Physics of optoelectronics

Contents......Page 5
1: Introduction to Semiconductor Lasers......Page 17
1.1 Basic Components and the Role of Feedback......Page 18
1.2 Basic Properties of Lasers......Page 20
1.2.1 Wavelength and Energy......Page 21
1.2.3 Monochromaticity and Brightness......Page 22
1.3.1 In-Plane and Edge-Emitting Lasers......Page 23
1.3.2 VCSEL......Page 24
1.3.7 Gas Laser......Page 25
1.4.1 Classification of Matter......Page 26
Solids......Page 27
1.4.2 Bonding and the Periodic Table......Page 28
1.5.1 Intuitive Origin of Bands......Page 30
1.5.3 Introduction to Transitions......Page 33
1.5.4 Introduction to Band Edge Diagrams......Page 34
1.5.5 Bandgap States and Defects......Page 36
1.5.6 Recombination Mechanisms......Page 37
1.6.1 Junction Technology......Page 40
1.6.2 Band-Edge Diagrams and the pn Junction......Page 41
1.6.3 Nonequilibrium Statistics......Page 42
1.7.1 Particle–Wave Nature of Light......Page 44
1.7.2 Classical Method of Controlling Light......Page 45
1.7.3 The Ridge Waveguide......Page 47
1.7.4 The Confinement Factor......Page 49
1.8.1 Brief Essay on Noise for Systems......Page 50
1.8.2 Johnson Noise......Page 51
1.8.4 The Origin of Shot Noise......Page 54
1.8.5 The Magnitude of the Shot Noise......Page 57
1.8.6 Introduction to Noise in Optics......Page 58
1.9 Review Exercises......Page 59
Principles and Systems......Page 61
Semiconductors......Page 62
2.1 Introduction to the Rate Equations......Page 63
2.1.1 The Simplest Rate Equations......Page 64
2.1.2 Optical Confinement Factor......Page 65
2.1.4 The Pump Term and the Internal Quantum Efficiency......Page 66
2.1.5 Recombination Terms......Page 67
2.1.6 Spontaneous Emission Term......Page 69
2.1.7 The Optical Loss Term......Page 70
2.2.1 Temporal Gain......Page 72
2.2.2 Single Pass Gain......Page 73
2.2.3 Material Gain......Page 74
2.2.5 Introduction to the Energy Dependence of Gain......Page 76
2.2.6 The Phenomenological Rate Equations......Page 77
Case 1 Below Lasing Threshold......Page 80
Case 2 Above Lasing Threshold......Page 81
2.3.2 Comment on the Threshold Density......Page 83
Case 1 P–I Below Threshold......Page 84
2.3.4 Power versus Voltage......Page 85
2.4.1 Internal Relations......Page 86
2.4.2 External Relations......Page 89
2.5.1 Introduction to the Response Function and Bandwidth......Page 90
2.5.2 Small Signal Analysis......Page 92
2.6 Introduction to RIN and the Weiner–Khintchine Theorem......Page 95
2.6.2 Basic Assumptions......Page 96
2.6.3 The Fluctuation-Dissipation Theorem......Page 98
2.6.4 Definition of Relative Intensity Noise as a Correlation......Page 100
2.6.5 The Weiner–Khintchine Theorem......Page 102
2.6.6 Alternate Derivations of the Weiner–Khintchine Formula......Page 103
2.6.8 Alternate Definitions for RIN......Page 105
2.7 Relative Intensity Noise for the Semiconductor Laser......Page 106
2.7.1 Rate Equations with Langevin Noise Sources and the Spectral Density......Page 107
2.7.2 Langevin–Noise Correlation......Page 110
2.7.3 The Relative Intensity Noise......Page 113
2.8 Review Exercises......Page 115
Stochastic Processes and Statistical Theory......Page 122
3: Classical Electromagnetics and Lasers......Page 123
3.1.1 Discussion of Maxwell’s Equations and Related Quantities......Page 124
3.1.2 Relation between Electric and Magnetic Fields in Vacuum......Page 127
