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Lectures in Scattering Theory
International Series of Monographs in Natural Philosophy
- 1st Edition - October 22, 2013
- Author: A. G. Sitenko
- Editor: P. J. Shepherd
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 7 1 0 1 - 2
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 8 6 8 2 - 5
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… Read more
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Request a sales quoteLectures 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.
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
- No. of pages: 280
- Language: English
- Edition: 1
- Published: October 22, 2013
- Imprint: Pergamon
- Paperback ISBN: 9781483171012
- eBook ISBN: 9781483186825
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