This book is based on the course in theoretical nuclear physics that has been given by the author for some years at the T. G. Shevchenko Kiev State University. This version is supplemented and revised to include new results obtained after 1971 and 1975 when the first and second editions were published. This text is intended as an introduction to the nonrelativistic theory of po tential scattering. The analysis is based on the scattering matrix concept where the relationship between the scattering matrix and observable physical quantities is considered. The stationary formulation of the…mehr
This book is based on the course in theoretical nuclear physics that has been given by the author for some years at the T. G. Shevchenko Kiev State University. This version is supplemented and revised to include new results obtained after 1971 and 1975 when the first and second editions were published. This text is intended as an introduction to the nonrelativistic theory of po tential scattering. The analysis is based on the scattering matrix concept where the relationship between the scattering matrix and observable physical quantities is considered. The stationary formulation of the scattering problem is presented; particle wave functions in the external field are obtained. A formulation of the optical theorem is given as well as a discussion on time inversion and the reci procity theorem. Analytic properties of the scattering matrix, dispersion relations, and complex moments are analyzed. The dispersion relations for an arbitrary di rection scattering amplitude are proven, and analytic properties of the amplitude in the plane of the complex cosine of the scattering angle are studied in detail.
1. Quantum Mechanical Description and Representations.- 1.1 Quantum Mechanical Description of Physical Systems.- 1.2 Schrödinger Representation.- 1.3 Heisenberg Representation.- 1.4 Interaction Representation.- 1.5 Time-Dependent Green Functions.- Problems.- 2. The Scattering Matrix and Transition Probability.- 2.1 The Scattering Matrix.- 2.2 Time Shift Operator in the Interaction Representation.- 2.3 Integrals of Motion and S Matrix Diagonalization.- 2.4 Transition Probability per Unit Time.- 2.5 Integral Equation for die t Matrix.- 2.6 Transformation of the Scattering Matrix. Cross Sections.- Problems.- 3 Stationary Scattering Theory.- 3.1 The Scattering Amplitude.- 3.2 The Lippmann-Schwinger Equation.- 3.3 The Möller Operators ?+ and ?-.- 3.4 The Green Functions G0 and G.- 3.5 The Scattering Amplitude and the Transition Matrix.- 3.6 Inelastic Scattering and Reactions.- 3.7 Born Approximation and Perturbation Theory.- 3.8 High Energy Approximation.- Problems.- 4. Particle Wave Functions in the External Field.- 4.1 Partial Wave Expansion.- 4.2 Square Well Potential.- 4.3 Coulomb Field.- 4.4 Partial Green Functions and the Scattering Matrix.- 4.5 Variable Phase Approach.- Problems.- 5. Optical Theorem.- 5.1 The Total Cross Section and the Elastic Scattering Amplitude.- 5.2 Unitarity Relation for the Elastic Scattering Amplitude.- Problems.- 6. Time Inversion and Reciprocity Theorem.- 6.1 Transformation of Wave Functions and Operators Under Inversion of Time.- 6.2 Time Inversion Operators for Particular Systems.- 6.3 Time-Inversed Wave Function.- 6.4 The Reciprocity Theorem and Detailed Balance.- Problems.- 7. Analytic Properties of the Scattering Matrix.- 7.1 Analytic Properties of Radial Wave Functions.- 7.2 Generalization to Include Nonzero Angular Momenta.- 7.3Jost Function Zeros and Bound States.- 7.4 Symmetry and Dislocation of Scattering Matrix Singularities in the Complex k Plane.- 7.5 Bound States and Extra Zeros.- 7.6 Quasistationary States and Resonances.- 7.7 Virtual States.- 7.8 The Scattering Matrix in the Case of a Square Well Potential.- Problems.- 8. Dispersion Relations.- 8.1 Integral Representations of the Jost Functions.- 8.2 Levinson Theorem.- 8.3 Complex Energy Shell.- 8.4 Analyticity of the Scattering Matrix and the Causality Principle.- 8.5 Dispersion Relations for the Forward Direction Scattering Amplitude.- 8.6 Dispersion Relations for the Arbitrary Direction Scattering Amplitude.- Problems.- 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 Analytic Properties of the Scattering Amplitude in the Complex z Plane.- 9.4 Asymptotic Behavior of the Scattering Amplitude for Large z.- 9.5 Momentum Transfer Dispersion Relations.- Problems.- 10. Double Dispersion Relations.- 10.1 Mandelstam Representation.- 10.2 Spectral Density and Unitarity Condition.- Problems.- 11. The Inverse Problem of Scattering Theory.- 11.1 Integral Representation of the Solutions of the Scattering Problem.- 11.2 Reproducing the Potential by the Scattering Phase Shifts.- Problems.- 12. Separable Representation of the Scattering Amplitude.- 12.1 The Scattering Amplitude off the Energy Shell.- 12.2 Hilbert-Schmidt Expansion of the Scattering Amplitude.- 12.3 Properties of Eigenvalues and Eigenfunctions of the Kernel of the Lippmann-Schwinger Equation.- Problems.- 13. Three-Particle Scattering.- 13.1 The Faddeev Equations.- 13.2 Coordinates and Momenta in the Three-Particle System.- 13.3 Momentum Representation.- 13.4 Partial Wave Expansion.- 13.5 Separable Expansion of the Two-Particle t Matrix and One-Dimensional Form of the Faddeev Equations.- Problems.- 14. Scattering of Spin-Possessing Particles.- 14.1 The Spin Wave Function and the Density Matrix.- 14.2 Spin-Tensor Expansion of the Density Matrix.- 14.3 The Scattering Amplitude in the Case of Spin-Possessing Particles.- 14.4 Addition of Spin and Angular Momentum and Diagonalization of the S Matrix.- 14.5 Spin ½ - Spin 0 Particle Scattering.- 14.6 Spin 1 - Spin 0 Particle Scattering.- Problems.- References.- General Reading.
