Classical and Quantum Dissipative Systems.
Title:
Classical and Quantum Dissipative Systems.
Author:
Razavy, Mohsen.
ISBN:
9781860949180
Personal Author:
Physical Description:
1 online resource (352 pages)
Contents:
Contents -- Dedication -- Preface -- 1 Introduction -- 2 Phenomenological Equations of Motion for Dissipative Systems -- 2.1 Frictional Forces Linear Velocity -- 2.2 Raleigh's Oscillator -- 2.3 One-Dimensional Motion and Bopp Transformation -- 2.4 The Classical Theory of Line Width -- 2.5 Frictional Forces Quadratic in Velocity -- 2.6 Non-Newtonian and Nonlocal Dissipative Forces -- 3 Lagrangian Formulations -- 3.1 Rayleigh and Lur'e Dissipative Functions -- 3.2 Inverse Problem of Analytical Dynamics -- 3.3 Some Examples of the Lagrangians for Dissipative Systems -- 3.4 Non-Uniqueness of the Lagrangian -- 3.5 Acceptable Lagrangians for Dissipative Systems -- 4 Hamiltonian Formulation -- 4.1 Inverse Problem for the Hamiltonian -- 4.2 Hamiltonians for Simple Dissipative Systems -- 4.3 Ostrogradsky's Method -- 4.4 Complex or Leaky Spring Constant -- 4.5 Dekker's Complex Coordinate Formulation -- 4.6 Hamiltonian Formulation of the Motion of a Particle with Variable Mass -- 4.7 Variable Mass Oscillator -- 4.8 Bateman's Damped-Amplified Harmonic Oscillators -- 4.9 Dissipative Forces Quadratic in Velocity -- 4.10 Resistive Forces Proportional to Arbitrary Powers of Velocity -- 4.11 Universal Lagrangian and Hamiltonian -- 4.12 Hamiltonian Formulation in Phase Space of N-Dimensions -- 4.13 Symmetric Phase Space Formulation of the Damped Harmonic Oscillator -- 4.14 Dynamical Systems Expressible as Linear Difference Equations -- 5 Hamilton-Jacobi Formulation -- 5.1 The Hamilton-Jacobi Equation for Linear Damping -- 5.2 Classical Action for an Oscillator with Leaky Spring Constant -- 5.3 More About the Hamilton-Jacobi Equation for the Damped Motion -- 6 Motion of a Charged Particle in an External Electromagnetic Field in the Presence of Damping.
7 Noether and Non-Noether Symmetries and Conservation Laws -- 7.1 Non-Noether Symmetries and Conserved Quantities -- 7.2 Noether's Theorem for a Scalar Field -- 8 Dissipative Forces Derived from Many-Body Problems -- 8.1 The Schrodinger Chain -- 8.2 A Particle Coupled to a Chain -- 8.3 Dynamics of a Non-Uniform Chain -- 8.4 Mechanical System Coupled to a Heat Bath -- 8.5 Euclidean Lagrangian -- 9 A Particle Coupled to a Field -- 9.1 Harmonically Bound Radiating Electron -- 9.2 An Oscillator Coupled to a String of Finite Length -- 9.3 An Oscillator Coupled to an Infinite String -- 10 Damped Motion of the Central Particle -- 10.1 Diagonalization of the Hamiltonian -- 11 Classical Microscopic Models of Dissipation and Minimal Coupling Rule -- 12 Quantization of Dissipative Systems -- 12.1 Early Attempts to Quantize the Damped Oscillator -- 12.2 Yang-Feldman Method of Quantization -- 12.3 Heisenberg's Equations of Motion for Dekker's Formulation -- 12.4 Quantization of the Bateman Hamiltonian -- 12.5 Fermi's Nonlinear Equation for Quantized Radiation Reaction -- 12.6 Attempts to Quantize Systems with a Dissipative Force Quadratic in Velocity -- 12.7 Solution of the Wave Equation for Linear and Newtonian Damping Forces -- 12.8 The Classical Limit of the Schrodinger Equation with Velocity-Dependent Forces -- 12.9 Quadratic Damping as an Externally Applied Force -- 12.10 Motion in a Viscous Field of Force Proportional to an Arbitrary Power of Velocity -- 12.11 The Classical Limit and the Van Vleck Determinant -- 13 Quantization of Explicitly Time-Dependent Hamiltonian -- 13.1 Wave Equation for the Kanai-Caldirola Hamiltonian -- 13.2 Coherent States of a Damped Oscillator -- 13.3 Squeezed State of a Damped Harmonic Oscillator -- 13.4 Quantization of a System with Variable Mass.
