Waves and Oscillations: A Prelude to Quantum Mechanics

by Walter Fox Smith

Overview

Waves and oscillations permeate virtually every field of current physics research, are central to chemistry, and are essential to much of engineering. Furthermore, the concepts and mathematical techniques used for serious study of waves and oscillations form the foundation for quantum mechanics. Once they have mastered these ideas in a classical context, students will be ready to focus on the challenging concepts of quantum mechanics when they encounter them, rather than struggling with techniques.

This lively textbook gives a thorough grounding in complex exponentials and the key aspects of differential equations and matrix math; no prior experience is assumed. The parallels between normal mode analysis, orthogonal function analysis (especially Fourier analysis), and superpositions of quantum states are clearly drawn, without actually getting into the quantum mechanics. An in-depth, accessible introduction to Hilbert space and bra-ket notation begins in Chapter 5 (on symmetrical coupled oscillators), emphasizing the analogy with conventional dot products, and continues in subsequent chapters.

Connections to current physics research (atomic force microscopy, chaos, supersolids, micro electro-mechanical systems (MEMS), magnetic resonance imaging, carbon nanotubes, and more) are highlighted in the text and in end-of-chapter problems, and are frequently updated in the associated website.

The book actively engages readers with a refreshing writing style and a set of carefully applied learning tools, such as in-text concept tests, "your turn" boxes (in which the student fills in one or two steps of a derivation), concept and skill inventories for each chapter, and "wrong way" problems in which the student explains the flaw in a line of reasoning. These tools promote self-awareness of the learning process.

The associated website features custom-developed applets, video and audio recordings, additional problems, and links to related current research. The instructor-only part includes difficulty ratings for problems, optional hints, full solutions, and additional support materials.

Product Details

ISBN-13:	9780195393491
Publisher:	Oxford University Press
Publication date:	05/20/2010
Edition description:	New Edition
Pages:	432
Product dimensions:	7.20(w) x 10.00(h) x 1.00(d)

About the Author

Walter Fox Smith combines his passions for teaching and nanophysics research at Haverford College. His research centers on the photoelectronic properties of self-assembling molecular electronics. He is also known as "the singing physics professor", thanks to his compositions and performances in the classroom and at social gatherings of physicists.

Table of Contents

Learning Tools Used in This Book ix

1 Simple Harmonic Motion 1

1.1 Sinusoidal oscillations are everywhere 1

1.2 The physics and mathematics behind simple sinusoidal motion 3

1.3 Important parameters and adjustable constants of simple harmonic motion 5

1.4 Mass on a spring 8

1.5 Electrical oscillators 10

1.6 Review of Taylor series approximations 12

1.7 Euler's equation 13

1.8 Review of complex numbers 14

1.9 Complex exponential notation for oscillatory motion 16

1.10 The complex representation for AC circuits 18

1.11 Another important complex function: The quantum mechanical wavefunction 24

1.12 Pure sinusoidal oscillations and uncertainty principles 26

Concept and skill inventory 29

Problems 31

2 Examples of Simple Harmonic Motion 39

2.1 Requirements for harmonic oscillation 39

2.2 Pendulums 40

2.3 Elastic deformations and Young's modulus 42

2.4 Shear 47

2.5 Torsion and torsional oscillators 49

2.6 Bending and Cantilevers 52

Concept and skill inventory 56

Problems 58

3 Damped Oscillations 64

3.1 Damped mechanical oscillators 64

3.2 Damped electrical oscillators 68

3.3 Exponential decay of energy 69

3.4 The quality factor 70

3.5 Underdamped, overdamped, and critically damped behavior 72

3.6 Types of damping 74

Concept and skill inventory 76

Problems 77

4 Driven Oscillations and Resonance 84

4.1 Resonance 84

4.2 Effects of damping 91

4.3 Energy flow 95

4.4 Linear differential equations the superposition principle for driven systems, and the response to multiple drive forces 99

