Quantum Mechanics in Potential Representation and Applications

✍ Scribed by Arvydas Juozapas Janavičius, Donatas Jurgaitis

• Publisher: WSPC
• Year: 2020
• Tongue: English
• Leaves: 270
• Category: Library

✦ Synopsis

This book is written with a focus on new mathematical methods and physical modeling that lay the groundwork for an interpretation to various experimental results and phenomena in nuclear physics, quantum mechanics, and particle physics. Summarized in three parts, the main topics of the book are as follows.

The first part importantly addresses scattering theory and nuclear reactions, with the usage of new potential representation method. This perturbation method offers the wave function as a product of the free particle solution and a function which depends on the interaction potential, allowing handy analytical expressions and integral equations for finding scattering matrices. It is highly applicable to the study of scattering and absorption of neutrons in atomic reactors, as well as the interactions between protons and nuclei by scattering processes in, for example, cyclotrons. The second part of the book concerns the perturbation method by variation of free constants and the semi-relativistic shell model of heavy nuclei in order to understand their stability. The last part is then furnished with the semi-relativistic model of mesons and relates to the binding energies of quarks in charm and bottom mesons.

This book would be a valuable resource for students and researchers on new mathematical methods in the theoretical unravelling of experiments concerning nuclei and mesons, nuclear reactors, radioactive isotopes, particle accelerators, new materials in electronics and healthcare products, as well as other practical applications of nuclear physics and quantum mechanics.

✦ Table of Contents

Contents
Preface
Introduction
Chapter 1: Quantum Nature of the Matter
1.1 The Structure of Atoms
1.2 The Schr¨odinger Equation
1.3 The Fundamental Forces
References
Chapter 2: Quantum Waves and Particles Diffusion in Physical Vacuum
2.1 Introduction
2.2 Diffusion of Quantum Waves
2.3 The Quantum Diffusion of an Electron in the Hydrogen Atom
2.4 Solution of the Quantum Diffusion Equation for the Tunnel Effect for a Rectangular Barrier
2.5 Conclusions
References
Chapter 3: Nuclear Forces
3.1 The Interactions between Nucleons
3.2 The Shell Model and Mean Field Potentials
References
Chapter 4: Systems of Micro Particles
References
Chapter 5: The Scattering Theory and Nuclear Reactions
5.1 Introduction
5.2 Nuclear Reactions and the Optical Model
5.3 Inverse Tasks of Scattering
References
Chapter 6: The Schrodinger Equation in Potential Representation
6.1 Introduction
6.2 Solution in the Case of s-Waves
6.3 The Case of Large Nuclei
6.4 Numerical Results and Conclusions
References
Chapter 7: A General Solution of the Schrodinger Equation
7.1 Introduction
7.2 General Solution
7.3 Numerical Results and Conclusions
References
Chapter 8: The General Solutions for Positive and Negative Energies
8.1 Introduction
8.2 The Integral Equation for Positive Energies in the Potential Representation
8.3 The Integral Equation for Negative Energies in the Potential Representation
8.4 Numerical Results and Conclusions
References
Chapter 9: The Connection between Scattering Matrices for Different Potentials
9.1 Introduction
9.2 Integral Equations for Positive Energies
9.3 Connection of Potential Representation Method with Green’s Functions
9.4 The Scattering Matrix
References
Chapter 10: The Separation of the Scattering Matrix from the Coulomb Field
10.1 Introduction
10.2 Obtaining Integral Equations
10.3 Obtaining the Scattering Matrix
References
Chapter 11: The General Solution for Bound States of the Woods–Saxon Potential
11.1 Introduction
11.2 The Derivation of Integral Equations
11.3 The Accuracy and Convergence of the Obtained Solutions
11.4 Conclusions
References
Chapter 12: The Perturbation Theory for Bound States
12.1 Introduction
12.2 Standard Green’s Functions
References
Chapter 13: The Perturbation Method of Variation of Free Constants
13.1 Green’s and Undefined Functions
References
Chapter 14: Green’s Functions and Non-physical Solutions
14.1 Introduction
14.2 Non-physical Solutions of the Radial Schrodinger Equation
14.3 Derivation of the Integral Equation
14.4 Results and Conclusions
References
Chapter 15: The Potential Representation Method for Non-spherical Perturbations
15.1 Introduction
15.2 Integral Equations for Negative Energies in the Potential Representation
References
Chapter 16: Solutions with the Model Potential for the Potential Representation Method
16.1 Introduction
16.2 Modelling the Solutions of the Schrodinger Equation with the Harmonic Oscillator Potential
16.3 Green’s Functions for the Potential Representation
16.4 The Accuracy of the Solutions Obtained
16.5 Conclusions
References
Chapter 17: Potential Representation for the Coulomb Interactions
17.1 Introduction
17.2 Bounded Systems: The Two-Body Task
17.3 The Many-Particles Task
17.4 Solution for the Ground State of the Helium Atom
17.5 Variation According to the Parameter Z
References
Chapter 18: Transformations of the Hamiltonian for Jastrow’s Correlation Method
18.1 Introduction
18.2 Transformation of the Hamiltonian for He Atom
18.3 Conclusions
References
Chapter 19: Stability of Nuclei
References
Chapter 20: Relativistic Corrections for Neutrons in the Harmonic Oscillator Well
20.1 Introduction
20.2 The Semi-relativistic Hamiltonian
20.3 Results and Conclusions
References
Chapter 21: Relativistic Corrections to One-Nucleon Energy Levels for 208Pb
21.1 Introduction
21.2 Semi-relativistic Equation
21.3 Methods and Results
21.4 Conclusions
References
Chapter 22: Solutions for the Semi-relativistic Equations for the Heaviest Nuclei
22.1 Introduction
22.2 The Integral–Differential Semi-relativistic Equation
22.3 Results and Conclusions
References
Chapter 23: Stability of the Shells of the Heaviest Atomic Nuclei in the Semi-relativistic Model
23.1 Introduction
23.2 The Integral–Differential Semi-relativistic Equation
23.3 Results and Conclusions
References
Chapter 24: The Semi-relativistic Nuclear Shell Model for the Many-Particles Case
24.1 Introduction
24.2 The Solutions of Integral–Differential Semi-relativistic Equation for the Singular Potentials
24.3 The System of Integral Semi-relativistic Equations in the Hartree–Fock Approach
24.4 Conclusions
References
Chapter 25: Relativistic Corrections for Different States of the Charmed and Bottom Mesons
25.1 Introduction
25.2 The Solutions of the Integral–Differential Semi-relativistic Equation
25.3 The Approximate System of Integral Semi-relativistic Equations
25.4 Semi-relativistic Model for Charmonium and Bottomonium
25.5 Conclusions
References
Bibliography of the Authors
Index