3.1.3 Relation between Electric and Magnetic Fields in Dielectrics......Page 128
3.1.4 General Form of the Complex Traveling Wave......Page 129
3.2.1 Derivation of the Wave Equation......Page 130
3.2.2 The Complex Wave Vector......Page 131
3.2.3 Definitions for Complex Index, Permittivity and Wave Vector......Page 132
3.2.4 The Meaning of kn......Page 133
3.2.5 Approximate Expression for the Wave Vector......Page 134
3.2.6 Approximate Expressions for the Refractive Index and Permittivity......Page 135
3.2.7 The Susceptibility and the Pump......Page 136
Case 1 No Free Surface Charge......Page 138
3.3.2 Electric Fields Parallel to the Surface......Page 140
3.3.4 The Tangential Magnetic Field......Page 141
3.3.5 Magnetic Field Perpendicular to the Interface (Without Magnetization)......Page 142
3.3.7 General Relations and Summary......Page 143
3.4.1 The Boundary Conditions......Page 145
3.4.2 The Law of Reflection......Page 147
3.4.3 Fresnel Reflectivity and Transmissivity for TE Fields......Page 148
3.4.5 Graph of the Reflectivity Versus Angle......Page 150
3.5.1 Introduction to Power Transport for Real Fields......Page 151
3.5.2 Power Transport and Energy Storage Mechanisms......Page 153
3.5.3 Poynting Vector for Complex Fields......Page 156
3.5.4 Power Flow Across a Boundary......Page 157
3.6.1 Introduction to Scattering Theory......Page 160
3.6.2 The Power-Amplitudes......Page 162
3.6.3 Reflection and Transmission Coefficients......Page 164
3.6.4 Scattering Matrices......Page 167
3.6.5 The Transfer Matrix......Page 170
3.6.6 Examples Using Scattering and Transfer Matrices......Page 172
3.7.1 Implications of the Transfer Matrix for the Fabry-Perot Laser......Page 176
3.7.2 Longitudinal Modes and the Threshold Condition......Page 180
Case 1 Imaginary Part Produces the Wavelength of the Longitudinal Modes......Page 181
3.7.3 Line Narrowing......Page 184
3.8 Introduction to Waveguides......Page 186
3.8.2 Introduction to EM Waves for Waveguiding......Page 187
3.8.3 The Triangle Relation......Page 188
3.8.4 The Cut-off Condition from Geometric Optics......Page 189
3.8.5 The Waveguide Refractive Index......Page 191
3.9.1 The Wave Equations......Page 192
3.9.2 The General Solutions......Page 194
3.9.4 The Solutions......Page 195
3.9.5 An Expression for Cut-off......Page 198
3.10 Dispersion in Waveguides......Page 199
3.10.1 The Dispersion Diagram......Page 200
3.10.2 A Formula for Dispersion......Page 201
3.11 The Displacement Current and Photoconduction......Page 202
3.11.1 Displacement Current......Page 203
3.11.3 Voltage Induced by Moving Charge......Page 205
3.12 Review Exercises......Page 206
Waveguides and Optical Filters......Page 212
4.1 Vector and Hilbert Spaces......Page 213
4.1.2 Inner Product, Norm, and Metric......Page 214
4.2 Dirac Notation and Euclidean Vector Spaces......Page 215
4.2.1 Kets, Bras, and Brackets for Euclidean Space......Page 216
4.2.2 Basis, Completeness, and Closure for Euclidean Space......Page 217
4.2.3 The Euclidean Dual Vector Space......Page 218
4.3.1 Hilbert Space of Functions with Discrete Basis Vectors......Page 219
4.3.2 The Continuous Basis Set of Functions......Page 221
4.3.3 Projecting Functions into Coordinate Space......Page 222
4.3.4 The Sine Basis Set......Page 226
4.3.6 The Fourier Series Basis Set......Page 227
4.3.7 The Fourier Transform......Page 228
4.4 The Grahm–Schmidt Orthonormalization Procedure......Page 230
