Waves and Oscillations: A Prelude to Quantum Mechanics

Contents......Page 12
Learning Tools Used in This Book......Page 10
1.1 Sinusoidal oscillations are everywhere......Page 18
1.2 The physics and mathematics behind simple sinusoidal motion......Page 20
1.3 Important parameters and adjustable constants of simple harmonic motion......Page 22
1.4 Mass on a spring......Page 25
1.5 Electrical oscillators......Page 27
1.6 Review of Taylor series approximations......Page 29
1.7 Euler’s equation......Page 30
1.8 Review of complex numbers......Page 31
1.9 Complex exponential notation for oscillatory motion......Page 33
1.10 The complex representation for AC circuits......Page 35
1.11 Another important complex function: The quantum mechanical wavefunction......Page 41
1.12 Pure sinusoidal oscillations and uncertainty principles......Page 43
Concept and skill inventory......Page 46
Problems......Page 48
2.1 Requirements for harmonic oscillation......Page 56
2.2 Pendulums......Page 57
2.3 Elastic deformations and Young’s modulus......Page 59
2.4 Shear......Page 64
2.5 Torsion and torsional oscillators......Page 66
2.6 Bending and Cantilevers......Page 69
Concept and skill inventory......Page 73
Problems......Page 75
3.1 Damped mechanical oscillators......Page 81
3.2 Damped electrical oscillators......Page 85
3.3 Exponential decay of energy......Page 86
3.4 The quality factor......Page 87
3.5 Underdamped, overdamped, and critically damped behavior......Page 89
3.6 Types of damping......Page 91
Concept and skill inventory......Page 93
Problems......Page 94
4.1 Resonance......Page 101
4.2 Effects of damping......Page 108
4.3 Energy flow......Page 112
4.4 Linear differential equations, the superposition principle for driven systems, and the response to multiple drive forces......Page 116
4.5 Transients......Page 118
4.6 Electrical resonance......Page 121
4.7 Other examples of resonance: MRI and other spectroscopies......Page 124
4.8 Nonlinear oscillators and chaos......Page 131
Concept and skill inventory......Page 145
Problems......Page 146
5.1 Beats: An aside?......Page 154
5.2 Two symmetric coupled oscillators: Equations of motion......Page 156
5.3 Normal modes......Page 159
5.4 Superposing normal modes......Page 163
5.5 Normal mode analysis, and normal modes as an alternate description of reality......Page 166
5.6 Hilbert space and bra-ket notation......Page 170
5.7 The analogy between coupled oscillators and molecular energy levels......Page 180
5.8 Nonzero initial velocities......Page 182
5.9 Damped, driven coupled oscillators......Page 183
Concept and skill inventory......Page 185
Problems......Page 187
6.1 Matrix math......Page 196
6.2 Equations of motion and the eigenvalue equation......Page 199
6.3 Procedure for solving the eigenvalue equation......Page 203
6.4 Systems with more than two objects......Page 208
6.5 Normal mode analysis for multi-object, asymmetrical systems......Page 211
6.6 More matrix math......Page 215
6.7 Orthogonality of normal modes, normal mode coordinates, degeneracy, and scaling of Hilbert space for unequal masses......Page 218
Concept and skill inventory......Page 225
Problems......Page 227
7.1 The beaded string......Page 233
7.2 Standing wave guess: Boundary conditions quantize the allowed frequencies......Page 236
7.3 The highest possible frequency; connection to waves in a crystalline solid......Page 239
7.4 Normal mode analysis for the beaded string......Page 243
7.5 Longitudinal oscillations......Page 244
7.6 The continuous string......Page 247
7.7 Normal mode analysis for continuous systems......Page 248
7.8 k-space......Page 251
Problems......Page 253
8.1 Introduction......Page 263
8.2 The Fourier Expansion......Page 264
8.3 Expansions using nonnormalized orthogonal basis functions......Page 267
8.4 Finding the coefficients in the Fourier series expansion......Page 268
8.5 Fourier Transforms and the meaning of negative frequency......Page 271
8.6 The Discrete Fourier Transform (DFT)......Page 275
8.7 Some applications of Fourier Analysis......Page 282
Concept and skill inventory......Page 284
Problems......Page 285
9.2 The wave equation......Page 297
9.3 Traveling sinusoidal waves......Page 301
9.4 The superposition principle for traveling waves......Page 302
9.5 Electromagnetic waves in vacuum......Page 304
9.6 Electromagnetic waves in matter......Page 313
9.7 Waves on transmission lines......Page 318
9.8 Sound waves......Page 322
9.9 Musical instruments based on tubes......Page 331
9.10 Power carried by rope and electromagnetic waves; RMS amplitudes......Page 333
9.11 Intensity of sound waves; decibels......Page 337
9.12 Dispersion relations and group velocity......Page 340
Concept and skill inventory......Page 349
Problems......Page 351
10.1 Reflections and the idea of boundary conditions......Page 360
10.2 Transmitted waves......Page 366
10.3 Characteristic impedances for mechanical systems......Page 369
10.4 “Universal” expressions for transmission and reflection......Page 373
10.5 Reflected and transmitted waves for transmission lines......Page 376
10.6 Reflection and transmission for electromagnetic waves in matter: Normal incidence......Page 381
10.7 Reflection and transmission for sound waves, and summary of isomorphisms......Page 384
10.8 Snell’s Law......Page 385
10.9 Total internal reflection and evanescent waves......Page 388
Concept and skill inventory......Page 395
Problems......Page 396
Appendix A: Group Velocity for an Arbitrary Envelope Function......Page 405
C......Page 410
E......Page 411
I......Page 412
M......Page 413
Q......Page 414
S......Page 415
Z......Page 416