Scattering Theory of Waves and Particles

Scattering Theory of Waves and Particles

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The observation and analysis of particle and wave scattering plays a crucial role in physics; numerous important discoveries, including nuclear fission, are the direct result of collision experiments. This concise volume crosses the boundaries of physics' traditional subdivisions to treat scattering theory within the context of classical electromagnetic radiation, classical particle mechanics, and quantum mechanics. An enlarged and improved edition of Roger G. Newton's text on the theory of scattering electromagnetic waves, this text explores classical particles and quantum-mechanic particles, including multiparticle collisions. This edition's updates include coverage of developments in three-particle collisions, scattering by noncentral potentials, and inverse scattering problems. Numerous problems, examples, notes, and references augment the text.

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Product details

  • Paperback | 768 pages
  • 136 x 208 x 42mm | 798.32g
  • Dover Publications Inc.
  • New York, United States
  • English
  • Revised
  • 2nd Revised edition
  • Bibliography, index
  • 0486425355
  • 9780486425351
  • 814,365

Table of contents

PART I SCATTERING OF ELECTROMAGNETIC WAVES1 Formalism and General Results  1.1 The Maxwell Equations  1.2 Stokes Parameters and Polarization    1.2.1 Definition of the Stokes Parameters    1.2.2 Significance of the Parameters    1.2.3 Partially Polarized Beams    1.2.4 Stokes Vectors    1.2.5 Relation to the Density Matrix  1.3 Scattering    1.3.1 The Scattering Amplitude    1.3.2 Change to a Reference Plane through a Fixed Direction    1.3.3 Relation of Circular to Linear Poloarization Components in the Scattering Amplitude    1.3.4 Stokes Vectors of the Scattered Wave    1.3.5 The Differential Cross Section    1.3.6 The Density Matrix of the Scattered Wave    1.3.7 Azimuthal Dependence of Forward and Backward Scattering    1.3.8 Effects of Rotational or Reflectional Symmetry    1.3.9 Forward Scattering; the Optical Theorem  1.4 Double Scattering  1.5 Scattering by a Cloud of Many Particles    1.5.1 Addition of Cross Sections    1.5.2 Index of Refraction    1.5.3 More than One Kind of ParticleNotes and ReferencesProblems2 Sperically Symmetric Scatterers  2.1 Spherical Harmonics    2.1.1 Legendre Polynomials    2.1.2 Associated Legendre Functions    2.1.3 Spherical Harmonics    2.1.4 Vector Spherical Harmonics    2.1.5 Transverse and Longitudinal Vector Spherical Harmonics    2.1.6 Rotationally Invariant Tensor Functions    2.1.7 Complex Conjugation Properties    2.1.8 q and j Components    2.1.9 The z Axis along r  2.2 Multipole Expansions    2.2.1 Expansion of a Plane Wave; Spherical Bessel Functions    2.2.2 Expansion of the Electric Field    2.2.3 The Magnetic Field    2.2.4 The K Matrix    2.2.5 The Scattering Amplitude    2.2.6 The z Axis along k  2.3 Unitarity and Reciprocity    2.3.1 Energy Conservation and Unitarity    2.3.2 Phase Shifts    2.3.3 Time Reversal and Reciprocity    2.3.4 The Generalized Optical Theorem    2.3.5 Generalization to Absence of Spherical Symmetry  2.4 Scattering by a Uniform Sphere (Mie Theory)    2.4.1 Calculation of the K Matrix    2.4.2 The Scattering AmplitudeNotes and ReferencesProblems3 Limiting Cases and Approximations  3.1 "Small Spheres, Not Too Dense (Rayleigh Scattering)"  3.2 "Low Optical Density, Not Too Large (Rayleigh-Gans; Born Approximation)"  3.3 Small Dense Spheres    3.3.1 Resonance Scattering    3.3.2 Totally Reflecting Spheres  3.4 Large Diffuse Spheres (Van de Hulst Scattering)    3.4.1 Forward Scattering    3.4.2 Small-Angle Scattering  3.5 Large Spheres (Geometrical-Optics Limit)    3.5.1 Fraunhofer Diffraction    3.5.2 Nonforward and Nonbackward Scattering; Real Index of Refraction    3.5.3 Large Diffuse Spheres    3.5.4 Large Dense Spheres    3.5.5 Complex Index of Refraction  3.6 The Rainbow  3.7 The Glory  3.8 