Qualitative Analysis of Physical Problems

Qualitative Analysis of Physical Problems

1st Edition - January 28, 1981

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  • Author: M Gitterman
  • eBook ISBN: 9780323157506

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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


    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


    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


    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


    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


    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


    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




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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