Scattering Theory of Waves and Particles: Second Edition(Dover Books on Physics)

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.

Table of Contents:
PART I SCATTERING OF ELECTROMAGNETIC WAVES 1 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 Particle Notes and References Problems 2 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 Amplitude Notes and References Problems 3 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 Questions Appendix: Saddle-Point Integration (The Method of Steepest Descent) Notes and References Problems 4 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 References Problems Additional References for Part I PART II SCATTERING OF CLASSICAL PARTICLES 5 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 Problem Notes and References Problems PART III QUANTUM SCATTERING THEORY 6 Time-Dependent Formal Scattering Theory 6.1 The Schrodinger Equation 6.2 Time Development of State Vectors in the Schrodinger Picture 6.3 The Mfller Wave Operator in the Schrodinger 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 Schrodinger Picture 6.8.4 The Limits in the Interaction Picture 6.8.5 The Limits in the Heisenberg Picture Notes and References Problems 7 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 Resolvent Appendix Notes and References Problems 8 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 Wave Notes and References Problems 9 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 Inverse Notes and References Problems 10 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 Length Appendix: Proof of the Hylleraas-Undheim Theorem Notes and References Problems 12 "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 Example Notes and References General References Problems 13 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 Representation Notes and References Problems 14 Examples 14.1 The Zero-Range Potential 14.2 The Repulsive Core 14.3 The Exponential Potential 14.4 The Hulthen 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 Cases Notes and References Problems 15 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 Schrodinger 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 Remarks Notes and References Problems 16 "Inelastic Scattering and Reactions (Multichannel Theory), I" 16.1 Descriptive Introduction 16.2 Time-Dependent Theory 16.2.1 The Schrodinger 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 Theory Appendix Notes and References Problems 17 "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 Effect Notes and References Problems 18 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 Approximation Notes and References Problems 19 The Decay of Unstable States 19.1 Qualitative Introduction 19.2 Exponential Decay and Its Limitations 19.3 Multiple Poles of the S Matrix Notes and References Problems 20 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 Backscattering Notes and References Problems Bibliography Index Errata

About the Author :
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.