Classical And Quantum Dissipative Systems

by Mohsen Razavy

Overview

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.

Product Details

ISBN-13:	9781860945250
Publisher:	Imperial College Press
Publication date:	01/17/2006
Pages:	352
Product dimensions:	6.70(w) x 9.80(h) x 1.20(d)

Table of Contents

	Dedication	v
	Preface	vii
1	Introduction	1
2	Phenomenological Equations of Motion for Dissipative Systems	5
2.1	Frictional Forces Linear Velocity	5
2.2	Raleigh's Oscillator	8
2.3	One-Dimensional Motion and Bopp Transformation	8
2.4	The Classical Theory of Line Width	11
2.5	Frictional Forces Quadratic in Velocity	12
2.6	Non-Newtonian and Nonlocal Dissipative Forces	13
3	Lagrangian Formulations	15
3.1	Rayleigh and Lur'e Dissipative Functions	15
3.2	Inverse Problem of Analytical Dynamics	20
3.3	Some Examples of the Lagrangians for Dissipative Systems	24
3.4	Non-Uniqueness of the Lagrangian	27
3.5	Acceptable Lagrangians for Dissipative Systems	30
4	Hamiltonian Formulation	33
4.1	Inverse Problem for the Hamiltonian	33
4.2	Hamiltonians for Simple Dissipative Systems	36
4.3	Ostrogradsky's Method	39
4.4	Complex or Leaky Spring Constant	42
4.5	Dekker's Complex Coordinate Formulation	42
4.6	Hamiltonian Formulation of the Motion of a Particle with Variable Mass	44
4.7	Variable Mass Oscillator	45
4.8	Bateman's Damped-Amplified Harmonic Oscillators	47
4.9	Dissipative Forces Quadratic in Velocity	48
4.10	Resistive Forces Proportional to Arbitrary Powers of Velocity	48
4.11	Universal Lagrangian and Hamiltonian	49
4.12	Hamiltonian Formulation in Phase Space of N-Dimensions	52
4.13	Symmetric Phase Space Formulation of the Damped Harmonic Oscillator	55
4.14	Dynamical Systems Expressible as Linear Difference Equations	57
5	Hamilton-Jacobi Formulation	63
5.1	The Hamilton-Jacobi Equation for Linear Damping	64
5.2	Classical Action for an Oscillator with Leaky Spring Constant	66
5.3	More About the Hamilton-Jacobi Equation for the Damped Motion	67
6	Motion of a Charged Particle in an External Electromagnetic Field in the Presence of Damping	71
7	Noether and Non-Noether Symmetries and Conservation Laws	77
7.1	Non-Noether Symmetries and Conserved Quantities	84
7.2	Noether's Theorem for a Scalar Field	86
8	Dissipative Forces Derived from Many-Body Problems	91
8.1	The Schrodinger Chain	91
8.2	A Particle Coupled to a Chain	93
8.3	Dynamics of a Non-Uniform Chain	94
8.4	Mechanical System Coupled to a Heat Bath	98
8.5	Euclidean Lagrangian	104
9	A Particle Coupled to a Field	107
9.1	Harmonically Bound Radiating Electron	107
9.2	An Oscillator Coupled to a String of Finite Length	109
9.3	An Oscillator Coupled to an Infinite String	112
10	Damped Motion of the Central Particle	117
10.1	Diagonalization of the Hamiltonian	117
11	Classical Microscopic Models of Dissipation and Minimal Coupling Rule	125
12	Quantization of Dissipative Systems	129
12.1	Early Attempts to Quantize the Damped Oscillator	129
12.2	Yang-Feldman Method of Quantization	136
12.3	Heisenberg's Equations of Motion for Dekker's Formulation	138
12.4	Quantization of the Bateman Hamiltonian	139
12.5	Fermi's Nonlinear Equation for Quantized Radiation Reaction	142
12.6	Attempts to Quantize Systems with a Dissipative Force Quadratic in Velocity	145
12.7	Solution of the Wave Equation for Linear and Newtonian Damping Forces	147
12.8	The Classical Limit of the Schrodinger Equation with Velocity-Dependent Forces	150
12.9	Quadratic Damping as an Externally Applied Force	151
