The first part of this book reviews some key topics on multi-variable advanced calculus. The approach presented includes detailed and rigorous studies on surfaces in Rn which comprises items such as differential forms and an abstract version of the Stokes Theorem in Rn. The conclusion section introduces readers to Riemannian geometry, which is used in the subsequent chapters.
The second part reviews applications, specifically in variational quantum mechanics and relativity theory. Topics such as a variational formulation for the relativistic Klein-Gordon equation, the derivation of a variational formulation for relativistic mechanics firstly through (semi)-Riemannian geometry are covered. The second part has a more general context. It includes fundamentals of differential geometry.
The later chapters describe a new interpretation for the Bohr atomic model through a semi-classical approach. The book concludes with a classical description of the radiating cavity model in quantum mechanics.
Author(s): Fabio Silva Botelho
Publisher: CRC Press
Year: 2021
Language: English
Pages: 335
Cover
Title Page
Copyright Page
Preface
Contents
Section I: Advanced Calculus
1. The Implicit Function Theorem and Related Results
1.1 Introduction
1.2 Implicit function theorem, scalar case
1.3 Vectorial functions in ℝn
1.3.1 Limit proprieties
1.4 Implicit function theorem for the vectorial case
1.5 Lagrange multipliers
1.6 Lagrange multipliers, the general case
1.7 Inverse function theorem
1.8 Some topics on differential geometry
1.8.1 Arc length
1.8.2 The arc length function
1.9 Some notes about the scalar curvature in a surface in ℝ3
2. Manifolds in ℝn
2.1 Introduction
2.2 The local form of sub-immersions
2.3 The local form of immersions
2.4 Parameterizations and surfaces in ℝn
2.4.1 Change of coordinates
2.4.2 Differentiable functions defined on surfaces
2.5 Oriented surfaces
2.6 Surfaces in ℝn with boundary
2.6.1 Parameterizations for surfaces in ℝn with boundary
2.7 The tangent space
2.8 Vector fields
2.9 The generalized derivative
2.9.1 On the integral curve existence
2.10 Differential forms
2.11 Integration of differential forms
2.12 A simple example to illustrate the integration process
2.13 Volume (area) of a surface
2.14 Change of variables, the general case
2.15 The Stokes Theorem
2.15.1 Recovering the classical results on vector calculus in ℝ3 from the general Stokes Theorem
Section II: Applications to Variational Quantum Mechanics and Relativity Theory
3. A Variational Formulation for the Relativistic Klein-Gordon Equation
3.1 Introduction
3.2 The Newtonian approach
3.3 A brief note on the relativistic context, the Klein-Gordon equation
3.3.1 Obtaining the Klein-Gordon equation
3.4 About the role of normal field as the wave function in quantum mechanics
3.5 A more general proposal for the energy, one more example
3.6 Another example, generalizing the standard quantum approach
3.7 Recovering the standard quantum approach
3.7.1 About the role of the angular momentum in quantum mechanics
3.7.2 A brief note concerning the relation between classical rotations and the quantum angular momentum
3.7.3 The eigenvalues of Lz
3.8 A numerical example
3.9 A brief note on the many body quantum approach
3.10 A brief note on a formulation similar to those of density functional theory
4. Some Numerical Results and Examples
4.1 Some preliminary results
4.2 A second numerical example
4.3 A third example, the hydrogen atom
4.4 A related control problem
4.4.1 A numerical example
4.4.2 Another numerical example
4.5 An approximation for the standard many body problem
4.6 Conclusion
5. A Variational Formulation for Relativistic Mechanics Based on Riemannian Geometry and its Application to the Quantum Mechanics Context
5.1 Introduction
5.2 Some introductory topics on vector analysis and Riemannian geometry
5.3 A relativistic quantum mechanics action
5.3.1 The kinetics energy
5.3.2 The energy part relating the curvature and wave function
5.4 Obtaining the relativistic Klein-Gordon equation as an approximation of the previous action
5.5 A note on the Einstein field equations in a vacuum
5.6 Conclusion
6. A General Variational Formulation for Relativistic Mechanics Based on Fundamentals of Differential Geometry
6.1 Introduction
6.2 The system energy
6.3 The final energy expression
6.4 Causal structure
6.5 Dependence domains and hyperbolicity
6.6 Existence of minimizer for the previous general functional
6.7 Conclusion
7. A New Interpretation for the Bhor Atomic Model
7.1 Introduction
7.2 The Newtonian approach and a concerning extension
7.3 A brief note on the relativistic context, the Klein-Gordon equation
7.4 A second model and the respective energy expression
7.5 A brief note on the relativistic context for such a second model
7.6 A brief note on the case including electro-magnetic effects
7.6.1 About a specific Lorentz transformation
7.6.2 Describing the self interaction energy and obtaining a final variational formulation
7.7 A new interpretation of the Bohr atomic model
7.8 A system with a large number of interacting atoms
7.8.1 A proposal for the case in which N is very large
7.9 About the definition of Temperature
7.10 A note on the Entropy concept
7.11 About modeling a chemical reaction
7.11.1 About the variational formulation modeling such a chemical reaction
7.11.2 The final variational formulation
7.12 A note on the Spin operator
8. Existence and Duality for Superconductivity and Related Models
8.1 Introduction
8.2 A global existence result for the full complex Ginzburg-Landau system
8.3 A brief initial description of our proposal for duality
8.4 The duality principle for a local extremal context
8.5 A second duality principle
8.6 A third duality principle
8.7 A criterion for global optimality
8.7.1 The concerning duality principle
8.8 Numerical results
8.8.1 A numerical example
9. A Classical Description of the Radiating Cavity Model in Quantum Mechanics
9.1 Introduction
9.2 Regarding an approximate classical description of the last section previous result
9.3 Conclusion
References
Index