1. Quantum Mechanical Description and Representations.- 1.1 Quantum Mechanical Description of Physical Systems.- 1.2 Schrödinger Representation.- 1.3 Heisenberg Representation.- 1.4 Interaction Representation.- 1.5 Time-Dependent Green Functions.- Problems.- 2. The Scattering Matrix and Transition Probability.- 2.1 The Scattering Matrix.- 2.2 Time Shift Operator in the Interaction Representation.- 2.3 Integrals of Motion and S Matrix Diagonalization.- 2.4 Transition Probability per Unit Time.- 2.5 Integral Equation for die t Matrix.- 2.6 Transformation of the Scattering Matrix. Cross Sections.- Problems.- 3 Stationary Scattering Theory.- 3.1 The Scattering Amplitude.- 3.2 The Lippmann-Schwinger Equation.- 3.3 The Möller Operators ?+ and ?-.- 3.4 The Green Functions G0 and G.- 3.5 The Scattering Amplitude and the Transition Matrix.- 3.6 Inelastic Scattering and Reactions.- 3.7 Born Approximation and Perturbation Theory.- 3.8 High Energy Approximation.- Problems.- 4. Particle Wave Functions in the External Field.- 4.1 Partial Wave Expansion.- 4.2 Square Well Potential.- 4.3 Coulomb Field.- 4.4 Partial Green Functions and the Scattering Matrix.- 4.5 Variable Phase Approach.- Problems.- 5. Optical Theorem.- 5.1 The Total Cross Section and the Elastic Scattering Amplitude.- 5.2 Unitarity Relation for the Elastic Scattering Amplitude.- Problems.- 6. Time Inversion and Reciprocity Theorem.- 6.1 Transformation of Wave Functions and Operators Under Inversion of Time.- 6.2 Time Inversion Operators for Particular Systems.- 6.3 Time-Inversed Wave Function.- 6.4 The Reciprocity Theorem and Detailed Balance.- Problems.- 7. Analytic Properties of the Scattering Matrix.- 7.1 Analytic Properties of Radial Wave Functions.- 7.2 Generalization to Include Nonzero Angular Momenta.- 7.3Jost Function Zeros and Bound States.- 7.4 Symmetry and Dislocation of Scattering Matrix Singularities in the Complex k Plane.- 7.5 Bound States and Extra Zeros.- 7.6 Quasistationary States and Resonances.- 7.7 Virtual States.- 7.8 The Scattering Matrix in the Case of a Square Well Potential.- Problems.- 8. Dispersion Relations.- 8.1 Integral Representations of the Jost Functions.- 8.2 Levinson Theorem.- 8.3 Complex Energy Shell.- 8.4 Analyticity of the Scattering Matrix and the Causality Principle.- 8.5 Dispersion Relations for the Forward Direction Scattering Amplitude.- 8.6 Dispersion Relations for the Arbitrary Direction Scattering Amplitude.- Problems.- 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 Analytic Properties of the Scattering Amplitude in the Complex z Plane.- 9.4 Asymptotic Behavior of the Scattering Amplitude for Large z.- 9.5 Momentum Transfer Dispersion Relations.- Problems.- 10. Double Dispersion Relations.- 10.1 Mandelstam Representation.- 10.2 Spectral Density and Unitarity Condition.- Problems.- 11. The Inverse Problem of Scattering Theory.- 11.1 Integral Representation of the Solutions of the Scattering Problem.- 11.2 Reproducing the Potential by the Scattering Phase Shifts.- Problems.- 12. Separable Representation of the Scattering Amplitude.- 12.1 The Scattering Amplitude off the Energy Shell.- 12.2 Hilbert-Schmidt Expansion of the Scattering Amplitude.- 12.3 Properties of Eigenvalues and Eigenfunctions of the Kernel of the Lippmann-Schwinger Equation.- Problems.- 13. Three-Particle Scattering.- 13.1 The Faddeev Equations.- 13.2 Coordinates and Momenta in the Three-Particle System.- 13.3 Momentum Representation.- 13.4 Partial Wave Expansion.- 13.5 Separable Expansion of the Two-Particle t Matrix and One-Dimensional Form of the Faddeev Equations.- Problems.- 14. Scattering of Spin-Possessing Particles.- 14.1 The Spin Wave Function and the Density Matrix.- 14.2 Spin-Tensor Expansion of the Density Matrix.- 14.3 The Scattering Amplitude in the Case of Spin-Possessing Particles.- 14.4 Addition of Spin and Angular Momentum and Diagonalization of the S Matrix.- 14.5 Spin ½ - Spin 0 Particle Scattering.- 14.6 Spin 1 - Spin 0 Particle Scattering.- Problems.- References.- General Reading.
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