4.5.1 Definition of a Linear Operator and Matrices......Page 231
4.5.2 A Matrix Equation......Page 232
4.5.4 Introduction to the Inverse of an Operator......Page 233
4.5.6 Trace......Page 234
4.5.8 Basis Vector Expansion of a Linear Operator......Page 235
4.5.9 The Hilbert Space of Linear Operators......Page 236
4.5.10 A Note on Matrices......Page 237
4.6 An Algebra of Operators and Commutators......Page 238
4.7.1 Tensor Product Spaces......Page 240
4.7.2 Operators......Page 241
Case 1:…......Page 242
4.7.4 The Matrix Representation of Basis Vectors for Direct Product Space......Page 243
4.8.1 Orthogonal Rotation Matrices......Page 245
4.8.2 Unitary Transformations......Page 246
4.8.3 Visualizing Unitary Transformations......Page 247
4.8.4 Similarity Transformations......Page 248
4.8.5 Trace and Determinant......Page 249
4.9.1 Adjoint, Self-Adjoint and Hermitian Operators......Page 250
4.9.3 Eigenvectors and Eigenvalues for Hermitian Operators......Page 252
4.9.4 The Heisenberg Uncertainty Relation......Page 255
4.10 A Relation Between Unitary and Hermitian Operators......Page 257
4.11.1 The Exponential Form of the Translation Operator......Page 258
4.11.3 Translation of the Position-Coordinate Ket......Page 260
4.12.1 Rotating Functions......Page 261
4.12.2 The Rotation Operator......Page 262
4.13 Dyadic Notation......Page 264
4.14 Minkowski Space......Page 265
4.15 Review Exercises......Page 268
Involved......Page 272
5.1 Introduction to Generalized Coordinates......Page 273
5.1.2 Generalized Coordinates......Page 274
5.1.3 Phase Space Coordinates......Page 276
5.2.1 Lagrange’s Equation from a Variational Principle......Page 277
5.2.3 Hamilton’s Canonical Equations......Page 280
5.3 Classical Commutation Relations......Page 282
5.4.1 Concepts for the Lagrangian and Hamiltonian Density......Page 284
5.4.2 The Lagrange Density for 1-D Wave Motion......Page 287
5.5 Schrodinger Equation from a Lagrangian......Page 289
5.6 Linear Algebra and the Quantum Theory......Page 291
5.6.1 Observables and Hermitian Operators......Page 292
5.6.2 The Eigenstates......Page 293
5.6.4 Probability Interpretation......Page 295
5.6.5 The Average and Variance......Page 297
5.6.6 Motion of the Wave Function......Page 299
5.6.7 Collapse of the Wave Function......Page 300
5.6.8 Noncommuting Operators and the Heisenberg Uncertainty Relation......Page 302
5.6.9 Complete Sets of Observables......Page 304
5.7.1 Summary of Elementary Facts......Page 305
5.7.3 The Momentum Operator......Page 306
5.7.4 Developing the Hamiltonian Operator and the Schrodinger Wave Equation......Page 307
5.7.5 Infinitely Deep Quantum Well......Page 308
5.8.1 Introduction to the Classical and Quantum Harmonic Oscillators......Page 311
5.8.2 The Hamiltonian for the Quantum Harmonic Oscillator......Page 314
5.8.3 Introduction to the Operator Solution of the Harmonic Oscillator......Page 315
5.8.4 Ladder Operators in the Hamiltonian......Page 316
5.8.5 Properties of the Raising and Lowering Operators......Page 318
5.8.7 The Energy Eigenfunctions......Page 320
5.9.1 Discussion of the Schrodinger, Heisenberg and Interaction Representations......Page 322
5.9.2 The Schrodinger Representation......Page 324
5.9.3 Rate of Change of the Average of an Operator in the Schrodinger Picture......Page 326
5.9.4 Ehrenfest’s Theorem for the Schrodinger Representation......Page 327
