Scattering Theory of Waves and Particles Second Edition

by Newton, Roger G.

• ISBN13: 9780486425351
• ISBN10: 0486425355
• Edition: 2nd
• Format: Paperback
• Copyright: 2013-06-19
• Publisher: Dover Publications

Summary

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 Waves	p. 1
Formalism and General Results	p. 3
The Maxwell Equations	p. 3
Stokes Parameters and Polarization	p. 4
Definition of the Stokes Parameters	p. 4
Significance of the Parameters	p. 6
Partially Polarized Beams	p. 8
Stokes Vectors	p. 9
Relation to the Density Matrix	p. 10
Scattering	p. 11
The Scattering Amplitude	p. 11
Change to a Reference Plane through a Fixed Direction	p. 12
Relation of Circular to Linear Polarization Components in the Scattering Amplitude	p. 12
Stokes Vectors of the Scattered Wave	p. 13
The Differential Cross Section	p. 14
The Density Matrix of the Scattered Wave	p. 15
Azimuthal Dependence of Forward and Backward Scattering	p. 16
Effects of Rotational or Reflectional Symmetry	p. 16
Forward Scattering; the Optical Theorem	p. 18
Double Scattering	p. 20
Scattering by a Cloud of Many Particles	p. 23
Addition of Cross Sections	p. 23
Index of Refraction	p. 24
More than One Kind of Particle	p. 26
Notes and References	p. 27
Problems	p. 28
Spherically Symmetric Scatterers	p. 30
Spherical Harmonics	p. 30
Legendre Polynomials	p. 30
Associated Legendre Functions	p. 31
Spherical Harmonics	p. 31
Vector Spherical Harmonics	p. 32
Transverse and Longitudinal Vector Spherical Harmonics	p. 34
Rotationally Invariant Tensor Functions	p. 35
Complex Conjugation Properties	p. 36
[theta] and [phi] Components	p. 36
The z Axis along r	p. 37
Multipole Expansions	p. 38
Expansion of a Plane Wave; Spherical Bessel Functions	p. 38
Expansion of the Electric Field	p. 40
The Magnetic Field	p. 41
The [characters not reproducible] Matrix	p. 42
The Scattering Amplitude	p. 42
The z Axis along k	p. 42
Unitarity and Reciprocity	p. 44
Energy Conservation and Unitarity	p. 44
Phase Shifts	p. 45
Time Reversal and Reciprocity	p. 46
The Generalized Optical Theorem	p. 47
Generalization to Absence of Spherical Symmetry	p. 48
Scattering by a Uniform Sphere (Mie Theory)	p. 48
Calculation of the [characters not reproducible] Matrix	p. 48
The Scattering Amplitude	p. 50
Notes and References	p. 51
Problems	p. 52
Limiting Cases and Approximations	p. 54
Small Spheres, Not Too Dense (Rayleigh Scattering)	p. 54
Low Optical Density, Not Too Large (Rayleigh-Gans; Born Approximation)	p. 56
Small Dense Spheres	p. 61
Resonance Scattering	p. 61
Totally Reflecting Spheres	p. 65
Large Diffuse Spheres (Van de Hulst Scattering)	p. 66
Forward Scattering	p. 66
Small-Angle Scattering	p. 69
Large Spheres (Geometrical-Optics Limit)	p. 70
Fraunhofer Diffraction	p. 71
Nonforward and Nonbackward Scattering; Real Index of Refraction	p. 72
Large Diffuse Spheres	p. 76
Large Dense Spheres	p. 77
Complex Index of Refraction	p. 77
The Rainbow	p. 78
The Glory	p. 82
Grazing Rays (The Watson Method)	p. 83
The Watson Transform	p. 84
Convergence Questions	p. 90
Saddle-Point Integration (The Method of Steepest Descent)	p. 94
Notes and References	p. 96
Problems	p. 96
Miscellaneous	p. 98
Other Methods	p. 98
Debye Potentials	p. 98
The Green's-Function Method	p. 100
Causality and Dispersion Relations	p. 103
Introduction	p. 103
Forward-Dispersion Relations	p. 104
Nonforward-Dispersion Relations	p. 107
Partial-Wave-Dispersion Relations	p. 109
Intensity-Fluctuation Correlations (Hanbury Brown and Twiss Effect)	p. 110
Notes and References	p. 116
Problems	p. 117
Additional References for Part I	p. 118
Scattering of Classical Particles	p. 119
Particle Scattering in Classical Mechanics	p. 121
The Orbit Equation and the Deflection Angle	p. 121
The Nonrelativistic Case	p. 121
The Relativistic Case	p. 123
The Scattering Cross Section	p. 124
The Rutherford Cross Section	p. 126
Orbiting (Spiral Scattering)	p. 127
Glory and Rainbow Scattering	p. 129
Singular Potentials	p. 130
