# 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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