Scattering Theory of Waves and Particles Second Edition

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  • Edition: 2nd
  • Format: Paperback
  • Copyright: 2013-06-19
  • Publisher: Dover Publications

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This concise volume represents an enlarged and improved edition of the author's original text on the theory of scattering electromagnetic waves, of classical particles, and of quantum-mechanic particles, including multiparticle collisions. Includes updates on developments in three-particle collisions, in scattering by noncentral potentials, and in inverse scattering problems. 1982 edition.

Author Biography

Roger G. Newton is Professor Emeritus at Indiana University. A Harvard PhD, he worked for several years at the legendary Institute for Advanced Study at Princeton. His research areas include field theory, scattering theories, nuclear and high-energy physics, elementary particles, quantum mechanics, and mathematical physics.

Table of Contents

Scattering of Electromagnetic Wavesp. 1
Formalism and General Resultsp. 3
The Maxwell Equationsp. 3
Stokes Parameters and Polarizationp. 4
Definition of the Stokes Parametersp. 4
Significance of the Parametersp. 6
Partially Polarized Beamsp. 8
Stokes Vectorsp. 9
Relation to the Density Matrixp. 10
Scatteringp. 11
The Scattering Amplitudep. 11
Change to a Reference Plane through a Fixed Directionp. 12
Relation of Circular to Linear Polarization Components in the Scattering Amplitudep. 12
Stokes Vectors of the Scattered Wavep. 13
The Differential Cross Sectionp. 14
The Density Matrix of the Scattered Wavep. 15
Azimuthal Dependence of Forward and Backward Scatteringp. 16
Effects of Rotational or Reflectional Symmetryp. 16
Forward Scattering; the Optical Theoremp. 18
Double Scatteringp. 20
Scattering by a Cloud of Many Particlesp. 23
Addition of Cross Sectionsp. 23
Index of Refractionp. 24
More than One Kind of Particlep. 26
Notes and Referencesp. 27
Problemsp. 28
Spherically Symmetric Scatterersp. 30
Spherical Harmonicsp. 30
Legendre Polynomialsp. 30
Associated Legendre Functionsp. 31
Spherical Harmonicsp. 31
Vector Spherical Harmonicsp. 32
Transverse and Longitudinal Vector Spherical Harmonicsp. 34
Rotationally Invariant Tensor Functionsp. 35
Complex Conjugation Propertiesp. 36
[theta] and [phi] Componentsp. 36
The z Axis along rp. 37
Multipole Expansionsp. 38
Expansion of a Plane Wave; Spherical Bessel Functionsp. 38
Expansion of the Electric Fieldp. 40
The Magnetic Fieldp. 41
The [characters not reproducible] Matrixp. 42
The Scattering Amplitudep. 42
The z Axis along kp. 42
Unitarity and Reciprocityp. 44
Energy Conservation and Unitarityp. 44
Phase Shiftsp. 45
Time Reversal and Reciprocityp. 46
The Generalized Optical Theoremp. 47
Generalization to Absence of Spherical Symmetryp. 48
Scattering by a Uniform Sphere (Mie Theory)p. 48
Calculation of the [characters not reproducible] Matrixp. 48
The Scattering Amplitudep. 50
Notes and Referencesp. 51
Problemsp. 52
Limiting Cases and Approximationsp. 54
Small Spheres, Not Too Dense (Rayleigh Scattering)p. 54
Low Optical Density, Not Too Large (Rayleigh-Gans; Born Approximation)p. 56
Small Dense Spheresp. 61
Resonance Scatteringp. 61
Totally Reflecting Spheresp. 65
Large Diffuse Spheres (Van de Hulst Scattering)p. 66
Forward Scatteringp. 66
Small-Angle Scatteringp. 69
Large Spheres (Geometrical-Optics Limit)p. 70
Fraunhofer Diffractionp. 71
Nonforward and Nonbackward Scattering; Real Index of Refractionp. 72
Large Diffuse Spheresp. 76
Large Dense Spheresp. 77
Complex Index of Refractionp. 77
The Rainbowp. 78
The Gloryp. 82
Grazing Rays (The Watson Method)p. 83
The Watson Transformp. 84
Convergence Questionsp. 90
Saddle-Point Integration (The Method of Steepest Descent)p. 94
Notes and Referencesp. 96
Problemsp. 96
Miscellaneousp. 98
Other Methodsp. 98
Debye Potentialsp. 98
The Green's-Function Methodp. 100
Causality and Dispersion Relationsp. 103
Introductionp. 103
Forward-Dispersion Relationsp. 104
Nonforward-Dispersion Relationsp. 107
Partial-Wave-Dispersion Relationsp. 109