13.5 The Schrodinger-Langevin Equation for Linear Damping -- 13.6 An Extension of the Madelung Formulation -- 13.7 Quantization of a Modified Hamilton-Jacobi Equation for Damped Systems -- 13.8 Exactly Solvable Cases of the Schrodinger-Langevin Equation -- 13.9 Harmonically Bound Radiating Electron and the Schrodinger-Langevin Equation -- 13.10 Other Phenomenological Nonlinear Potentials for Dissipative Systems -- 13.11 Scattering in the Presence of Frictional Forces -- 13.12 Application of the Noether Theorem: Linear and Nonlinear Wave Equations for Dissipative Systems -- 13.13 Wave Equation for Impulsive Forces Acting at Certain Intervals -- 13.14 Classical Limit for the Time-Dependent Problems -- 14 Density Matrix and the Wigner Distribution Function -- 14.1 Classical Distribution Function for Nonconservative Motions -- 14.2 The Density Matrix -- 14.3 Phase Space Quantization of Dekker's Hamiltonian -- 14.4 Density Operator and the Fokker-Planck Equation -- 14.5 The Density Matrix Formulation of a Solvable Model -- 14.6 Wigner Distribution Function for the Damped Oscillator -- 14.7 Density Operator for a Particle Coupled to a Heat Bath -- 15 Path Integral Formulation of a Damped Harmonic Oscillator -- 15.1 Propagator for the Damped Harmonic Oscillator -- 15.2 Path Integral Quantization of a Harmonic Oscillator with Complex Spring Constant -- 15.3 Modified Classical Action and the Propagator for the Damped Harmonic Oscillator -- 15.4 Path Integral Formulation of a System Coupled to a Heat Bath -- 16 Quantization of the Motion of an Infinite Chain -- 16.1 Quantum Mechanics of a Uniform Chain -- 16.2 Ground State of the Central Particle -- 16.3 Wave Equation for a Non-Uniform Chain -- 16.4 Connection with Other Phenomenological Frictional Forces.
16.5 Fokker-Planck Equation for the Probability Density -- 17 The Heisenberg Equations of Motion for a Particle Coupled to a Heat Bath -- 17.1 Heisenberg Equations for a Damped Harmonic Oscillator -- 17.2 Density Matrix for the Motion of a Particle Coupled to a Field -- 17.3 Equations of Motion for the Central Particle -- 17.4 Wave Equation for the Motion of the Central Particle -- 17.5 Motion of the Center-of-Mass in Viscous Medium -- 17.6 Invariance Under Galilean Transformation -- 17.7 Velocity Coupling and Coordinate Coupling -- 17.8 Equation of Motion for a Harmonically Bound Radiating Electron -- 18 Quantum Mechanical Models of Dissipative Systems -- 18.1 Forced Vibration with Damping -- 18.2 The Wigner-Weisskopf Model -- 18.3 Quantum Theory of Line Width -- 18.4 The Optical Potential -- 18.5 Gisin's Nonlinear Wave Equation -- 18.6 Nonlinear Generalization of the Wave Equation -- 18.7 Dissipation Arising from the Motion of the Boundaries -- 18.8 Decaying States in a Many-Boson System -- 19 More on the Concept of Optical Potential -- 19.1 The Classical Analogue of the Nonlocal Interaction -- 19.2 Minimal and/or Maximal Coupling -- 19.3 Damped Harmonic Oscillator and Optical Potential -- 19.4 Quantum Mechanical Analogue of the Raleigh Oscillator -- Index.
Abstract:
This book discusses issues associated with the quantum mechanical formulation of dissipative systems. It begins with an introductory review of phenomenological damping forces, and the construction of the Lagrangian and Hamiltonian for the damped motion. It is shown, in addition to these methods, that classical dissipative forces can also be derived from solvable many-body problems. A detailed discussion of these derived forces and their dependence on dynamical variables is also presented. The second part of this book investigates the use of classical formulation in the quantization of dynamical systems under the influence of dissipative forces. The results show that, while a satisfactory solution to the problem cannot be found, different formulations represent different approximations to the complete solution of two interacting systems. The third and final part of the book focuses on the problem of dissipation in interacting quantum mechanical systems, as well as the connection of some of these models to their classical counterparts. A number of important applications, such as the theory of heavy-ion scattering and the motion of a radiating electron, are also discussed. Sample Chapter(s). Chapter 1: Introduction (87 KB). Contents: Phenomenological Equations of Motion for Dissipative Systems; Lagrangian Formulations; Hamiltonian Formulation; Hamilton-Jacobi Formulation; Motion of a Charged Particle in an External Electromagnetic Field in the Presence of Damping; Noether and Non-Noether Symmetries and Conservation Laws; Dissipative Forces Derived from Many-Body Problems; A Particle Coupled to a Field; Damped Motion of the Central Particle; Classical Microscopic Models of Dissipation and Minimal Coupling Rule; Quantization of Dissipative Systems; Quantization of Explicitly Time-Dependent Hamiltonian; Density Matrix and the Wigner Distribution
Function; Path Integral Formulation of a Damped Harmonic Oscillator; Quantization of the Motion of an Infinite Chain; The Heisenberg Equations of Motion for a Particle Coupled to a Heat Bath; Quantum Mechanical Models of Dissipative Systems; More on the Concept of Optical Potential. Readership: Researchers and graduate students in applied mathematics and theoretical physics.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
Genre:
Electronic Access:
Click to View