The Quantum Mechanical Three-Body Problem

The Quantum Mechanical Three-Body Problem deals with the three-body problem in quantum mechanics. Topics include the two- and three-particle problem, the Faddeev equations and their solution, separable potentials, and variational methods.
This book has eight chapters; the first of which introduces the reader to the quantum mechanical three-body problem, its difficulties, and its importance in nuclear physics. Scattering experiments with three-particle breakup are presented. Attention then turns to some concepts of quantum mechanics, with emphasis on two-particle scattering and the Hamiltonian for three particles. The chapters that follow are devoted to the Faddeev equations, including those for scattering states and transition operators, and how such equations can be solved in practice. The solution of the Faddeev equations for separable potentials and local potentials is presented, along with the use of Padé approximation to solve the Faddeev equations. This book concludes with an appraisal of variational methods for bound states, elastic and rearrangement scattering, and the breakup reaction. A promising variational method for solving the Faddeev equations is described.
This book will be of value to students interested in three-particle physics and to experimentalists who want to understand better how the theoretical data are derived.

Inhalt

1. Introduction

2. Scattering Experiments with Three-Particle Breakup

3. Difficulties of the Theory

4. Importance of the Three-Body Problem in Nuclear Physics

2. Some Concepts of Quantum Mechanics

   1. The Two-Particle Problem

   1.1. The Hamiltonian

   1.2. Boundary Condition of the Scattering State

   1.3. The Mfller Operator

   1.4. Resolvent Equation and Lippmann-Schwinger Equation

   1.5. The S-Matrix

   1.6. The T-Matrix

   1.7. The Unitarity Relation

   2. The Three-Particle Problem

   2.1. The Hamiltonian

   2.2. Two-Particle Subsystems in Three-Particle Space

   2.3. Boundary Conditions and Mfller Operators

   2.4. Resolvent Equation and Lippmann-Schwinger Equation

3. The Faddeev Equations

   1. The Faddeev Equations for the T-Matrix

   2. The Faddeev Equations for the Resolvent

   3. The Faddeev Equations for the Scattering States

   4. The S-Matrix

   5. The Faddeev Equations for Transition Operators

   6. The Unitarity Relation

4. Solution Methods for the Faddeev Equations

   1. Partial Wave Decomposition of the Faddeev Equations

   2. Some Concepts of the Theory of Integral Equations

   3. Application to the Faddeev Equations

5. Separable Potentials

   1. Separable Potentials in the Two-Particle Problem

   2. Solution of the Faddeev Equations for Separable Potentials

   2.1. Rearrangement Scattering

   2.2. Three-Particle Breakup

   2.3. Identical Particles

   2.4. Numerical Solution of the Faddeev Equations for Separable Potentials

   2.5. Results for Three Identical Particles

   2.6. The Watson Model

   2.7. Results for the Three-Nucleon System

6. Solution of the Faddeev Equations for Local Potential

   1. Direct Solution of the Faddeev Equations for Local Potential

   2. The Schmidt Method (Weinberg's Quasiparticle Method)

   3. The Quasiparticle Method in the Three-Particle Problem

   3.1. The Alt-Grassberger-Sandhas Equations

   3.2. Application of the Quasiparticle Method to the Three-Particle Resolvent Equation

   3.3. Practical Calculations with the Quasiparticle Method

7. Solution of the Faddeev Equations by Padé Approximation

   1. The Technique of Padé Approximation

   2. Padé Approximation and Integral Equations

   3. Padé Approximation and the Faddeev Equations

8. Variational Methods

   1. Variational Methods for Bound States

   2. Variational Methods for Elastic Scattering and for Multichannel Scattering

   3. Variational Methods for Multichannel Scattering with Three-Particle Breakup

   3.1. Hyperspherical Functions

   3.2. Sequential Decay States

   3.3. The Method of Pieper, Schlessinger, and Wright

References

Index