Mathematical Analysis of Physical Problems 1st Edition by Philip Wallace ISBN 978-0486646763 0486646769

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Product details: 

ISBN 10:  0486646769

ISBN 13: 978-0486646763

Author: Philip R. Wallace

Intended for the advanced undergraduate or beginning graduate student, this lucid work links classical and modern physics through common techniques and concepts and acquaints the reader with a variety of mathematical tools physicists use to describe and comprehend the physical universe.
For the physicist, mathematics is a language, or shorthand, for constructing workable models (necessarily approximate and incomplete) of aspects of physical reality. The present text, by a noted professor of physics at McGill University, Montreal, deals in an exceptionally well-organized way with some of the crucial mathematical tools used to construct such models.
Contents include: I: The Vibrating String; II. Linear Vector Spaces; III. The Potential Equation; IV: Fourier and Laplace Transforms and Their Applications; V. Propagation and Scattering of Waves; VI. Problems of Diffusion and Attenuation; VII. Probability and Stochastic Processes; VIII. Fundamental Principles of Quantum Mechanics; IX. Some Soluble Problems of Quantum Mechanics; X. Quantum Mechanics of Many-body Problems.
A special helpful feature of this volume is a Prelude to each chapter, which outlines the topics with which the chapter deals. In addition to providing a guide to the organization of its contents, it indicates the mathematical background assumed and calls attention to those methods and concepts which have an application in different physical problems. Relevant test problems are interspersed throughout the text to test the student’s grasp of the material, while brief bibliographies at the chapter ends suggest further reading.
Ideal as a primary or supplementary text, Mathematical Analysis of Physical Problems will reward any reader seeking a firmer grasp of the mathematical procedures by which physicists unlock the secrets of the universe.

Table of contents: 

PRELUDE TO CHAPTER 1
THE VIBRATING STRING

1. Introduction

2. Derivation of the Equation of Motion

3. Solution of the Equation

4. Energy of the String

5. Energy in the Harmonics

6. The “Loaded” String

7. Reflection and Transmission at a Fixed Mass

8. Propagation on a String with Regularly Spaced Masses Attached

9. Reflection by and Transmission through a Section of Different Density

10. Inhomogeneous String and the Method of Separation of Variables

11. Boundary Conditions and the Eigenvalue Problem

12. Orthogonality of Eigenfunctions

13. Rayleigh-Ritz Variational Principle

14. Approximate Calculation of Eigenvalues from the Variational Principle

15. Expansion into Eigenfunctions

16. The Inhomogeneous Problem for the Vibrating String

17. Green’s Function

18. Effect of a Perturbation of Density

19. The JWKB Method

20. An Example

21. Lagrangian and Hamiltonian Formulations of the Vibrating String Problem

PRELUDE TO CHAPTER 2
LINEAR VECTOR SPACES

1. Introduction

2. Vector Spaces

3. Linear Independence, Dimensionality, and Bases

4. Scalar Products

5. Schmidt Inequality and Orthogonalization

6. An Example of the Schmidt Procedure

7. Matrix Representation of Vectors and Transformation of Basis

8. Linear Operators and Their Matrix Representations

9. The Eigenvalue Problem for Hermitian Operators

10. Another Example

11. Sturm-Liouville Problem and Linear Vector Spaces

PRELUDE TO CHAPTER 3
THE POTENTIAL EQUATION

1. Introduction: Electrostatic Potential

2. Solution of Laplace’s Equation in Spherical Coordinates

3. The Equation and the Factorization Method

4. Spherical Harmonics

5. Radial Solution and the General Solution of Laplace’s Equation

6. Legendre Polynomials

7. An Alternative Derivation of Legendre Polynomials: Multipoles

8. Multipole without Axial Symmetry and Associated Legendre Functions

9. An Addition Theorem for Spherical Harmonics

10. Potential of a Given Charge Distribution

11. Potential of an Axially Symmetric Charge Distribution

12. Potentials of Charge Distributions under Various Boundary Conditions – Green’s Functions

13. Further Problems Involving the Potential Equation
    Appendix 3A: A Recursion Relation for P™ (cos θ)
    Appendix 3B: Review of Theory of Linear Differential Equations of the Second Order

PRELUDE TO CHAPTER 4
FOURIER AND LAPLACE TRANSFORMS AND THEIR APPLICATIONS

1. Introduction: Fourier Transform in One Dimension

2. The Convolution Theorem

3. Causality and Dispersion Relations

4. Linear Response Functions

5. Cross-Correlation and Autocorrelation Functions

6. Fourier Transform in Three Dimensions

7. Solution of Poisson’s Equation by Fourier Transforms

8. Poisson’s Summation Formula in One and Three Dimensions

9. A Note on Delta Functions and Three-Dimensional Transforms

10. Two- and Three-Center Integrals

11. Laplace Transforms

12. Transforms of Derivatives

13. “Shifting” Theorem

14. Convolution Theorem

15. Some Simple Transforms

16. Laplace Transform of a Periodic Function

17. Resonance

18. Use of Laplace and Fourier Transforms to Solve the Vibrating String Problem

19. The Gamma Function

20. Stirling’s Formula

21. The Beta Function

22. Use of Transforms for Equations with Linear Coefficients

23. The Confluent Hypergeometric Function

24. Laguerre Functions

25. Hermite Functions

26. Equations Reducible to the Confluent Hypergeometric: The Bessel Equation

27. General Properties of Bessel Functions

28. Second Solution of the Bessel Equation

29. Zeros of Bessel Functions

30. Hankel Functions

31. Further Formulas Involving Bessel Functions

PRELUDE TO CHAPTER 5
PROPAGATION AND SCATTERING OF WAVES

1. Introduction

2. Sound Waves. Derivation of the Equations

3. Dynamics of Sound Waves

4. Lagrangian and Hamiltonian Formulation

5. Guided Waves

6. Wave Equation with Sources

7. Spherical Waves

8. Expansion of a Plane Wave in Spherical Waves

9. Radiation from a Periodic Source

10. Time-Varying Source

11. Radiation from a Moving Source

12. Solution with Initial and Boundary Conditions

13. Waves in Guides and Enclosures

14. Spherical Enclosure

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