Grazing Rays (The Watson Method)    3.8.1 The Watson Transform    3.8.2 Convergence QuestionsAppendix: Saddle-Point Integration (The Method of Steepest Descent)Notes and ReferencesProblems4 Miscellaneous  4.1 Other Methods    4.1.1 Debye Potentials    4.1.2 The Green's-Function Method  4.2 Causality and Dispersion Relations    4.2.1 Introduction    4.2.2 Forward-Dispersion Relations    4.2.3 Nonforward-Dispersion Relations    4.2.4 Partial-Wave-Dispersion Relations  4.3 Intensity-Fluctuation Correlations (Hanbury Brown and Twiss Effect)Notes and ReferencesProblemsAdditional References for Part IPART II SCATTERING OF CLASSICAL PARTICLES5 Particle Scattering in Classical Mechanics  5.1 The Orbit Equation and the Deflection Angle    5.1.1 The Nonrelativistic Case    5.1.2 The Relativistic Case  5.2 The Scattering Cross Section  5.3 The Rutherford Cross Section  5.4 Orbiting (Spiral Scattering)  5.5 Glory and Rainbow Scattering  5.6 Singular Potentials  5.7 Transformation Between Laboratory and Center-of-Mass Coordinate Systems  5.8 Identical Particles  5.9 The Inverse ProblemNotes and ReferencesProblemsPART III QUANTUM SCATTERING THEORY6 Time-Dependent Formal Scattering Theory  6.1 The Schrödinger Equation  6.2 Time Development of State Vectors in the Schrödinger Picture  6.3 The Mfller Wave Operator in the Schrödinger Picture  6.4 The S Matrix  6.5 The Interaction Picture  6.6 The Heisenberg Picture  6.7 Scattering into Cones  6.8 Mathematical Questions    6.8.1 Convergence of Vectors    6.8.2 Operator Convergence    6.8.3 Convergences in the Schrödinger Picture    6.8.4 The Limits in the Interaction Picture    6.8.5 The Limits in the Heisenberg PictureNotes and ReferencesProblems7 Time-Independent Formal Scattering Theory  7.1 Green's Functions and State Vectors    7.1.1 The Green's Functions    7.1.2 The State Vectors    7.1.3 Expansion of the Green's Functions  7.2 The Wave Operator and the S Matrix    7.2.1 "The Operators W, S, and S'"    7.2.2 The T Matrix    7.2.3 The K Matrix    7.2.4 Unitarity and Reciprocity    7.2.5 Additive Interactions  7.3 Mathematical Questions    7.3.1 The Spectrum    7.3.2 Compact Operators    7.3.3 Hermitian and Unitary Operators    7.3.4 Analyticity of the ResolventAppendixNotes and ReferencesProblems8 Cross Section  8.1 General Definition of Differential Cross Sections  8.2 Relativistic Generalization  8.3 Scattering of Incoherent Beams    8.3.1 The Density Matrix    8.3.2 Particles with Spin    8.3.3 The Cross Section and the Density Matrix of the Scattered WaveNotes and ReferencesProblems9 Formal Methods of Solution and Approximations  9.1 Perturbation Theory    9.1.1 The Born Series    9.1.2 The Born Approximation    9.1.3 The Distorted-Wave Born Approximatin    9.1.4 Bound States from the Born Approximation  9.2 The Schmidt Process (Quasi Particles)  9.3 The Fredholm Method  9.4 Singularities of an Operator InverseNotes and ReferencesProblems10 Single-Channel Scattering (Three-Dimensional Analysis in Specific Representations)  10.1 The Scattering Equation in the One-Particle Case    10.1.1 Preliminaries    10.1.2 The Coordinate Representation    10.1.3 The Momentum Representation    10.1.4 Separable Interactions  10.2 The Scattering Equations in the Two-Particle Case (Elimination of Center-of-Mass Motion)  10.3 Three-Dimensional Analysis of Potential Scattering&n    11.3.2 "The T Matrix, K Matrix, and the Green's Function"    11.3.3 Variational Formulations of the Phase Shift    11.3.4 The s-Wave Scattering LengthAppendix: Proof of the Hylleraas-Undheim TheoremNotes and ReferencesProblems12 "Single-Channel Scattering of Spin 0 Particles, II"  12.1 Rigorous Discussion of s -Wave Scattering    12.1.1 The Regular and Irregular Solutions    12.1.2 The Jost Function and the Complete Green's Function    12.1.3 The S Matrix    12.1.4 The Poles of S    12.1.5 Completeness  12.2 Higher Angular Momenta  12.3 Continuous Angular Momenta  12.4 Singular Potentials    12.4.1 The Difficulties    12.4.2 Singular Repulsive Potentials    12.4.3 An