# Mathematical Description of a Classical Physical System # Classical Observables as Abelian C*-Algebra # States as Positive Linear Functionals on the Algebra of Observables # Gelfand-Naimark Theorem: The Observables as Hilbert Space Operators # Heisenberg Uncertainty Relations # The Weyl Algebra # The Quantum Hamiltonian # Double Slit Interference and Particle Wave Duality # Quantum Harmonic Oscillator # The Hydrogen Atom # and other topics
Mathematical Description of a Classical Physical System # Classical Observables as Abelian C*-Algebra # States as Positive Linear Functionals on the Algebra of Observables # Gelfand-Naimark Theorem: The Observables as Hilbert Space Operators # Heisenberg Uncertainty Relations # The Weyl Algebra # The Quantum Hamiltonian # Double Slit Interference and Particle Wave Duality # Quantum Harmonic Oscillator # The Hydrogen Atom # and other topics
Preface * Part I. Introductory Concepts in Quantum Theory * Units of Measurement * Quantum Mechanics---Some Remarks and Themes * Equations of Motion in Classical Mechanics * Quantization and the Schrodinger Equation * Hypergeometric Equations and Special Functions * Hydrogen-like Atoms * Heisenberg's Uncertainty Principle * Group Representations and Selection Rules * A Charged Particle in an Electro-magnetic Field * Spin Wave Functions * Introduction to Multi-electron Atoms * Part II. Some Selected Topics * Fresnel Integrals and Feynman Integrals * Path Integral for the Harmonic Oscillator * Euclidean Path Integrals * The Density Matrix and Partition Function * Zeta Regularization * Helmholtz Free Energy and the Selberg Trace Formula * The Zeta Function of a Product * Two Guest Lectures * Some Further Electron Configurations * Mendeleev Periodic Table * Another Example of Zeta Regularization * Integral Evaluation * Some Informal Comments on QFT * References * Index
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.
Structured as a dialogue between a mathematician and a physicist, Symmetry and Quantum Mechanics unites the mathematical topics of this field into a compelling and physically-motivated narrative that focuses on the central role of symmetry. Aimed at advanced undergraduate and beginning graduate students in mathematics with only a minimal background in physics, this title is also useful to physicists seeking a mathematical introduction to the subject. Part I focuses on spin, and covers such topics as Lie groups and algebras, while part II offers an account of position and momentum in the context of the representation theory of the Heisenberg group, along the way providing an informal discussion of fundamental concepts from analysis such as self-adjoint operators on Hilbert space and the Stone-von Neumann Theorem. Mathematical theory is applied to physical examples such as spin-precession in a magnetic field, the harmonic oscillator, the infinite spherical well, and the hydrogen atom.
(NOTE: All chapters conclude with Final Thoughts, and Exercises and Problems most also include Annotated Selected Readings.)I. WHAT'S IT ALL ABOUT?: A QUALITATIVE INTRODUCTION. 1. Introductory: Why Quantum Physics? 2. Into the Microworld: Duality and the Double Slit. II. STARTING FROM SCRATCH. 3. First Postulates: The State Function and Its Interpretation. 4. An Illustrative Interlude: Wave Packets in One Dimension. 5. Observables in Quantum Physics: A Pragmatist's Approach. 6. A Quantum Equation of Motion: The Schroedinger Equation At Last. 7. Simplifying Matters: The Time-Independent Schroedinger Equation. III. EXAMPLES, EXAMPLES, EXAMPLES. 8. States of a Particle in One Dimension I: Mainly Piecewise-Constant Potentials. 9. States of a Particle in One Dimension II: Mainly Continuous Potentials. IV. TOOLS OF THE TRADE MORE FORMALISM. 10. Operators in Quantum Mechanics I: The Importance of Being Hermitian. 11. Operators in Quantum Mechanics II: To Commute or Not to Commute. 12. Eigenfunction Expansions: How to Solve Any Problem in Quantum Mechanics. 13. Altered States: The Great Measurement Mystery. Appendixes. A. SI Units (The International System of Units. B. Fundamental Constants in SI (and CGS) Units. C. Order-of-Magnitude Estimates. D. Conversion Tables. E. Prefix Dictionary. F. A User's Guide to Atomic Units. G. The Greek Alphabet. H. Handy Trigonometric identities. I. Integrals You Should Know. J. Integrals Involving the Gaussian Function. K. Review of Complex Numbers. L. The Dirac Delta Function. Bibliography. Index.
A Brief Survey of Analytical Dynamics, Discovery of Matrix Mechanics, Mathematical Preliminaries Postulates of Quantum Theory Equations of Motion, Hamiltonian Operator and the Commutation Relations Symmetries and Conservation Laws, Classical and Quantum Integrability Classical and Quantum Mechanical Degeneracies Bound State Energies for One-Dimensional Problems Exactly Solvable Potentials, Supersymmetry and Shape Invariance Supersymmetry and Superpotential Shape Invariant Potentials Two-Body Problem Determination of the Angular Momentum Eigenvalues Methods of Integration of the Heisenberg Equations of Motion Discrete-Time Formulation of the Heisenberg's Equations of Motion Quantum Tunneling Using Discrete-Time Formulation Integration of the Operator Differential Equations Perturbation Theory Other Methods of Approximation Heisenberg Correspondence Principle, Quantization of the Equations of Motion with Higher Derivatives Determinantal Method in Potential Scattering Time-Dependent Scattering Theory The Scattering Matrix Impact Parameter Representation of the Scattering Amplitude Transition Probability Quantum Diffraction, Motion of a Charged Particle in Electromagnetic Field The Aharanov-Bohm Effect Neutron Interferometry Quantum Many-Body Problem Hartree and Hartree-Fock Approximations Second Quantization Bogoliubov Transformation for a Many-Boson System Bogoliubov Transformation for Fermions Interacting Through Pairing Forces Quantum Theory of Free Electromagnetic Field Casimir Force Interaction of Radiation with Matter Heisenberg Equations for Interaction of an Atom with Radiation Bell's Inequalities