Transformation Between Laboratory and Center-of-Mass Coordinate Systems	p. 132
Indentical Particles	p. 136
The Inverse Problem	p. 137
Notes and References	p. 139
Problems	p. 140
Quantum Scattering Theory	p. 141
Time-Dependent Formal Scattering Theory	p. 143
The Schrodinger Equation	p. 145
Time Development of State Vectors in the Schrodinger Picture	p. 146
The Moller Wave Operator in the Schrodinger Picture	p. 151
The S Matrix	p. 156
The Interaction Picture	p. 158
The Heisenberg Picture	p. 160
Scattering into Cones	p. 162
Mathematical Questions	p. 164
Convergence of Vectors	p. 164
Operator Convergence	p. 166
Convergences in the Schrodinger Picture	p. 169
The Limits in the Interaction Picture	p. 171
The Limits in the Heisenberg Picture	p. 172
Notes and References	p. 172
Problems	p. 174
Time-Independent Formal Scattering Theory	p. 175
Green's Functions and State Vectors	p. 176
The Green's Functions	p. 176
The State Vectors	p. 178
Expansion of the Green's Functions	p. 181
The Wave Operator and the S Matrix	p. 182
The Operators [Omega], S, and S'	p. 182
The T Matrix	p. 183
The K Matrix	p. 187
Unitarity and Reciprocity	p. 189
Additive Interactions	p. 191
Mathematical Questions	p. 194
The Spectrum	p. 195
Compact Operators	p. 198
Hermitian and Unitary Operators	p. 200
Analyticity of the Resolvent	p. 204
Appendix	p. 208
Notes and References	p. 208
Problems	p. 209
Cross Sections	p. 211
General Definition of Differential Cross Sections	p. 211
Relativistic Generalization	p. 215
Scattering of Incoherent Beams	p. 217
The Density Matrix	p. 217
Particles with Spin	p. 221
The Cross Section and the Density Matrix of the Scattered Wave	p. 225
Notes and References	p. 226
Problems	p. 227
Formal Methods of Solution and Approximations	p. 228
Perturbation Theory	p. 228
The Born Series	p. 228
The Born Approximation	p. 238
The Distorted-Wave Born Approximation	p. 239
Bound States from the Born Approximation	p. 240
The Schmidt Process (Quasi Particles)	p. 241
The Fredholm Method	p. 247
Singularities of an Operator Inverse	p. 255
Notes and References	p. 257
Problems	p. 258
Single-Channel Scattering (Three-Dimensional Analysis in Specific Representations)	p. 260
The Scattering Equation in the One-Particle Case	p. 260
Preliminaries	p. 260
The Coordinate Representation	p. 261
The Momentum Representation	p. 266
Separable Interactions	p. 268
The Scattering Equations in the Two-Particle Case (Elimination of Center-of-Mass Motion)	p. 270
Three-Dimensional Analysis of Potential Scattering	p. 273
Born Series	p. 274
Fredholm Theory	p. 277
Scattering Amplitude, Cross Section, and S Matrix	p. 282
Dispersion Relations	p. 289
An Example (the Yukawa Potential)	p. 291
Notes and References	p. 295
Problems	p. 297
Single-Channel Scattering of Spin 0 Particles, I	p. 298
Partial-Wave Expansion	p. 298
The S Matrix and Traveling Waves	p. 298
The K Matrix and Standing Waves	p. 303
Time Delay	p. 304
Heuristic Survey of Phase-Shift Behavior	p. 305
General Properties	p. 305
Discussion of Low-Energy Phase-Shift Behavior	p. 306
Variational Approaches	p. 318
General Introduction	p. 318
The T Matrix, K Matrix, and the Green's Function	p. 319
Variational Formulations of the Phase Shift	p. 321
The s-Wave Scattering Length	p. 322
Proof of the Hylleraas-Undheim Theorem	p. 326
Notes and References	p. 327
Problems	p. 328
Single-Channel Scattering of Spin 0 Particles, II	p. 331
Rigorous Discussion of s-Wave Scattering	p. 331
The Regular and Irregular Solutions	p. 331
The Jost Function and the Complete Green's Function	p. 341
The S Matrix	p. 350
The Poles of S	p. 357
Completeness	p. 368
Higher Angular Momenta	p. 371
Continuous Angular Momenta	p. 380
Singular Potentials	p. 389
The Difficulties	p. 389
Singular Repulsive Potentials	p. 392
An Example	p. 394
Notes and References	p. 396
General References	p. 399
Problems	p. 399
The Watson-Regge Method (Complex Angular Momentum)	p. 402
The Watson Transform	p. 402
Uniqueness of the Interpolation	p. 408