ExampleNotes and ReferencesGeneral ReferencesProblems13 The Watson-Regge Method (Complex Angular Momentum)  13.1 The Watson Transform  13.2 Uniqueness of the Interpolation  13.3 Regge Poles  13.4 The Mandelstam RepresentationNotes and ReferencesProblems14 Examples  14.1 The Zero-Range Potential  14.2 The Repulsive Core  14.3 The Exponential Potential  14.4 The Hulthén Potential  14.5 Potentials of the Yukawa Type  14.6 The Coulomb Potential    14.6.1 The Pure Coulomb Field    14.6.2 Coulomb Admixtures  14.7 Bargmann Potentials and Generalizations    14.7.1 General Procedure    14.7.2 Special CasesNotes and ReferencesProblems15 Elastic Scattering of Particles with Spin  15.1 Partial-Wave Analysis    15.1.1 Expansion in j and s    15.1.2 Amplitudes for Individual Spins    15.1.3 "Unitarity, Reciprocity, Time-Reversal Invariance, and Parity Conservation"    15.1.4 Special Cases    15.1.5 Cross Sections    15.1.6 Double Scattering  15.2 Solution of the Coupled Schrödinger Equations    15.2.1 The Matrix Equation    15.2.2 Solutions    15.2.3 Jost Matrix and S Matrix    15.2.4 Bound States    15.2.5 Miscellaneous RemarksNotes and ReferencesProblems16 "Inelastic Scattering and Reactions (Multichannel Theory), I"  16.1 Descriptive Introduction  16.2 Time-Dependent Theory    16.2.1 The Schrödinger Picture    16.2.2 The Heisenberg Picture    16.2.3 Two-Hilbert-Space Formulation  16.3 Time-Independent Theory    16.3.1 Formal Theory    16.3.2 Distorted-Wave Rearrangement Theory    16.3.3 Identical Particles    16.3.4 Large-Distance Behavior of the Two-Cluster Wave Function  16.4 Partial-Wave Analysis    16.4.1 The Coupled Equations    16.4.2 The S Matrix    16.4.3 Rearrangements  16.5 General Scattering Rates  16.6 Formal Resonance TheoryAppendixNotes and ReferencesProblems17 "Inelastic Scattering and Reactions (Multichannel Theory), II"  17.1 Analyticity in Many-Channel Problems    17.1.1 The Coupled Equations    17.1.2 An Alternative Procedure    17.1.3 Analyticity Properties    17.1.4 Bound States    17.1.5 The Riemann Surface of the Many-Channel S Matrix  17.2 Threshold Effects    17.2.1 Threshold Branch Points    17.2.2 Physical Threshold Phenomena; General Arguments    17.2.3 Details of the Anomaly    17.2.4 The Threshold Anomaly for Charged Particles  17.3 Examples    17.3.1 The Square Well    17.3.2 Potentials of Yukawa Type    17.3.3 The Wigner-Weisskopf Model  17.4 The Three-Body Problem    17.4.1 Failure of the Multichannel Method and of the Lippmann-Schwinger Equation    17.4.2 The Faddeev Method    17.4.3 Other Methods    17.4.4 Fredholm Properties and Spurious Solutions    17.4.5 The Asymptotic Form of Three-Particle Wave Functions    17.4.6 Angular Momentum Couplings    17.4.7 The S Matrix    17.4.8 The Efimov EffectNotes and ReferencesProblems18 Short-Wavelength Approximations  18.1 Introduction    18.1.1 Diffraction from the Optical Theorem  18.2 The WKB Method    18.2.1 The WKB Phase Shifts    18.2.2 The Scattering Amplitude    18.2.3 The Rainbow    18.2.4 The Glory    18.2.5 Orbiting (Spiral Scattering)  18.3 The Eikonal Approximation  18.4 The Impulse ApproximationNotes and ReferencesProblems19 The Decay of Unstable States  19.1 Qualitative Introduction  19.2 Exponential Decay and Its Limitations  19.3 Multiple Poles of the S MatrixNotes and ReferencesProblems20 The Inverse Scattering Problem  20.1 Introduction  20.2 The Phase of the Amplitude  20.3 The Central Potential Obtained from a Phase Shfit    20.3.1 The Gel'fand-Levitan Equations    20.3.2 Infinitesimal Variations    20.3.3 The Marchenko Equation  20.4 The Central Potential Obtained from All Phase Shifts at One Energy    20.4.1 The Construction Procedure    20.4.2 Examples  20.5 The Inverse Scattering Problem for Noncentral Potentials    20.5.1 Introduction    20.5.2 The Generalized Marchenko Equation    20.5.3 A Generalized Gel'fand-Levitan Equation    20.5.4 Potential Obtained from BackscatteringNotes and ReferencesProblemsBibliographyIndexErrata

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