# Lectures in Scattering Theory

## 1st Edition

### International Series of Monographs in Natural Philosophy

**Authors:**A. G. Sitenko

**Editors:**P. J. Shepherd

**eBook ISBN:**9781483186825

**Imprint:**Pergamon

**Published Date:**1st January 1971

**Page Count:**280

## Description

Lectures in Scattering Theory discusses problems in quantum mechanics and the principles of the non-relativistic theory of potential scattering. This book describes in detail the properties of the scattering matrix and its connection with physically observable quantities. This text presents a stationary formulation of the scattering problem and the wave functions of a particle found in an external field. This book also examines the analytic properties of the scattering matrix, dispersion relations, complex angular moments, as well as the separable representation of the scattering amplitude. The text also explains the method of factorizing the potential and the two-particle scattering amplitude, based on the Hilbert-Schmidt theorem for symmetric integral equations. In investigating the problem of scattering in a three-particle system, this book notes that the inapplicability of the Lippman-Schwinger equations can be fixed by appropriately re-arranging the equations. Faddeev equations are the new equations formed after such re-arrangements. This book also cites, as an example, the scattering of a spin-1/2 particle by a spinless particle (such as the scattering of a nucleon by a spinless nucleus). This text is suitable for students and professors dealing with quantum mechanics, theoretical nuclear physics, or other fields of advanced physics.

## Table of Contents

Preface

Chapter 1. Quantum-Mechanical Description and Representations

1.1. Quantum-Mechanical Description of Physical Systems

1.2. The Schrödinger Representation

1.3. The Heisenberg Representation

1.4. The Interaction Representation

Problems

Chapter 2. The Scattering Matrix and Transition Probability

2.1. The Scattering Matrix

2.2. The Time-Shift Operator in the Interaction Representation

2.3. Integrals of Motion and Diagonalization of the S-Matrix

2.4. The Transition Probability Per Unit Time

2.5. An Integral Equation for the T-Matrix

2.6. Transformation of the Scattering Matrix. Cross-Sections

Problems

Chapter 3. Stationary Scattering Theory

3.1. The Scattering Amplitude

3.2. The Lippmann-Schwinger Equation

3.3. The Relation between the Scattering Amplitude and the Transition Matrix

3.4. Inelastic Scattering and Reactions

3.5. The Born Approximation

Problems

Chapter 4. Wave Function of a Particle in an External Field

4.1. Scattering in a Central Field. Expansion in Partial Waves

4.2. The Rectangular Potential Well

4.3. The Coulomb Field

Problems

Chapter 5. The Optical Theorem

5.1. The Relation between the Total Cross-Section and the Elastic Scattering Amplitude

5.2. The Unitarity Relation for the Elastic Scattering Amplitude

Problem

Chapter 6. Time Reversal and the Reciprocity Theorem

6.1. Transformation of the Wave Functions and Operators on Time Reversal

6.2. The Time-Reversal Operator for Specific Systems

6.3. The Time-Reversed Wave Function

6.4. The Reciprocity Theorem and Detailed Balance

Problems

Chapter 7. Analytic Properties of the Scattering Matrix

7.1. Analytic Properties of the Radial Wave Functions

7.2. The Case of Non-Zero Angular Momenta

7.3. Zeros of the Jost Function and Bound States

7.4. The Symmetry and Location of the Scattering Matrix Singularities in the Complex Plane

7.5. Bound States and Redundant Zeros

7.6. Quasi-Stationary States and Resonances

7.7. Virtual States

7.8. The Scattering Matrix in the Case of a Rectangular Potential Well

Problems

Chapter 8. Dispersion Relations

8.1. Integral Representations of the Jost Functions

8.2. Levinson's Theorem

8.3. The Complex Energy Surface

8.4. Analyticity of the Scattering Matrix and the Causality Principle

8.5. Dispersion Relations for the Scattering Amplitude

Chapter 9. Complex Angular Momenta

9.1. Analytic Properties of the Scattering Matrix in the Complex Angular Momentum Plane

9.2. Poles of the Scattering Matrix in the Complex Angular Momentum Plane

9.3. Asymptotic Behaviour of the Scattering Amplitude when cos θ → ∞

Problems

Chapter 10. Separable Representation of the Scattering Amplitude

10.1. The Scattering Amplitude off the Energy Surface

10.2. The Hilbert-Schmidt Expansion for the Scattering Amplitude

10.3. Properties of the Eigenvalues and Eigenfunctions of the Kernel of the Lippmann-Schwinger Equation

Problems

Chapter 11. Scattering in a Three-Particle System

11.1. The Faddeev Equations

11.2. Positions and Momenta in a Three-Particle System

11.3. The Momentum Representation

11.4. Expansion in Partial Waves

11.5. Separable Expansion of the Two-Particle T-Matrix and Reduction of the Faddeev Integral Equations to One-Dimensional Form

Problem

Chapter 12. Scattering of Particles with Spin

12.1. The Spin Wave Function and Density Matrix

12.2. Expansion of the Density Matrix in Spin-Tensors

12.3. The Scattering Amplitude of Particles with Spin

12.4. Coupling of Spin and Orbital Angular Momenta and Diagonalization of the S-Matrix

12.5. Scattering of a Spin-1/2 Particle by a Spinless Particle

12.6. Scattering of a Spin-1 Particle by a Spinless Particle

Problems

Appendix

Bibliography

Index

## Details

- No. of pages:
- 280

- Language:
- English

- Copyright:
- © Pergamon 1971

- Published:
- 1st January 1971

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483186825