Regge Poles	p. 410
The Mandelstam Representation	p. 412
Notes and References	p. 415
Problems	p. 416
Examples	p. 417
The Zero-Range Potential	p. 418
The Repulsive Core	p. 419
The Exponential Potential	p. 420
The Hulthen Potential	p. 421
Potentials of the Yukawa Type	p. 422
The Coulomb Potential	p. 424
The Pure Coulomb Field	p. 424
Coulomb Admixtures	p. 431
Bargmann Potentials and Generalizations	p. 433
General Procedure	p. 433
Special Cases	p. 437
Notes and References	p. 440
Problems	p. 441
Elastic Scattering of Particles with Spin	p. 444
Partial-Wave Analysis	p. 444
Expansion in j and s	p. 444
Amplitudes for Individual Spins	p. 450
Unitarity, Reciprocity, Time-Reversal Invariance, and Parity Conservation	p. 452
Special Cases	p. 457
Cross Sections	p. 459
Double Scattering	p. 460
Solution of the Coupled Schrodinger Equations	p. 461
The Matrix Equation	p. 461
Solutions	p. 462
Jost Matrix and S Matrix	p. 464
Bound States	p. 468
Miscellaneous Remarks	p. 471
Notes and References	p. 472
Problems	p. 472
Inelastic Scattering and Reactions (Multichannel Theory), I	p. 474
Descriptive Introduction	p. 474
Time-Dependent Theory	p. 477
The Schrodinger Picture	p. 477
The Heisenberg Picture	p. 482
Two-Hilbert-Space Formulation	p. 483
Time-Independent Theory	p. 485
Formal Theory	p. 485
Distorted-Wave Rearrangement Theory	p. 487
Identical Particles	p. 488
Large-Distance Behavior of the Two-Cluster Wave Function	p. 489
Partial-Wave Analysis	p. 492
The Coupled Equations	p. 492
The S Matrix	p. 494
Rearrangements	p. 495
General Scattering Rates	p. 495
Formal Resonance Theory	p. 500
Appendix	p. 508
Notes and References	p. 510
Problems	p. 513
Inelastic Scattering and Reactions (Multichannel Theory), II	p. 514
Analyticity in Many-Channel Problems	p. 514
The Coupled Equations	p. 514
An Alternative Procedure	p. 519
Analyticity Properties	p. 521
Bound States	p. 526
The Riemann Surface of the Many-Channel S Matrix	p. 528
Threshold Effects	p. 534
Threshold Branch Points	p. 534
Physical Threshold Phenomena; General Arguments	p. 538
Details of the Anomaly	p. 539
The Threshold Anomaly for Charged Particles	p. 544
Examples	p. 548
The Square Well	p. 548
Potentials of Yukawa Type	p. 551
The Wigner-Weisskopf Model	p. 553
The Three-Body Problem	p. 555
Failure of the Multichannel Method and of the Lippmann-Schwinger Equation	p. 555
The Faddeev Method	p. 558
Other Methods	p. 560
Fredholm Properties and Spurious Solutions	p. 563
The Asymptotic Form of Three-Particle Wave Functions	p. 565
Angular Momentum Couplings	p. 571
The S Matrix	p. 579
The Efimov Effect	p. 580
Notes and References	p. 581
Problems	p. 586
Short-Wavelength Approximations	p. 588
Introduction	p. 588
Diffraction from the Optical Theorem	p. 590
The WKB Method	p. 591
The WKB Phase Shifts	p. 591
The Scattering Amplitude	p. 594
The Rainbow	p. 597
The Glory	p. 598
Orbiting (Spiral Scattering)	p. 600
The Eikonal Approximation	p. 600
The Impulse Approximation	p. 605
Notes and References	p. 609
Problems	p. 611
The Decay of Unstable States	p. 612
Qualitative Introduction	p. 612
Exponential Decay and Its Limitations	p. 614
Multiple Poles of the S Matrix	p. 625
Notes and References	p. 626
Problems	p. 626
The Inverse Scattering Problem	p. 629
Introduction	p. 629
The Phase of the Amplitude	p. 633
The Central Potential Obtained from a Phase Shift	p. 637
The Gel'fand-Levitan Equations	p. 637
Infinitesimal Variations	p. 645
The Marchenko Equation	p. 648
The Central Potential Obtained from All Phase Shifts at One Energy	p. 650
The Construction Procedure	p. 650
Examples	p. 657
The Inverse Scattering Problem for Noncentral Potentials	p. 659
Introduction	p. 659
The Generalized Marchenko Equation	p. 659
A Generalized Gel'fand-Levitan Equation	p. 665
Potential Obtained from Backscattering	p. 666
Notes and References	p. 667
Problems	p. 670
Bibliography	p. 671
Index	p. 727
Errata	p. 745
Table of Contents provided by Syndetics. All Rights Reserved.