Waves and Oscillations
A Prelude to Quantum Mechanics
Walter Fox Smith
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.
Readership : The primary readership for this text is sophomore- and junior-level majors in physics. The book is written to serve as the primary text for a sophomore- or junior-level physics course on waves and oscillations, or perhaps on mathematical methods.
Courses on waves and oscillations at the sophomore level are offered by about 40% of the top ten universities and also the top ten liberal arts colleges (as ranked by US News & World Report magazine). In particular, such courses are offered at Harvard, MIT, Stanford, CalTech, Columbia, Williams,
Swarthmore, Wellesley, Middlebury, Wesleyan, and Haverford. Enrolments in these courses are not readily available, but I estimate the typical enrollment at liberal arts colleges to be 15 students, and the typical enrollment at universities to be 25 students.
The principal markets include all countries where textbooks in English are used at the sophomore level in physics. If the text is sufficiently successful, we could of course consider translated editions.
Learning tools used in this book
1. Simple Harmonic Motion
1.1 Sinusoidal Oscillations are Everywhere
1.2 The physics and mathematics behind simple harmonic motion
1.3 Important parameters and adjustable constant of simple harmonic motion
1.4 Mass on a spring
1.5
Electrical oscillators
1.6 Review of Taylor Series Approximations
1.7 Euler's equation
1.8 Review of complex numbers
1.9 Complex exponential notation for oscillatory motion
1.10 The complex representation for AC circuits
1.11 Another important complex function: The quantum
mechanical wavefunction
1.12 Pure sinusoidal oscillations and uncertainty principles
Concept and skill inventory
Problems
2. Examples of Simple Harmonic Motion
2.1 Requirements for harmonic oscillation
2.2 Pendulums
2.3 Elastic deformations and Young's modulus
2.4
Shear
2.5 Torsion and Torsional Oscillators
2.6 Bending and Cantilevers
Concept and skill inventory
Problems
3. Damped oscillations
3.1 Damped mechanical oscillators
3.2 Damped electrical oscillators
3.3 Exponential decay of energy
3.4 The Quality Factor
3.5
Underdamped, overdamped, and critically damped behavior
3.6 Types of damping
Concept and skill inventory
Problems
4. Driven Oscillations and Resonance
4.1 Resonance
4.2 Effects of damping
4.3 Energy flow
4.4 Linear differential equations, the superposition principle
for driven systems, and the response to multiple drive forces
4.5 Transients
4.6 Electrical resonance
4.7 Other examples of resonance: MRI and other spectroscopies
4.8 Non-linear oscillators and chaos
Concept and skill inventory
Problems
5. Symmetric coupled oscillators
and Hilbert space
5.1 Beats: An aside?
5.2 Two symmetric coupled oscillators: equations of motion
5.3 Normal modes
5.4 Superposing normal modes
5.5 Normal mode analysis, and normal modes as an alternate description of reality
5.6 Hilbert Space and bra-ket notation
5.7 The
analogy between coupled oscillators and molecular energy levels
5.8 Non-zero initial velocities
5.9 Damped, driven coupled oscillators
Concept and skill inventory
Problems
6. Asymmetric coupled oscillators and the eigenvalue equation
6.1 Matrix math
6.2 Equations of
motion and the eigenvalue equation
6.3 Procedure for solving the eigenvalue equation
6.4 Systems with more than two objects
6.5 Normal mode analysis for mulit-object, asymmetrical systems
6.6 More matrix math
6.7 Orthogonality of normal modes, normal mode coordinates, degeneracy,
and scaling of Hilbert space for unequal masses Concept and skill inventory
Concept and skill inventory
Problems
7. String theory
7.1 The beaded string
7.2 Standing wave guess: Boundary conditions quantize the allowed frequencies
7.3 The highest possible frequency; connection
to waves in a crystalline solid
7.4 Normal mode analysis for the beaded string
7.5 Longitudinal oscillations
7.6 The continuous string
7.7 Normal mode analysis for continuous systems
7.8 k-space
Concept and skill inventory
Problems
8. Fourier analysis
8.1
Introduction
8.2 The Fourier Expansion
8.3 Expansions using non-normalized orthogonal basis functions
8.4 Finding the coefficients in the Fourier expansion
8.5 Fourier Transforms and the meaning of negative frequency
8.6 The Discrete Fourier Transform (DFT)
8.7 Some
applications of Fourier analysis
Concept and skill inventory
Problems
9. Traveling waves
9.1 Introduction
9.2 The Wave Equation
9.3 Traveling sinusoidal waves
9.4 The Superposition Principle for traveling waves
9.5 Electromagnetic waves in vacuum
9.6
Electromagnetic waves in matter
9.7 Waves on transmission lines
9.8 Sound Waves
9.9 Musical Instruments based on tubes
9.10 Power carried by rope and electromagnetic waves; RMS amplitudes
9.11 Intensity of sound waves; decibels
9.12 Dispersion relations and group
velocity
Concept and skill inventory
Problems
10. Waves at interfaces
10.1 Reflections and the idea of boundary conditions
10.2 Transmitted waves
10.3 Characteristic impedances for mechanical systems
10.4 "Universal" expressions for transmission and reflection
10.5
Reflected and transmitted waves for transmission lines
10.6 Reflection and transmission for electromagnetic waves in matter: normal incidence
10.7 Reflection and transmission for sound waves, and summary of isomorphisms
10.8 Snell's Law
10.9 Total Internal Reflection and evanescent
waves
Concept and skill inventory
Problems
Appendix A: Group velocity for an arbitrary envelope function
Index
There are no Instructor/Student Resources available at this time.
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.
Special Features
- Unique emphasis on the connections between quantum mechanics and classical waves and oscillations.
- Provides a strong grounding in the concepts and techniques that students will need in a subsequent study of quantum mechanics.
- Actively engages the reader through in-text concept
tests, and other tools that promote metacognition.
