Scattering Theory: Quantum Theory on Nonrelativistic Collisions

✍ Scribed by John R. Taylor

• Publisher: Wiley
• Year: 1972
• Tongue: English
• Leaves: 495
• Edition: 99
• Category: Library

✦ Synopsis

This graduate-level text is intended for any student of physics who requires a thorough grounding in the quantum theory of nonrelativistic scattering. It is designed for readers who are already familiar with the general principles of quantum mechanics and who have some small acquaintance with scattering theory. Study of this text will allow students of atomic or nuclear physics to begin reading the literature and tackling real problems, with a complete grasp of the underlying principles. For students of high-energy physics, it provides the necessary background for later study of relativistic problems.
Topics are presented in terms of the simplest relevant example, so that scattering theory can be learned by becoming familiar with all of the basic concepts — the S operator, cross sections, the T matrix, and so forth — in their simplest context. The time-dependent approach to the subject is emphasized, starting with the use of time-dependent formalism to define all of the basic concepts and the subsequent introduction of the time-independent theory as a tool for computation and for establishing certain general properties. Problems at the end of each chapter improve and supplement readers' grasp of the material.

✦ Table of Contents

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