5.9.5 The Heisenberg Representation......Page 328
5.9.6 The Heisenberg Equation......Page 329
5.9.8 The Interaction Representation......Page 331
5.10.1 Physical Concept......Page 333
5.10.2 Time Dependent Perturbation Theory Formalism in the Schrodinger Picture......Page 335
Case m=i......Page 338
Case…......Page 339
0th Order Approximation......Page 340
5.10.4 An Evolution Operator in the Interaction Representation......Page 341
5.11 Density Operator......Page 342
5.11.1 Introduction to the Density Operator......Page 343
5.11.2 The Density Operator and the Basis Expansion......Page 347
5.11.3 Ensemble and Quantum Mechanical Averages......Page 351
5.11.4 Loss of Coherence......Page 354
5.11.5 Some Properties......Page 356
5.12 Review Exercises......Page 357
Density Operator......Page 367
6.1 A Brief Overview of the Quantum Theory of Electromagnetic Fields......Page 369
6.2 The Classical Vector Potential and Gauges......Page 372
6.2.1 Relation between the Electromagnetic Fields and the Potential Functions......Page 373
6.2.2 The Fields in a Source-Free Region of Space......Page 376
6.2.3 Gauge Transformations......Page 377
6.2.4 Coulomb Gauge......Page 378
6.3 The Plane Wave Expansion of the Vector Potential and the Fields......Page 380
6.3.1 Boundary Conditions......Page 381
6.3.2 The Plane Wave Expansion......Page 383
6.3.4 Spatial-Temporal Modes......Page 386
6.4 The Quantum Fields......Page 388
6.4.1 The Quantized Vector Potential......Page 390
6.4.2 Quantizing the Electric and Magnetic Fields......Page 391
6.4.3 Other Basis Sets......Page 392
6.4.4 EM Fields with Quadrature Operators......Page 393
6.4.5 An Alternate Set of Quadrature Operators......Page 394
6.4.6 Phase Rotation Operator for the Quantized Electric Field......Page 395
6.4.7 Trouble with Amplitude and Phase Operators......Page 397
6.4.8 The Operator for the Poynting Vector......Page 398
6.5.1 The Classical Free-Field Hamiltonian......Page 399
6.5.4 The Schrodinger Equation for the EM Field......Page 405
6.5.2 The Quantum Mechanical Free-Field Hamiltonian......Page 402
6.5.3 The EM Hamiltonian in Terms of the Quadrature Operators......Page 404
6.6 Introduction to Fock States......Page 407
6.6.1 Introduction to Fock States......Page 408
6.6.2 The Fabry–Perot Resonator as an Example......Page 410
6.6.3 Creation and Annihilation Operators......Page 411
6.6.4 Comparison between Creation–Annihilation and Ladder Operators......Page 412
6.6.5 Introduction to the Fermion Fock States......Page 413
6.7 Fock States as Eigenstates of the EM Hamiltonian......Page 414
6.7.1 Coordinate Representation of Boson Wavefunctions......Page 415
6.7.2 Fock States as Energy Eigenstates......Page 418
6.8 Interpretation of Fock States......Page 420
6.8.2 Interpretation of the Coordinate Representation of Fock States......Page 421
6.8.3 Comparison between the Electron and EM Harmonic Oscillator......Page 422
6.8.4 An Uncertainty Relation between the Quadratures......Page 423
6.8.5 Fluctuations of the Electric and Magnetic Fields in Fock States......Page 425
6.9.1 The Electric Field in the Coherent State......Page 426
6.9.3 Normalized Quadrature Operators and the Wigner Plot......Page 429
6.9.4 Introduction to the Coherent State as a Displaced Vacuum in Phase Space......Page 430
6.9.5 Introduction to the Nature of Quantum Noise in the Coherent State......Page 433
6.9.6 Comments on the Theory......Page 434