4.5 Transients 101

4.6 Electrical resonance 104

4.7 Other examples of resonance: MRT and other spectroscopies 107

4.8 Nonlinear oscillators and chaos 114

Concept and skill inventory 128

Problems 129

5 Symmetric Coupled Oscillators and Hilbert Space 137

5.1 Beats: An aside? 137

5.2 Two symmetric coupled oscillators: Equations of motion 139

5.3 Normal modes 142

5.4 Superposing normal modes 146

5.5 Normal mode analysis, and normal modes as an alternate description of reality 149

5.6 Hilbert space and bra-ket notation 153

5.7 The analogy between coupled oscillators and molecular energy levels 163

5.8 Nonzero initial velocities 165

5.9 Damped, driven coupled oscillators 166

Concept and skill inventory 168

Problems 170

6 Asymmetric Coupled Oscillators and the Eigenvalue Equation 179

6.1 Matrix math 179

6.2 Equations of motion and the eigenvalue equation 182

6.3 Procedure for solving the eigenvalue equation 186

6.4 Systems with more than two objects 191

6.5 Normal mode analysis for multi-object, asymmetrical systems 194

6.6 More matrix math 198

6.7 Orthogonality of normal modes, normal mode coordinates, degeneracy, and scaling of Hilbert space for unequal masses 201

Concept and skill inventory 208

Problems 210

7 String Theory 216

7.1 The beaded string 216

7.2 Standing wave guess: Boundary conditions quantize the allowed frequencies 219

7.3 The highest possible frequency; connection to waves in a crystalline solid 222

7.4 Normal mode analysis for the beaded string 226

7.5 Longitudinal oscillations 227

7.6 The continuous string 230

7.7 Normal mode analysis for continuous systems 231

7.8 k-space 234

Concept and skill inventor 236

Problems 236

8 Fourier Analysis 246

8.1 Introduction 246

8.2 The Fourier Expansion 247

8.3 Expansions using nonnormalized orthogonal basis functions 250

8.4 Finding the coefficients in the Fourier series expansion 251

8.5 Fourier Transforms and the meaning of negative frequency 254

8.6 The Discrete Fourier Transform (DFT) 258

8.7 Some applications of Fourier Analysis 265

Concept and skill inventory 267

Problems 268

9 Traveling Waves 280

9.1 Introduction 280

9.2 The wave equation 280

9.3 Traveling sinusoidal waves 284

9.4 The superposition principle for traveling waves 285

9.5 Electromagnetic waves in vacuum 287

9.6 Electromagnetic waves in matter 296

9.7 Waves on transmission lines 301

9.8 Sound waves 305

9.9 Musical instruments based on tubes 314

9.10 Power carried by rope and electromagnetic waves; RMS amplitudes 316

9.11 Intensity of sound waves; decibels 320

9.12 Dispersion relations and group velocity 323

Concept and skill inventory 332

Problems 334

10 Waves at Interfaces 343

10.1 Reflections and the idea of boundary conditions 343

10.2 Transmitted waves 349

10.3 Characteristic impedances for mechanical systems 352

10.4 "Universal" expressions for transmission and reflection 356

10.5 Reflected and transmitted waves for transmission lines 359

10.6 Reflection and transmission for electromagnetic waves in matter: Normal incidence 364

10.7 Reflection and transmission for sound waves, and summary of isomorphisms 367

10.8 Snell's Law 368

10.9 Total internal reflection and evanescent waves 371

Concept and skill inventory 378

Problems 379

Appendix A Group Velocity for an Arbitrary Envelope Function 388

Index 393

Editorial Reviews

"This book provides a rigorous introduction to a host of wave and oscillation phenomena and their real-world and research applications, while at the same time laying the mathematical and conceptual groundwork for upper level physics classes. I strongly recommend this book for use in a course that serves as a bridge for students who are making the transition from introductory courses into an upper-level curriculum."
Prof. Nathan Harshman, American University

From the Publisher