12.10	Motion in a Viscous Field of Force Proportional to an Arbitrary Power of Velocity	153
12.11	The Classical Limit and the Van Vleck Determinant	154
13	Quantization of Explicitly Time-Dependent Hamiltonian	157
13.1	Wave Equation for the Kanai-Caldirola Hamiltonian	157
13.2	Coherent States of a Damped Oscillator	164
13.3	Squeezed State of a Damped Harmonic Oscillator	167
13.4	Quantization of a System with Variable Mass	169
13.5	The Schrodinger-Langevin Equation for Linear Damping	172
13.6	An Extension of the Madelung Formulation	175
13.7	Quantization of a Modified Hamilton-Jacobi Equation for Damped Systems	180
13.8	Exactly Solvable Cases of the Schrodinger -Langevin Equation	184
13.9	Harmonically Bound Radiating Electron and the Schrodinger-Langevin Equation	187
13.10	Other Phenomenological Nonlinear Potentials for Dissipative Systems	189
13.11	Scattering in the Presence of Frictional Forces	192
13.12	Application of the Noether Theorem: Linear and Nonlinear Wave Equations for Dissipative Systems	193
13.13	Wave Equation for Impulsive Forces Acting at Certain Intervals	196
13.14	Classical Limit for the Time-Dependent Problems	197
14	Density Matrix and the Wigner Distribution Function	201
14.1	Classical Distribution Function for Nonconservative Motions	201
14.2	The Density Matrix	205
14.3	Phase Space Quantization of Dekker's Hamiltonian	207
14.4	Density Operator and the Fokker-Planck Equation	209
14.5	The Density Matrix Formulation of a Solvable Model	213
14.6	Wigner Distribution Function for the Damped Oscillator	216
14.7	Density Operator for a Particle Coupled to a Heat Bath	219
15	Path Integral Formulation of a Damped Harmonic Oscillator	223
15.1	Propagator for the Damped Harmonic Oscillator	224
15.2	Path Integral Quantization of a Harmonic Oscillator with Complex Spring Constant	230
15.3	Modified Classical Action and the Propagator for the Damped Harmonic Oscillator	233
15.4	Path Integral Formulation of a System Coupled to a Heat Bath	235
16	Quantization of the Motion of an Infinite Chain	239
16.1	Quantum Mechanics of a Uniform Chain	239
16.2	Ground State of the Central Particle	242
16.3	Wave Equation for a Non-Uniform Chain	244
16.4	Connection with Other Phenomenological Frictional Forces	246
16.5	Fokker-Planck Equation for the Probability Density	247
17	The Heisenberg Equations of Motion for a Particle Coupled to a Heat Bath	249
17.1	Heisenberg Equations for a Damped Harmonic Oscillator	249
17.2	Density Matrix for the Motion of a Particle Coupled to a Field	260
17.3	Equations of Motion for the Central Particle	263
17.4	Wave Equation for the Motion of the Central Particle	264
17.5	Motion of the Center-of-Mass in Viscous Medium	269
17.6	Invariance Under Galilean Transformation	272
17.7	Velocity Coupling and Coordinate Coupling	273
17.8	Equation of Motion for a Harmonically Bound Radiating Electron	274
18	Quantum Mechanical Models of Dissipative Systems	279
18.1	Forced Vibration with Damping	279
18.2	The Wigner-Weisskopf Model	282
18.3	Quantum Theory of Line Width	286
18.4	The Optical Potential	291
18.5	Gisin's Nonlinear Wave Equation	295
18.6	Nonlinear Generalization of the Wave Equation	298
18.7	Dissipation Arising from the Motion of the Boundaries	301
18.8	Decaying States in a Many-Boson System	308
19	More on the Concept of Optical Potential	315
19.1	The Classical Analogue of the Nonlocal Interaction	315
19.2	Minimal and/or Maximal Coupling	319
19.3	Damped Harmonic Oscillator and Optical Potential	323
19.4	Quantum Mechanical Analogue of the Raleigh Oscillator	326
	Index	331