6.10.1 The Coherent State in the Fock Basis Set......Page 435
6.10.2 The Poisson Distribution......Page 437
6.10.3 The Average and Variance of the Photon Number......Page 438
6.10.4 Signal-to-Noise Ratio......Page 439
6.10.5 Poisson Distribution from a Binomial Distribution......Page 440
6.11 Coherent States as Displaced Vacuum States......Page 441
6.11.1 The Displacement Operator......Page 442
6.11.2 Properties of the Displacement Operator......Page 443
Method 1 Coordinate Representation of the Coherent State Using the Annihilation Operator......Page 445
Method 2 Coordinate Representation of the Coherent State Using the Displacement Operator......Page 447
6.12 Quasi-Orthonormality, Closure and Trace for Coherent States......Page 449
6.12.2 Normalization......Page 450
6.12.3 Quasi-Orthogonality......Page 451
6.12.4 Closure......Page 452
6.12.5 Coherent State Expansion of a Fock State......Page 453
6.12.7 Trace of an Operator Using Coherent States......Page 454
6.13 Field Fluctuations in the Coherent State......Page 455
6.13.1 The Quadrature Uncertainty Relation for Coherent States......Page 456
6.13.2 Comparison of Variance for Coherent and Fock States......Page 457
6.14 Introduction to Squeezed States......Page 458
6.15 The Squeezing Operator and Squeezed States......Page 461
6.15.3 The Squeezed Creation and Annihilation Operators......Page 462
6.15.4 The Squeezed EM Quadrature Operators......Page 464
6.15.5 Variance of the EM Quadrature......Page 465
6.15.6 Coordinate Representation of Squeezed States......Page 467
6.16.1 The Average Electric Field in a Squeezed Coherent State......Page 469
6.16.2 The Variance of the Electric Field in a Squeezed Coherent State......Page 470
6.16.3 The Average of the Hamiltonian in a Squeezed Coherent State......Page 471
6.16.4 Photon Statistics for the Squeezed State......Page 473
6.17 The Wigner Distribution......Page 474
6.17.1 The Wigner Formula and an Example......Page 475
6.17.2 Derivation of the Wigner Formula......Page 477
Step 0 The Rotations......Page 478
Step 2 The Characteristic Function......Page 479
Step 3 Fourier Transform of the Quantum Probability Density Function......Page 480
Step 4 Equate the Results of Steps 2 and 3......Page 481
6.17.3 Example of the Wigner Function......Page 482
6.18 Measuring the Noise in Squeezed States......Page 483
6.19 Review Exercises......Page 486
Vector Potential and Gauges......Page 492
7: Matter–Light Interaction......Page 495
7.1.1 Comparison of the Classical and Quantum Mechanical Dipole......Page 496
7.1.2 The Quantum Mechanical Dipole Moment......Page 498
7.1.3 A Comment on Visualizing an Oscillating Electron in a Harmonic Potential......Page 500
7.2.1 The EM Interaction Potential......Page 502
7.2.2 The Integral for the Probability Amplitude......Page 503
7.2.3 Rotating Wave Approximation......Page 505
7.2.4 Absorption......Page 506
7.2.5 Emission......Page 507
7.2.6 Discussion of the Results......Page 509
7.3 Fermi’s Golden Rule......Page 510
7.3.1 Definition of the Density of States......Page 511
7.3.2 Equations for Fermi’s Golden Rule......Page 513
7.3.3 Introduction to Laser Gain......Page 516
Mechanics......Page 517
7.4.2 The EM Lagrangian and Hamiltonian......Page 518
7.4.3 4-Vector Form of Maxwell’s Equations from the Lagrangian......Page 519
Gauss’ Law......Page 521
Magnetic Monopole Relation......Page 522
