Waves and oscillations : a prelude to quantum mechanics

Select an Action

Place Hold(s)
Add to My Lists
Email
Print

Title:

Personal Author:

Smith, Walter Fox

Publication Information:

New York : Oxford University Press, 2010

Physical Description:

xiv, 399 p. : ill. ; 26 cm.

ISBN:

9780195393491

Subject Term:

Wave equation

Mathematical physics

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010293106	QC174.26.W28 S655 2010	Open Access Book	Book	Searching... Unknown

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.

Author Notes

Walter Fox Smith is Professor of Physics at Haverford at Haverford College and a member of the American Physical Society and the American Association of Physics Teachers.

Learning Tools Used in This Book	p. ix
1 Simple Harmonic Motion	p. 1
1.1 Sinusoidal oscillations are everywhere	p. 1
1.2 The physics and mathematics behind simple sinusoidal motion	p. 3
1.3 Important parameters and adjustable constants of simple harmonic motion	p. 5
1.4 Mass on a spring	p. 8
1.5 Electrical oscillators	p. 10
1.6 Review of Taylor series approximations	p. 12
1.7 Euler's equation	p. 13
1.8 Review of complex numbers	p. 14
1.9 Complex exponential notation for oscillatory motion	p. 16
1.10 The complex representation for AC circuits	p. 18
1.11 Another important complex function: The quantum mechanical wavefunction	p. 24
1.12 Pure sinusoidal oscillations and uncertainty principles	p. 26
Concept and skill inventory	p. 29
Problems	p. 31
2 Examples of Simple Harmonic Motion	p. 39
2.1 Requirements for harmonic oscillation	p. 39
2.2 Pendulums	p. 40
2.3 Elastic deformations and Young's modulus	p. 42
2.4 Shear	p. 47
2.5 Torsion and torsional oscillators	p. 49
2.6 Bending and Cantilevers	p. 52
Concept and skill inventory	p. 56
Problems	p. 58
3 Damped Oscillations	p. 64
3.1 Damped mechanical oscillators	p. 64
3.2 Damped electrical oscillators	p. 68
3.3 Exponential decay of energy	p. 69
3.4 The quality factor	p. 70
3.5 Underdamped, overdamped, and critically damped behavior	p. 72
3.6 Types of damping	p. 74
Concept and skill inventory	p. 76
Problems	p. 77
4 Driven Oscillations and Resonance	p. 84
4.1 Resonance	p. 84
4.2 Effects of damping	p. 91
4.3 Energy flow	p. 95
4.4 Linear differential equations the superposition principle for driven systems, and the response to multiple drive forces	p. 99
4.5 Transients	p. 101
4.6 Electrical resonance	p. 104
4.7 Other examples of resonance: MRT and other spectroscopies	p. 107
4.8 Nonlinear oscillators and chaos	p. 114
Concept and skill inventory	p. 128
Problems	p. 129
5 Symmetric Coupled Oscillators and Hilbert Space	p. 137
5.1 Beats: An aside?	p. 137
5.2 Two symmetric coupled oscillators: Equations of motion	p. 139
5.3 Normal modes	p. 142
5.4 Superposing normal modes	p. 146
5.5 Normal mode analysis, and normal modes as an alternate description of reality	p. 149
5.6 Hilbert space and bra-ket notation	p. 153
5.7 The analogy between coupled oscillators and molecular energy levels	p. 163
5.8 Nonzero initial velocities	p. 165
5.9 Damped, driven coupled oscillators	p. 166
Concept and skill inventory	p. 168
Problems	p. 170
6 Asymmetric Coupled Oscillators and the Eigenvalue Equation	p. 179
6.1 Matrix math	p. 179
6.2 Equations of motion and the eigenvalue equation	p. 182
6.3 Procedure for solving the eigenvalue equation	p. 186
6.4 Systems with more than two objects	p. 191
6.5 Normal mode analysis for multi-object, asymmetrical systems	p. 194
6.6 More matrix math	p. 198
6.7 Orthogonality of normal modes, normal mode coordinates, degeneracy, and scaling of Hilbert space for unequal masses	p. 201
Concept and skill inventory	p. 208
Problems	p. 210
7 String Theory	p. 216
7.1 The beaded string	p. 216
7.2 Standing wave guess: Boundary conditions quantize the allowed frequencies	p. 219
7.3 The highest possible frequency; connection to waves in a crystalline solid	p. 222
7.4 Normal mode analysis for the beaded string	p. 226
7.5 Longitudinal oscillations	p. 227
7.6 The continuous string	p. 230
7.7 Normal mode analysis for continuous systems	p. 231
7.8 k-space	p. 234
Concept and skill inventor	p. 236
Problems	p. 236
8 Fourier Analysis	p. 246
8.1 Introduction	p. 246
8.2 The Fourier Expansion	p. 247
8.3 Expansions using nonnormalized orthogonal basis functions	p. 250
8.4 Finding the coefficients in the Fourier series expansion	p. 251
8.5 Fourier Transforms and the meaning of negative frequency	p. 254
8.6 The Discrete Fourier Transform (DFT)	p. 258
8.7 Some applications of Fourier Analysis	p. 265
Concept and skill inventory	p. 267
Problems	p. 268
9 Traveling Waves	p. 280
9.1 Introduction	p. 280
9.2 The wave equation	p. 280
9.3 Traveling sinusoidal waves	p. 284
9.4 The superposition principle for traveling waves	p. 285
9.5 Electromagnetic waves in vacuum	p. 287
9.6 Electromagnetic waves in matter	p. 296
9.7 Waves on transmission lines	p. 301
9.8 Sound waves	p. 305
9.9 Musical instruments based on tubes	p. 314
9.10 Power carried by rope and electromagnetic waves; RMS amplitudes	p. 316
9.11 Intensity of sound waves; decibels	p. 320
9.12 Dispersion relations and group velocity	p. 323
Concept and skill inventory	p. 332
Problems	p. 334
10 Waves at Interfaces	p. 343
10.1 Reflections and the idea of boundary conditions	p. 343
10.2 Transmitted waves	p. 349
10.3 Characteristic impedances for mechanical systems	p. 352
10.4 ôUniversalö expressions for transmission and reflection	p. 356
10.5 Reflected and transmitted waves for transmission lines	p. 359
10.6 Reflection and transmission for electromagnetic waves in matter: Normal incidence	p. 364
10.7 Reflection and transmission for sound waves, and summary of isomorphisms	p. 367
10.8 Snell's Law	p. 368
10.9 Total internal reflection and evanescent waves	p. 371
Concept and skill inventory	p. 378
Problems	p. 379
Appendix A Group Velocity for an Arbitrary Envelope Function	p. 388
Index	p. 393

Available:*

On Order

Summary

Summary

Author Notes

Table of Contents