Mathematics for the physical sciences / James B. Seaborn.

Format:

Book

Author/Creator:

Seaborn, James B.

Contributor:

Emma Louise McClellan Fund.

Language:

English

Subjects (All):

Mathematical physics.

Physical Description:

xi, 245 pages : illustrations ; 24 cm

Place of Publication:

New York : Springer, [2002]

Summary:

This book is intended to furnish a bridge from courses in general physics to the intermediate-level courses in classical mechanics, electrodynamics, and quantum mechanics.

It begins with a short review of some topics familiar from general physics that are then used throughout the book to provide the physical contexts for the mathematical methods that are developed. The concept of magnetic flux serves to give physical meaning to the integral theorems of vector calculus. Examples from both classical and quantum physics illustrate the chapters on ordinary and partial differential equations and eigenvalue problems. These include a conducting sphere in a uniform electric field, a vibrating drumhead, classical and quantum harmonic oscillators, and a particle in a box. The eigenvalue problem leads naturally to a discussion of orthogonal functions, which again uses the quantum harmonic oscillator to provide the physical insight, and to matrices, where coupled oscillators and the principal axes of a rotating rigid body supply the physical context. The text concludes with a brief discussion of variational methods.

Problems at the end of each chapter furnish the student with experience in applying the mathematics, and illustrative exercises throughout give guidance. Many of the exercises call for graphical representations, and some are particularly amenable to the use of numerical methods, but the treatment avoids the implication that computers are necessary to solve the problems.

Contents:

1 A Review 1

1.1 Electrostatics 1

1.2 Electric Current 3

1.3 Magnetic Flux 4

1.4 Simple Harmonic Motion 5

1.5 A Rigid Rotator 7

2 Vectors 13

2.1 Representations of Vectors 13

2.2 The Scalar Product of Two Vectors 16

2.3 The Vector Product of Two Vectors 18

3 Vector Calculus 29

3.1 Partial Derivatives 30

3.2 A Vector Differential Operator 32

3.3 Components of the Gradient 34

3.4 Flux 40

4 Complex Numbers 69

4.1 Why Study Complex Numbers? 69

4.2 Roots of a Complex Number 72

5 Differential Equations 77

5.1 Infinite Series 79

5.2 Analytic Functions 80

5.3 The Classical Harmonic Oscillator 82

5.4 Boundary Conditions 84

5.5 Polynomial Solutions 88

5.6 Elementary Functions 90

5.7 Singularities 91

6 Partial Differential Equations 103

6.1 The Method of Separation of Variables 103

6.2 The Quantum Harmonic Oscillator 105

6.3 A Conducting Sphere in an Electric Field 109

6.4 The Schrodinger Equation for a Central Field 115

7 Eigenvalue Problems 127

7.1 Boundary Value Problems 127

7.2 A Vibrating Drumhead 128

7.3 A Particle in a One-Dimensional Box 131

8 Orthogonal Functions 137

8.1 The Failure of Classical Physics 137

8.2 Observables and Their Measurement 138

8.3 Mathematical Operators 139

8.4 Eigenvalue Equations 139

8.5 The Quantum Harmonic Oscillator 143

8.6 Sturm-Liouville Theory 145

8.7 The Dirac Delta Function 148

8.8 Fourier Integrals 153

8.9 Fourier Series 156

8.10 Periodic Functions 161

9 Matrix Formulation of the Eigenvalue Problem 179

9.1 Reformulating the Eigenvalue Problem 180

9.2 Systems of Linear Equations 180

9.3 Back to the Eigenvalue Problem 182

9.4 Coupled Harmonic Oscillators 193

9.5 A Rotating Rigid Body 195

10 Variational Principles 207

10.1 Fermat's Principle 207

10.2 Another Variational Calculation 209

10.3 The Euler-Lagrange Equation 211

Appendix A Vector Relations 227

A.1 Vector Identities 227

A.2 Integral Theorems 227

A.3 The Functions of Vector Calculus 228

Appendix B Fundamental Equations of Physics 229

B.1 Poisson's Equation 229

B.2 Laplace's Equation 229

B.3 Maxwell's Equations 229

B.4 Time-Dependent Schrodinger Equation 230

Appendix C Some Useful Integrals and Sums 231

C.1 Integrals 231

C.2 Sums 234

Appendix D Algebraic Equations 235

D.1 Quadratic Equation 235

D.2 Cubic Equation 235.

Notes:

Includes bibliographical references (pages [237]-240) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Emma Louise McClellan Fund.

ISBN:

0387953426

OCLC:

47667272