7.5 The Classical Hamiltonian for Fields, Particles and Interactions......Page 523
7.5.1 The EM Hamiltonian Density......Page 524
7.5.2 The Canonical Field Momentum......Page 525
7.5.4 The Field and Interaction Hamiltonian......Page 526
7.5.5 The Hamiltonian for Fields, Particles and their Interactions......Page 528
7.5.6 Discussion......Page 530
7.6.1 Discussion of the Classical Interaction Energy......Page 532
7.6.2 Schrodinger’s Equation with the Matter–Light Interaction......Page 534
7.6.3 The Origin of the Dipole Operator......Page 535
7.7 Stimulated and Spontaneous Emission Using Fock States......Page 538
7.7.1 Restatement of Fermi’s Golden Rule......Page 539
7.7.2 The Dipole Approximation......Page 540
7.7.3 Calculate Matrix Elements......Page 541
7.7.4 Stimulated and Spontaneous Emission......Page 542
7.8 Introduction to Matter and Light as Systems......Page 543
7.8.1 The Complete System......Page 544
7.8.2 Introduction to Homogeneous Broadening......Page 546
7.9 Liouville Equation for the Density Operator......Page 547
7.9.1 The Liouville Equation Using the Full Hamiltonian......Page 548
7.9.2 The Liouville Equation Using a Phenomenological Relaxation Term......Page 552
7.10.1 Preliminaries......Page 554
7.10.2 Assumptions for the Density Matrix......Page 555
7.10.3 Liouville’s Equation for the Density Matrix without Thermal Equilibrium......Page 557
7.10.4 The Liouville Equation for the Density Matrix with Thermal Equilibrium......Page 559
7.10.6 The Carrier Relaxation Time......Page 560
7.11.1 Evaluating the Commutator......Page 561
7.11.2 Two Independent Equations......Page 564
7.11.3 The Optical Bloch Equations......Page 565
7.11.4 The Solutions......Page 567
7.12 Gain, Absorption and Index for Independent Two Level Atoms......Page 570
7.12.1 The Quantum Polarization and the Polarization Envelope Functions......Page 571
7.12.2 The Quantum Polarization and Macroscopic Quantities......Page 572
7.12.3 Comparing the Classical and Quantum Mechanical Polarization......Page 573
7.12.5 Quantum Mechanical Gain......Page 574
7.12.6 Discussion of Results......Page 576
7.13 Broadening Mechanisms......Page 578
7.13.1 Homogeneous Broadening......Page 579
7.13.2 Inhomogeneous Broadening......Page 580
7.13.3 Hole Burning......Page 581
7.14 Introduction to Jaynes-Cummings’ Model......Page 582
7.14.1 The Pauli Operators......Page 583
7.14.2 The Atomic Hamiltonian......Page 584
7.14.3 The Free-Field Hamiltonian......Page 585
7.14.4 The Interaction Hamiltonian......Page 586
7.14.5 Atomic and Interaction Hamiltonians Using Fermion Operators......Page 589
7.15 The Interaction Representation for the Jaynes-Cummings’ Model......Page 590
7.15.1 Atomic Creation and Annihilation Operators......Page 591
7.15.2 The Boson Creation and Annihilation Operators......Page 592
7.15.3 Interaction Representation of the Subsystem Density Operators......Page 593
7.15.4 The Interaction Representation of the Direct-Product Density Operator......Page 594
7.15.5 Rate Equation for the Density Operator in the Interaction Representation......Page 596
7.16 The Master Equation......Page 597
7.16.1 The System......Page 598
7.16.2 Multiple Reservoirs......Page 599
7.16.3 Dynamics and the Perturbation Expansion......Page 600
7.16.5 Reservoir Correlation Time and the Course Grain Derivative......Page 603
7.16.6 The Relaxation Term......Page 605
7.16.7 The Pauli Master Equation......Page 607
