Scattering Theory The Quantum Theory of Nonrelativistic Collisions

Preface
Contents
Introduction
1 Mathematical Preliminaries
	1-a The Hilbert Space of State Vectors
	1-b Subspace
	1-c Operators and Inverses
	1-d Unitary Operators
	1-e Isometric Operators
	1-f Convergence of Vectors
	1-g Operator limits
2 The Scattering Operator for a Single Particle
	2-a Classical Scattering
	2-b Quantum Scattering
	2-c The Asymptotic Condition
	2-d Orthogonality and Asymptotic Completeness
	2-e The Scattering Operator
	2-f Unitarity
3 Cross Sections in Terms of the S Matrix
	3-a Conservation of Energy
	3-b The On-Shell T Matrix and Scattering Amplitude
	3-c The Classical Cross Section
	3-d Definition of the Quantum Cross Section
	3-e Calculation of the Quantum Cross Section
	3-f The Optical Theorem
4 Scattering of Two Spinless Particles
	4-a Two-Particle Wave Functions
	4-b The Two-Particle S Operator
	4-c Conservation of Energy-Momentum and the T Matrix
	4-d Cross Sections in Various Frames
	4-e The Center-of-Mass Cross Section
5 Scattering of Two Particles with Spin
	5-a The Hilbert Space for Particles with Spin
	5-b The S Operator for Particles with Spin
	5-c The Amplitudes and Amplitude Matrix
	5-d Sums and Averages Over Spins
	5-e The In and Out Spinors
6 Invariance Principles and Conservation Laws
	6-a Translational Invariance and Conservation of Momentum
	6-b Rotational Invariance and Conservation of Angular Momentum
	6-c The Partial-Wave Series for Spinless Particles
	6-d Parity
	6-e Time Reversal
	6-f Invariance Principles for Particles with Spin; Momentum-Space Analysis
	6-g Invariance Principles for Particles with Spin; Angular-Momentum Analysis
7 More About Particles with Spin
	7-a Polarization and the Density Matrix
	7-b The In and Out Density Matrices
	7-c Polarization Experiments in (Spin 1/2) — (Spin 0) Scattering
	7-d The Helicity Formalism
	7-e Some Useful Formulas
8 The Green's Operator and the T Operator
	8-a The Green's Operator
	8-b The T Operator
	8-c Relation to the Møller Operators
	8-d Relation to the Scattering Operator
9 The Born Series
	9-a The Born Series
	9-b The Born Approximation
	9-c The Yukawa Potential
	9-d Scattering of Electrons off Atoms
	9-e Interpretation of the Born Series in Terms of Feynman Diagrams
10 The Stationary Scattering States
	10-a Definition and Properties of the Stationary Scattering States
	10-b Equations for the Stationary Scattering Vectors
	10-c The Stationary Wave Functions
	10-d A Spatial Description of the Scattering Process
11 The Partial-Wave Stationary States
	11-a The Partial-Wave S Matrix
	11-b The Free Radial Wave Functions
	11-c The Partial-Wave Scattering States
	11-d The Partial-Wave Lippmann-Schwinger Equation
	11-e Properties of the Partial-Wave Amplitude
	11-f The Regular Solution
	11-g The Variable Phase Method
	11-h Iterative Solution for the Regular Wave Function
	11-i The Jost Function
	11-j The Partial-Wave Born Series
12 Analytic Properties of the Partial-Wave Amplitude
	12-a Analytic Functions of a Complex Variable
	12-b Analytic Properties of the Regular Solution
	12-c Analytic Properties of the Jost Function and S Matrix
	12-d Bound States and Poles of the S Matrix
	12-e Levinson's Theorem
	12-f Threshold Behavior and Effective Range Formulas
	12-g Zeros of the Jost Function at Threshold
13 Resonances
	13-a Resonances and Poles of the S Matrix
	13-b Bound States and Resonances
	13-c Time Delay
	13-d Decay of a Resonant State
14 Additional Topics in Single-Channel Scattering
	14-a Coulomb Scattering
	14-b Coulomb Plus Short-Range Potentials
	14-c The Distorted-Wave Born Approximation
	14-d Variational Methods
	14-e The K Matrix
15 Dispersion Relations and Complex Angular Momenta
	15-a Partial-Wave Dispersion Relations
	15-b Forward Dispersion Relations
	15-c Nonforward Dispersion Relations
	15-d The Mandelstam Representation
	I5-e Complex Angular Momenta
	15-f Regge Poles
	15-g The Watson Transform
16 The Scattering Operator in Multichannel Scattering
	16-a Channels
	16-b Channel Hamiltonians and Asymptotic States
	16-c Orthogonality and Asymptotic Completeness
	16-d A Little More Mathematics
	16-e The Scattering Operator
17 Cross Sections and Invariance Principles in Multichannel Scattering
	17-a The Momentum-Space Basis Vectors
	17-b Conservation of Energy and the On-Shell T Matrix
	17-c Cross Sections
	17-d Rotational Invariance
	17-e Time-Reversal Invariance
18 Fundamentals of Time-Independent Multichannel Scattering
	18-a The Stationary Scattering States
	18-b The Lippman—Schwinger Equations
	18-c The T Operators
	18-d The Born Approximation; Elastic Scattering
	18-e The Born Approximation; Excitation
19 Properties of the Multichannel Stationary Wave Functions
	19-a Asymptotic Form of the Stationary Wave Functions; Collisions Without Rearrangement
	19-b Asymptotic Form of the Stationary Wave Functions; Rearrangement Collisions
	19-c Expansion in Terms of Target States
	19-d The Optical Potential
20 Analytic Properties and Multichannel Resonances
	20-a Analytic Properties
	20-b Proof of Analytic Properties
	20-c Bound States
	20-d Resonances
	20-e Decay of a Multichannel Resonance
21 Two More Topics in Multichannel Scattering
	21-a The Distorted-Wave Born Approximation
	21-b Final-State Interactions
22 Identical Particles
	22-a The Formalism of Identical Particles
	22-b Scattering of Two Identical Particles
	22-c Multichannel Scattering with Identical Particles
	22-d Transition Probabilities and Cross Sections
	22-e Electron—Hydrogen Scattering
References
Index