-Independent Schrödinger and Riccati Equations -- 3.1 On Supersymmetry and Riccati Equations -- 3.2 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 Schrödinger Equation -- 4.4.1 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 ƒ -- 5.3 Linearization of the Riccati Equation and Dissipative Lagrange--Hamilton ƒ -- 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 -- 5.6 Dissipative Schrödinger Equation in Momentum Space -- 5.6.1 Friction Term in Momentum Space -- 5.6.2 Wave Packet Solutions in Momentum Space -- 5.6.3 Time-Dependent Green Function in Momentum Space -- 5.7 Wigner Function and Ermakov Invariant for the Dissipative Case -- 5.8 Algebraic Derivation of the Dissipative Ermakov Invariant -- 5.9 Generalized Creation and Annihilation Operators and Coherent States ƒ -- References -- 6 Dissipative Version of Time-Independent Nonlinear Quantum Mechanics -- References -- 7 Nonlinear Riccati Equations in Other Fields of Physics -- 7.1 Riccati Equations in Statistical Thermodynamics -- 7.2 The Logistic or Verhulst Equation -- 7.3 Nonlinear Dynamics with Hopf Bifurcation.

7.4 Solitons and Riccati Equations -- 7.4.1 Burgers Equation -- 7.4.2 Korteweg--de Vries Equation -- 7.4.3 Connections Between the Soliton Equations -- 7.5 Complex Riccati Equation in Classical Optics -- 7.6 Ermakov Equation for Bose--Einstein Condensates -- 7.7 Ermakov Equation in Cosmology -- 7.8 Complex Riccati Equation and Pythagorean Triples -- References -- 8 Summary, Conclusions and Perspectives -- References -- Appendix A Method of Linear and Quadratic Invariants -- Appendix B Position and Momentum Uncertainties in the Dissipative Case -- Appendix C Classical Lagrange--Hamilton Formalism in Expanding Coordinates -- Appendix D On the Connection Between the Bateman Hamiltonian and the Hamiltonian in Expanding Coordinates -- D.1 The Case c = 0 -- D.2 The Case a = 0 -- D.3 The Case b = 0 -- Appendix E Logarithmic Nonlinear Schrödinger Equation via Complex Hydrodynamic Equation of Motion -- Index.

Description based on publisher supplied metadata and other sources.

There are no comments for this item.

Log in to your account to post a comment.