This book presents a concise introduction to the mathematical structure of quantum mechanics from the perspective of operator algebras for systems with a finite number of degrees of freedom. It begins with a review of the mathematical tools required throughout the text, followed by an exposition of the standard Schrödinger formulation based on the Dirac–von Neumann axioms. The operator algebraic formulation is then developed from a set of physically motivated principles, providing an alternative and conceptually transparent description of quantum systems. The book concludes with an application comparing the standard and operator algebraic formalisms. Written in Spanish, this book is intended primarily for Spanish-speaking undergraduate and graduate students in physics, mathematics, and mathematical physics, aiming to provide accessible access to modern mathematical formulations of quantum mechanics.
This book presents a concise introduction to the mathematical structure of quantum mechanics from the perspective of operator algebras for systems with a finite number of degrees of freedom. It begins with a review of the mathematical tools required throughout the text, followed by an exposition of the standard Schrödinger formulation based on the Dirac–von Neumann axioms. The operator algebraic formulation is then developed from a set of physically motivated principles, providing an alternative and conceptually transparent description of quantum systems. The book concludes with an application comparing the standard and operator algebraic formalisms. Written in Spanish, this book is intended primarily for Spanish-speaking undergraduate and graduate students in physics, mathematics, and mathematical physics, aiming to provide accessible access to modern mathematical formulations of quantum mechanics.
This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. It addition to traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin, and it introduces the reader to functional methods in quantum mechanics. This includes the Feynman path integral approach to quantum mechanics, integration in functional spaces, the relation between Feynman and Wiener integrals, Gaussian integration and regularized determinants of differential operators, fermion systems and integration over anticommuting (Grassmann) variables, supersymmetry and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer formula for the index of the Dirac operator. Prior to this book, mathematicians could find these topics only in physics textbooks and in specialized literature. This book is written in a concise style with careful attention to precise mathematics formulation of methods and results.Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory. Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis.
Quantum mechanics is one of the principle pillars of modern physics. It also remains a topic of great interest to mathematicians. Since its discovery it has inspired and been inspired by many topics within modern mathematics, including functional analysis and operator algebras, Lie groups, Lie algebras and their representations, principle bundles, distribution theory, and much more. Written with beginning graduate students in mathematics in mind, this book provides a thorough treatment of (nonrelativistic) quantum mechanics in a style that is leisurely, without the usual theorem-proof grammar of pure mathematics, while remaining mathematically honest. The author takes the time to fully develop the required mathematics and employs a consistent mathematical presentation to clarify the often-confusing notation of physics texts. Along the way the reader encounters several topics requiring more advanced mathematics than found in many discussions of the subject, making for a fascinating course in how mathematics and physics interact.
Quantum mechanics is one of the principle pillars of modern physics. It also remains a topic of great interest to mathematicians. Since its discovery it has inspired and been inspired by many topics within modern mathematics, including functional analysis and operator algebras, Lie groups, Lie algebras and their representations, principle bundles, distribution theory, and much more. Written with beginning graduate students in mathematics in mind, this book provides a thorough treatment of (nonrelativistic) quantum mechanics in a style that is leisurely, without the usual theorem-proof grammar of pure mathematics, while remaining mathematically honest. The author takes the time to fully develop the required mathematics and employs a consistent mathematical presentation to clarify the often-confusing notation of physics texts. Along the way the reader encounters several topics requiring more advanced mathematics than found in many discussions of the subject, making for a fascinating course in how mathematics and physics interact.
Os trabalhos originais de Dirac, “The Principles of Quantum Mechanics’ e von Neumann,\n“The Mathematical Foundations of Quantum Mechanics”, estabeleceram as bases do\nformalismo matemático atual da mecânica quântica. Junto com a álgebra C*, essas são três\ndiferentes abordagens matemáticas e conceituais que podem ser utilizadas para a caracterização\nmatemática da mecânica quântica. Fizemos uma releitura comparativa e mais matemática\ndesses dois clássicos originais buscando clarificar suas semelhanças e diferenças, e em seguida\nexpusemos de forma didática a álgebra C*, apresentando todos os pré-requisitos de estruturas\nalgébricas e outros conceitos necessários para a sua compreensão. O trabalho é ainda uma\nexposição mais didática dos formalismos de Dirac, von Neumann, e Álgebra C*, permitindo\numa compreensão mais rigorosa da matemática da mecânica Quântica.
Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. This new edition has additions and improvements throughout the book to make the presentation more student friendly.
This original and innovative textbook takes the unique perspective of introducing and solving problems in quantum mechanics using linear algebra methods, to equip readers with a deeper and more practical understanding of this fundamental pillar of contemporary physics. Extensive motivation for the properties of quantum mechanics, Hilbert space, and the Schrödinger equation is provided through analysis of the derivative, while standard topics like the harmonic oscillator, rotations, and the hydrogen atom are covered from within the context of operator methods. Advanced topics forming the basis of modern physics research are also included, such as the density matrix, entropy, and measures of entanglement. Written for an undergraduate audience, this book offers a unique and mathematically self-contained treatment of this hugely important topic. Students are guided gently through the text by the author's engaging writing style, with an extensive glossary provided for reference and numerous homework problems to expand and develop key concepts. Online resources for instructors include a fully worked solutions manual and lecture slides.