Qualitative Analysis of Physical Problems
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Qualitative Analysis of Physical Problems reviews the essential features of all the main approaches used for the qualitative analysis of physical problems and demonstrates their application to problems from a wide variety of fields. Topics covered include model construction, dimensional analysis, symmetry, and the method of the small parameter. This book consists of six chapters and begins by looking at various approaches for the construction of models, along with nontrivial applications of dimensional analysis to some typical model systems. The following chapters focus on the application of symmetry to the microscopic and macroscopic properties of systems; the implications of analyticity and occurrence of singularities; and some methods of deriving the magnitude of the solutions (that is, approximate numerical values) for problems that usually cannot be solved exactly in closed form. The final chapter demonstrates the use of qualitative analysis to address the problem of second harmonic generation in nonlinear optics. This monograph will be a useful resource for graduate students, experimental and theoretical physicists, chemists, engineers, college and high school teachers, and those who are interested in obtaining a general perspective of modern physics.
Table of Contents
Preface
Chapter 1 The Construction of Models
1.1 Introduction
The Need for Models
Simplification of the Problem
Microscopic andMacroscopic Approaches
Ideal and Nonideal Gases
Systems of Interacting Particles
Examples of the Microscopic Approach
Examples of the Macroscopic Approach
Other Applications of Models
1.2 The Atomic Nucleus
The Need for Nuclear Models
The Liquid-Drop Model
The Shell Model
Compound Nucleus and Optical Models
Use of Conflicting Simple Models
1.3 The Quark Model of Elementary Particles
Definition of Elementary Particles
Classification of Particles
Symmetry Groupings
The Quark Model
Modifications of the Quark Model
Experimental Confirmation and Outstanding Problems
1.4 Elementary Excitation in Solids
The Free-Electron Model
Normal Coordinates
Quasi-particles
The Successes and Failures of the Free-Electron Model
Magnetic Properties of the Electron Gas
Different Types of Elementary Excitations in Solids
1.5 Steady-State Space-Charge-Limited Currents in Insulators
Description of the System
Construction and Analysis of an Idealized Model
Simplification of the Model
Solutions for Extreme Cases
1.6 Boundary Layer Theory in Hydrodynamics
The Equations of Motion for a Fluid
The Flow of Fluid past a Solid Body
Simplification of the Hydrodynamic Equations
Chapter 2 Dimensional Analysis
2.1 Introduction
Fundamental and Derived Units
Derivation of Formulas
Nonlinear Heat Conduction
Dimensionless Equations
Hydrodynamic Modeling
Phase Transitions
The Ising Model
Scaling Theory
2.2 The Derivation of Formulas by Dimensional Analysis
The II Theorem
Planetary Motion
Electrical Units
Space-Charge-Limited Currents
Vector Lengths
The Thermal Conductivity of a Gas
2.3 Simple Derivation of Physical Laws
Motion in a Potential Field
Statistical Physics
Equation of State of Fermi and Bose Gases
2.4 Dimensionless Equations and Physical Similarity
The Electrical Charge Distribution in Atoms—The Thomas-Fermi Equation
Heat Conduction in a Cubic Block
Equations Involving Parameters
Hydrodynamic Modeling
2.5 Modern Theory of Critical Phenomena
The Renormalization Group
An Application of the Renormalization
Group Theory
Problems
Chapter 3 Symmetry
3.1 Introduction
Classical Mechanics
Frames of Reference and Relativity
Quantum Mechanics
Classical Electrodynamics
Elementary Particles
Molecular Vibrations
Symmetry of Crystal Structures
Symmetry of the Properties of Crystals
The Symmetry of Kinetic Coefficients - Onsager's Principle
Order-Disorder Phase Transitions
3.2 Conservation Laws in Quantum Mechanics
Quantum-Mechanical Formulation of Conservation Laws
The Conservation of Energy, Momentum, and Angular Momentum