7.17.1 Some Introductory Comments......Page 610
7.17.2 Quantum Mechanical Fluctuation Dissipation Theorem......Page 611
7.17.3 The Average of the Langevin Noise Term......Page 615
7.18 Review Exercises......Page 617
Matter-Fields......Page 624
Reservoir Theory......Page 625
8.1 Effective Mass, Density of States and the Fermi Distribution......Page 627
8.1.1 Effective Mass......Page 628
8.1.3 The Fixed Endpoint Boundary Conditions......Page 630
8.1.4 The Periodic Boundary Condition......Page 632
8.1.5 The Density of k-States......Page 634
8.1.6 The Electron Density of Energy States for a 2-D Crystal......Page 635
8.1.7 Overlapping Bands......Page 637
8.1.8 Density of States from Fixed-Endpoint Boundary Conditions......Page 638
8.1.9 Changing Summations to Integrals......Page 639
8.1.10 A Brief Review of the Fermi-Dirac Distribution......Page 640
8.1.11 The Quasi-Fermi Levels......Page 641
8.2.1 Free Electron Model......Page 643
8.2.2 The Nearly Free Electron Model......Page 644
8.2.3 Introduction to the Bloch Wave Function......Page 645
8.2.4 Orthonormality Relation for the Bloch Wave Functions......Page 648
8.2.5 The Effective Mass Equation......Page 650
8.3.1 Envelope Function Approximation......Page 651
8.3.2 Summary of Solution to the Schrodinger Wave Equation for the Quantum Well......Page 653
8.3.3 Density of Energy States for the Quantum Well......Page 655
8.3.4 The Density of Energy States for the Quantum Wire......Page 657
8.3.5 The Quantum Box......Page 658
8.4.1 The Reduced Density of States......Page 659
8.4.2 Quantum Well Reduced Density of States......Page 661
8.4.3 The Quasi-Fermi Levels......Page 663
8.5 Fermi’s Golden Rule for Semiconductor Devices......Page 664
8.5.1 Vector Potential Form of the Interaction......Page 665
8.5.2 The Matrix Elements for the Homojunction Devices......Page 667
8.5.3 The Quantum Well System......Page 670
8.6.1 Homojunction Emitters and Detectors......Page 674
8.6.2 The Gain for Quantum Well Materials......Page 677
8.6.3 Gain for Quantum Dot Materials......Page 679
8.6.4 Example of a Homojunction 2-D Laser with Unequal Band Masses......Page 680
8.7.1 Homojunction Devices......Page 682
8.7.2 Quantum Well Material......Page 685
8.8 Review Exercises......Page 687
8.9 Further Reading......Page 695
1: Review of Integrating Factors......Page 697
2: Rate and Continuity Equations......Page 699
3: The Group Velocity......Page 701
A3.1 Simple Illustration of Group Velocity......Page 703
A3.3 Group Velocity and the Fourier Integral......Page 705
A3.4 The Group Velocity for a Plane Wave......Page 707
A4.1 Probability Density......Page 709
A4.2 Processes......Page 710
A4.3 Ensembles......Page 711
A4.4 Stationary and Ergodic Processes......Page 712
A4.5 Correlation......Page 714
A5.1 Introduction to the Dirac Delta Function......Page 717
A5.2 The Dirac Delta Function as Limit of a Sequence of Functions......Page 718
A5.3 The Dirac Delta Function from the Fourier Transform......Page 721
A5.4 Other Representations of the Dirac Delta Function......Page 722
A5.6 The Principal Part......Page 724
A5.7 Convergence Factors and the Dirac Delta Function......Page 727
6: Coordinate Representations of the Schrodinger Wave Equation......Page 729
7: Integrals with Two Time Scales......Page 731
8: The Dipole Approximation......Page 735
9: The Density Operator and the Boltzmann Distribution......Page 737