Lectures on the mathematics of quantum mechanics 1

Description

The first volume (General Theory) differs from most textbooks as it emphasizes the mathematical structure and mathematical rigor, while being adapted to the teaching the first semester of an advanced course in Quantum Mechanics (the content of the book are the lectures of courses actually delivered.). It differs also from the very few texts in Quantum Mechanics that give emphasis to the mathematical aspects because this book, being written as Lecture Notes, has the structure of lectures delivered in a course, namely introduction of the problem, outline of the relevant points, mathematical tools needed, theorems, proofs. This makes this book particularly useful for self-study and for instructors in the preparation of a second course in Quantum Mechanics (after a first basic course). With some minor additions it can be used also as a basis of a first course in Quantum Mechanics for students in mathematics curricula. The second part (Selected Topics) are lecture notes of a more advanced course aimed at giving the basic notions necessary to do research in several areas of mathematical physics connected with quantum mechanics, from solid state to singular interactions, many body theory, semi-classical analysis, quantum statistical mechanics. The structure of this book is suitable for a second-semester course, in which the lectures are meant to provide, in addition to theorems and proofs, an overview of a more specific subject and hints to the direction of research. In this respect and for the width of subjects this second volume differs from other monographs on Quantum Mechanics. The second volume can be useful for students who want to have a basic preparation for doing research and for instructors who may want to use it as a basis for the presentation of selected topics. Front Matter....Pages i-xxi Lecture 1: Elements of the History of Quantum Mechanics I....Pages 1-15 Lecture 2: Elements of the History of Quantum Mechanics II....Pages 17-37 Lecture 3: Axioms, States, Observables, Measurement, Difficulties....Pages 39-58 Lecture 4: Entanglement, Decoherence, Bell’s Inequalities, Alternative Theories....Pages 59-75 Lecture 5: Automorphisms; Quantum Dynamics; Theorems of Wigner, Kadison, Segal; Continuity and Generators....Pages 77-101 Lecture 6: Operators on Hilbert Spaces I; Basic Elements....Pages 103-124 Lecture 7: Quadratic Forms....Pages 125-149 Lecture 8: Properties of Free Motion, Anholonomy, Geometric Phase ....Pages 151-172 Lecture 9: Elements of \(C^*\) -algebras, GNS Representation, Automorphisms and Dynamical Systems....Pages 173-194 Lecture 10: Derivations and Generators. K.M.S. Condition. Elements of Modular Structure. Standard Form....Pages 195-216 Lecture 11: Semigroups and Dissipations. Markov Approximation. Quantum Dynamical Semigroups I....Pages 217-237 Lecture 12: Positivity Preserving Contraction Semigroups on \(C^*\) -algebras. Conditional Expectations. Complete Dissipations....Pages 239-259 Lecture 13: Weyl System, Weyl Algebra, Lifting Symplectic Maps. Magnetic Weyl Algebra....Pages 261-281 Lecture 14: A Theorem of Segal. Representations of Bargmann, Segal, Fock. Second Quantization. Other Quantizations (Deformation, Geometric)....Pages 283-312 Lecture 15: Semiclassical Limit; Coherent States; Metaplectic Group....Pages 313-334 Lecture 16: Semiclassical Approximation for Fast Oscillating Phases. Stationary Phase. W.K.B. Method. Semiclassical Quantization Rules....Pages 335-356 Lecture 17: Kato-Rellich Comparison Theorem. Rollnik and Stummel Classes. Essential Spectrum....Pages 357-382 Lecture 18: Weyl’s Criterium, Hydrogen and Helium Atoms....Pages 383-408 Lecture 19: Estimates of the Number of Bound States. The Feshbach Method....Pages 409-428 Lecture 20: Self-adjoint Extensions. Relation with Quadratic Forms. Laplacian on Metric Graphs. Boundary Triples. Point Interaction....Pages 429-454 Back Matter....Pages 455-459