Parity
Time-Reversal Symmetry in Classical Physics
Time-Reversal Symmetry and Irreversibility
Time-Reversal Symmetry in Quantum Mechanics
Indistinguishable Particles
Gauge Invariance and Charge Conservation
Charge Conjugation
3.3 Symmetry and the Microscopic Properties of Systems
The Symmetry of Eigenfunctions
Matrix Elements and Selection Rules
Irreducible Representations of Groups
One-Dimensional Representations
The Translational Symmetry of Crystals
Selection Rules for Crystals
Irreducible Representations of a Crystal's SpaceGroup
Structural Phase Transitions in Crystals
Integrals over the First Brillouin Zone
3.4 The Inversion Symmetry and Magnetic Symmetry of Crystal Properties
Inversion Symmetry—Polar and Axial Tensors
Optical Activity
Time-Reversal Symmetry—{-Tensors and c-Tensors; Magnetic Systems
Magnetic Point Groups
Pyromagnetism and Piezomagnetism
The Magnetoelectric Effect
Problems
Chapter 4 Analytical and Related Properties
4.1 Introduction
Phase Transition Points
Singularities and Analytical Relationships
Singularities in Quantum Mechanics
The Dielectric Constant of Model Systems
Dispersion Relations
Sum Rules
Causality and Time-Reversal Symmetry
Fluctuations and Dissipation
4.2 Analytic Properties of the Scattering Matrix
Scattering Amplitudes and the S-Matrix
Analytical Properties of the S-Matrix
Scattering by a Square Well Potential
Dispersion Relations
4.3 Dispersion Relations for Macroscopic Systems
Convergence Conditions
Applications of Dispersion Relations
Quantum-Mechanical Approach
Calculation of the Dielectric Constant
Oscillator Strengths and Quantum-Mechanical Sum Rules
Additional Sum Rules; The Physical Meaning of Sum Rules and Dispersion Relations
4.4 The Fluctuation-Dissipation Theorem
Fluctuations of Extensive Variables
Time Correlation Functions
The Fluctuation-Dissipation Theorem
Application of the Fluctuation-Dissipation Theorem: Energy Density of Radiation Field
Time-Dependent Correlation Functions and Transport Coefficients
The Electrical Conductivity
The Electrical Susceptibility of a Dielectric Medium
Problems
Chapter 5 The Method of the Small Parameter
5.1 Introduction
A Typical Problem
Perturbation Theory—The Series Expansion Technique
Solution for a Problem with Two Boundary Conditions at the Same Point
Renormalization Techniques
Eigenvalue Problems
Rayleigh-Schrodinger Perturbation Theory
Mathieu's Equation
Brillouin-Wigner Perturbation Theory
Choice of the Small Parameter
Density Expansion of Transport Coefficients
Low-Density Systems of Charged Particles
The High-Density Electron Gas
Breakdown of Perturbation Theory
Decrease of the Order of a Differential Equation
5.2 Integral Equation Formulations of Perturbation Theory
Integral Equations
Greens Functions
Brillouin-Wigner and Rayleigh-Schrodinger Perturbation Theory
Convergence of the Perturbation Series
Scattering Theory—The First Born Approximation
Dysons Equation
5.3 Choice of the Small Parameter
Quantum-Mechanical Description of a System of Nuclei and Electrons
Degenerate Systems with Two Perturbations
Flexible Choice of the Perturbation
5.4 Difficulties in the Use of the Small Parameter
A Small Parameter Multiplying the Highest Derivative
The Effective Mass Approximation
Magnetic Interactions of Nuclei through Conduction Electrons
Problems
Chapter 6 Epilogue—Example of the Application of the Above Methods to a Problem in Nonlinear Optics
6.1 Introduction 250
6.2 Model System 251
Analysis of the Model
The Models Limitations
6.3 Nonlinear Susceptibilities
Nonlinear Response Functions
Free Energy and Intrinsic Symmetry
Second Harmonic Generation in KDP
6.4 Use of Perturbation Theory
Preparation of the Problem for Perturbation Theory
Application of Perturbation Theory
Conclusions
References
Index
Product details
- No. of pages: 288
- Language: English
- Copyright: © Academic Press 1981
- Published: January 28, 1981
- Imprint: Academic Press
- eBook ISBN: 9780323157506
About the Author
M Gitterman
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