Advanced Calculus and its Applications in Variational Quantum Mechanics and Relativity Theory
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.
SECTION I: ADVANCED CALCULUS. The Implicit Function Theorem and Related Results. Manifolds in Rn. SECTION II: APPLICATIONS TO VARIATIONAL QUANTUM MECHANICS AND RELATIVITY THEORY. A Variational Formulation for the Relativistic Klein-Gordon Equation. Some Numerical Results and Examples. A variational formulation for relativistic mechanics based on Riemannian geometry and its application to the quantum mechanics context. A General Variational Formulation for Relativistic Mechanics Based on Fundamentals of Differential Geometry. A New Interpretation for the Bhor Atomic Model. Existence and Duality for Superconductivity and Related Models. A Classical Description of the Radiating Cavity Model in Quantum Mechanics.