Intensity-Fluctuation Correlations (Hanbury Brown and Twiss Effect)p. 110
Notes and Referencesp. 116
Problemsp. 117
Additional References for Part Ip. 118
Scattering of Classical Particlesp. 119
Particle Scattering in Classical Mechanicsp. 121
The Orbit Equation and the Deflection Anglep. 121
The Nonrelativistic Casep. 121
The Relativistic Casep. 123
The Scattering Cross Sectionp. 124
The Rutherford Cross Sectionp. 126
Orbiting (Spiral Scattering)p. 127
Glory and Rainbow Scatteringp. 129
Singular Potentialsp. 130
Transformation Between Laboratory and Center-of-Mass Coordinate Systemsp. 132
Indentical Particlesp. 136
The Inverse Problemp. 137
Notes and Referencesp. 139
Problemsp. 140
Quantum Scattering Theoryp. 141
Time-Dependent Formal Scattering Theoryp. 143
The Schrodinger Equationp. 145
Time Development of State Vectors in the Schrodinger Picturep. 146
The Moller Wave Operator in the Schrodinger Picturep. 151
The S Matrixp. 156
The Interaction Picturep. 158
The Heisenberg Picturep. 160
Scattering into Conesp. 162
Mathematical Questionsp. 164
Convergence of Vectorsp. 164
Operator Convergencep. 166
Convergences in the Schrodinger Picturep. 169
The Limits in the Interaction Picturep. 171
The Limits in the Heisenberg Picturep. 172
Notes and Referencesp. 172
Problemsp. 174
Time-Independent Formal Scattering Theoryp. 175
Green's Functions and State Vectorsp. 176
The Green's Functionsp. 176
The State Vectorsp. 178
Expansion of the Green's Functionsp. 181
The Wave Operator and the S Matrixp. 182
The Operators [Omega], S, and S'p. 182
The T Matrixp. 183
The K Matrixp. 187
Unitarity and Reciprocityp. 189
Additive Interactionsp. 191
Mathematical Questionsp. 194
The Spectrump. 195
Compact Operatorsp. 198
Hermitian and Unitary Operatorsp. 200
Analyticity of the Resolventp. 204
Appendixp. 208
Notes and Referencesp. 208
Problemsp. 209
Cross Sectionsp. 211
General Definition of Differential Cross Sectionsp. 211
Relativistic Generalizationp. 215
Scattering of Incoherent Beamsp. 217
The Density Matrixp. 217
Particles with Spinp. 221
The Cross Section and the Density Matrix of the Scattered Wavep. 225
Notes and Referencesp. 226
Problemsp. 227
Formal Methods of Solution and Approximationsp. 228
Perturbation Theoryp. 228
The Born Seriesp. 228
The Born Approximationp. 238
The Distorted-Wave Born Approximationp. 239
Bound States from the Born Approximationp. 240
The Schmidt Process (Quasi Particles)p. 241
The Fredholm Methodp. 247
Singularities of an Operator Inversep. 255
Notes and Referencesp. 257
Problemsp. 258
Single-Channel Scattering (Three-Dimensional Analysis in Specific Representations)p. 260
The Scattering Equation in the One-Particle Casep. 260
Preliminariesp. 260
The Coordinate Representationp. 261
The Momentum Representationp. 266
Separable Interactionsp. 268
The Scattering Equations in the Two-Particle Case (Elimination of Center-of-Mass Motion)p. 270
Three-Dimensional Analysis of Potential Scatteringp. 273
Born Seriesp. 274
Fredholm Theoryp. 277
Scattering Amplitude, Cross Section, and S Matrixp. 282
Dispersion Relationsp. 289
An Example (the Yukawa Potential)p. 291
Notes and Referencesp. 295
Problemsp. 297
Single-Channel Scattering of Spin 0 Particles, Ip. 298
Partial-Wave Expansionp. 298
The S Matrix and Traveling Wavesp. 298
The K Matrix and Standing Wavesp. 303
Time Delayp. 304
Heuristic Survey of Phase-Shift Behaviorp. 305
General Propertiesp. 305
Discussion of Low-Energy Phase-Shift Behaviorp. 306
Variational Approachesp. 318
General Introductionp. 318
The T Matrix, K Matrix, and the Green's Functionp. 319
Variational Formulations of the Phase Shiftp. 321
The s-Wave Scattering Lengthp. 322
Proof of the Hylleraas-Undheim Theoremp. 326
Notes and Referencesp. 327
Problemsp. 328
Single-Channel Scattering of Spin 0 Particles, IIp. 331
Rigorous Discussion of s-Wave Scatteringp. 331
The Regular and Irregular Solutionsp. 331
The Jost Function and the Complete Green's Functionp. 341
The S Matrixp. 350
The Poles of Sp. 357
Completenessp. 368
Higher Angular Momentap. 371
Continuous Angular Momentap. 380
