Enlarged, improved edition of 1982 text on theory of scattering electromagnetic waves, classical particles, and quantum-mechanic particles. Includes updates on developments in three-particle collisions, scattering by noncentral potentials, inverse scattering problems.
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.
Enlarged, improved edition of author's original graduate-level text on the theory of scattering electromagnetic waves, classical particles, and quantum-mechanic particles, including multiparticle collisions. 1982 edition.
This 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 version of the 1982 text, this edition includes updates on developments in three-particle collisions, scattering by noncentral potentials, and inverse scattering problems.
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 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 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
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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 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 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 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 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 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 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