Singular Potentialsp. 389
The Difficultiesp. 389
Singular Repulsive Potentialsp. 392
An Examplep. 394
Notes and Referencesp. 396
General Referencesp. 399
Problemsp. 399
The Watson-Regge Method (Complex Angular Momentum)p. 402
The Watson Transformp. 402
Uniqueness of the Interpolationp. 408
Regge Polesp. 410
The Mandelstam Representationp. 412
Notes and Referencesp. 415
Problemsp. 416
Examplesp. 417
The Zero-Range Potentialp. 418
The Repulsive Corep. 419
The Exponential Potentialp. 420
The Hulthen Potentialp. 421
Potentials of the Yukawa Typep. 422
The Coulomb Potentialp. 424
The Pure Coulomb Fieldp. 424
Coulomb Admixturesp. 431
Bargmann Potentials and Generalizationsp. 433
General Procedurep. 433
Special Casesp. 437
Notes and Referencesp. 440
Problemsp. 441
Elastic Scattering of Particles with Spinp. 444
Partial-Wave Analysisp. 444
Expansion in j and sp. 444
Amplitudes for Individual Spinsp. 450
Unitarity, Reciprocity, Time-Reversal Invariance, and Parity Conservationp. 452
Special Casesp. 457
Cross Sectionsp. 459
Double Scatteringp. 460
Solution of the Coupled Schrodinger Equationsp. 461
The Matrix Equationp. 461
Solutionsp. 462
Jost Matrix and S Matrixp. 464
Bound Statesp. 468
Miscellaneous Remarksp. 471
Notes and Referencesp. 472
Problemsp. 472
Inelastic Scattering and Reactions (Multichannel Theory), Ip. 474
Descriptive Introductionp. 474
Time-Dependent Theoryp. 477
The Schrodinger Picturep. 477
The Heisenberg Picturep. 482
Two-Hilbert-Space Formulationp. 483
Time-Independent Theoryp. 485
Formal Theoryp. 485
Distorted-Wave Rearrangement Theoryp. 487
Identical Particlesp. 488
Large-Distance Behavior of the Two-Cluster Wave Functionp. 489
Partial-Wave Analysisp. 492
The Coupled Equationsp. 492
The S Matrixp. 494
Rearrangementsp. 495
General Scattering Ratesp. 495
Formal Resonance Theoryp. 500
Appendixp. 508
Notes and Referencesp. 510
Problemsp. 513
Inelastic Scattering and Reactions (Multichannel Theory), IIp. 514
Analyticity in Many-Channel Problemsp. 514
The Coupled Equationsp. 514
An Alternative Procedurep. 519
Analyticity Propertiesp. 521
Bound Statesp. 526
The Riemann Surface of the Many-Channel S Matrixp. 528
Threshold Effectsp. 534
Threshold Branch Pointsp. 534
Physical Threshold Phenomena; General Argumentsp. 538
Details of the Anomalyp. 539
The Threshold Anomaly for Charged Particlesp. 544
Examplesp. 548
The Square Wellp. 548
Potentials of Yukawa Typep. 551
The Wigner-Weisskopf Modelp. 553
The Three-Body Problemp. 555
Failure of the Multichannel Method and of the Lippmann-Schwinger Equationp. 555
The Faddeev Methodp. 558
Other Methodsp. 560
Fredholm Properties and Spurious Solutionsp. 563
The Asymptotic Form of Three-Particle Wave Functionsp. 565
Angular Momentum Couplingsp. 571
The S Matrixp. 579
The Efimov Effectp. 580
Notes and Referencesp. 581
Problemsp. 586
Short-Wavelength Approximationsp. 588
Introductionp. 588
Diffraction from the Optical Theoremp. 590
The WKB Methodp. 591
The WKB Phase Shiftsp. 591
The Scattering Amplitudep. 594
The Rainbowp. 597
The Gloryp. 598
Orbiting (Spiral Scattering)p. 600
The Eikonal Approximationp. 600
The Impulse Approximationp. 605
Notes and Referencesp. 609
Problemsp. 611
The Decay of Unstable Statesp. 612
Qualitative Introductionp. 612
Exponential Decay and Its Limitationsp. 614
Multiple Poles of the S Matrixp. 625
Notes and Referencesp. 626
Problemsp. 626
The Inverse Scattering Problemp. 629
Introductionp. 629
The Phase of the Amplitudep. 633
The Central Potential Obtained from a Phase Shiftp. 637
The Gel'fand-Levitan Equationsp. 637
Infinitesimal Variationsp. 645
The Marchenko Equationp. 648
The Central Potential Obtained from All Phase Shifts at One Energyp. 650
The Construction Procedurep. 650
Examplesp. 657
The Inverse Scattering Problem for Noncentral Potentialsp. 659
Introductionp. 659
The Generalized Marchenko Equationp. 659
A Generalized Gel'fand-Levitan Equationp. 665
Potential Obtained from Backscatteringp. 666
Notes and Referencesp. 667
Problemsp. 670
Bibliographyp. 671
Indexp. 727
Erratap. 745
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