Physical systems have been traditionally described in terms of either classical or quantum mechanics. But in recent years, semiclassical methods have developed rapidly, providing deep physical insight and computational tools for quantum dynamics and spectroscopy. In this book, Eric Heller introduces and develops this subject, demonstrating its power with many examples. In the first half of the book, Heller covers relevant aspects of classical mechanics, building from them the semiclassical way through the semiclassical limit of the Feynman path integral. The second half of the book applies this approach to various kinds of spectroscopy, such as molecular spectroscopy and electron imaging and quantum dynamical systems with an emphasis on tunneling. Adopting a distinctly time-dependent viewpoint, Heller argues for semiclassical theories from experimental and theoretical vantage points valuable to research in physics and chemistry. Featuring more than two hundred figures, the book provides a geometric, phase-space, and coordinate-space pathway to greater understanding. Filled with practical examples and applications, The Semiclassical Way to Dynamics and Spectroscopy is a comprehensive presentation of the tools necessary to successfully delve into this unique area of quantum mechanics.
Physical systems have been traditionally described in terms of either classical or quantum mechanics. But in recent years, semiclassical methods have developed rapidly, providing deep physical insight and computational tools for quantum dynamics and spectroscopy. In this book, Eric Heller introduces and develops this subject, demonstrating its power with many examples. In the first half of the book, Heller covers relevant aspects of classical mechanics, building from them the semiclassical way through the semiclassical limit of the Feynman path integral. The second half of the book applies this approach to various kinds of spectroscopy, such as molecular spectroscopy and electron imaging and quantum dynamical systems with an emphasis on tunneling. Adopting a distinctly time-dependent viewpoint, Heller argues for semiclassical theories from experimental and theoretical vantage points valuable to research in physics and chemistry. Featuring more than two hundred figures, the book provides a geometric, phase-space, and coordinate-space pathway to greater understanding. Filled with practical examples and applications, The Semiclassical Way to Dynamics and Spectroscopy is a comprehensive presentation of the tools necessary to successfully delve into this unique area of quantum mechanics.
Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information
Providing a pedagogical introduction to the essential principles of path integrals and Hamiltonians, this book describes cutting-edge quantum mathematical techniques applicable to a vast range of fields, from quantum mechanics, solid state physics, statistical mechanics, quantum field theory, and superstring theory to financial modeling, polymers, biology, chemistry, and quantum finance. Eschewing use of the Schrödinger equation, the powerful and flexible combination of Hamiltonian operators and path integrals is used to study a range of different quantum and classical random systems, succinctly demonstrating the interplay between a system's path integral, state space, and Hamiltonian. With a practical emphasis on the methodological and mathematical aspects of each derivation, this is a perfect introduction to these versatile mathematical methods, suitable for researchers and graduate students in physics and engineering.
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.
"Feynman path integrals are ubiquitous in quantum physics, even if a large part of the scientific community still considers them as a heuristic tool that lacks a sound mathematical definition. Our book aims to refute this prejudice, providing an extensive and self-contained description of the mathematical theory of Feynman path integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. This second edition presents a detailed discussion of the general theory of complex integration on infinite dimensional spaces, providing on one hand a unified view of the various existing approaches to the mathematical construction of Feynman path integrals and on the other hand a connection with the classical theory of stochastic processes. Moreover, new chapters containing recent applications to several dynamical systems have been added. This book bridges between the realms of stochastic analysis and the theory of Feynman path integration. It is accessible to both mathematicians and physicists"--
The Quantum Chemist’s Companion: Background Classical and Quantum Mechanics is the first part of a comprehensive two volume reference designed for students and researchers seeking a deeper, more intuitive understanding of quantum chemistry. Filling a long standing gap in the traditional chemistry curriculum, this volume builds the essential bridge between classical mechanics, quantum mechanics and the mathematical tools that underpin them. Ideal as both a course companion and a long-term desk reference, this book brings clarity to concepts that often challenge chemists especially those rarely explored in standard chemistry training. Detailed treatments of Lagrangian and Hamiltonian mechanics, the structure of vector spaces and the role of linear algebra in quantum theory make this volume an indispensable resource. Every derivation is presented step by step, ensuring that even the most mathematically inexperienced reader can follow the logic with confidence. Inside this volume: • Part 1 provides a guided introduction to the central ideas of classical and quantum mechanics, emphasizing how the two theories connect; • Part 2 dives deeper into classical mechanics, exploring key systems, such as the harmonic oscillator, that form the backbone of later quantum discussions; • Part 3 covers systematic development of quantum mechanics built directly on top of the foundations established in Parts 1 and 2; and • Part 4 consolidates the tools and techniques needed for the mathematical material in this volume and beyond. Whether you are navigating quantum mechanics for the first time or revisiting the fundamentals with a fresh perspective, this book supports learning at every stage. Use it cover to cover or consult it as a precise, cross referenced guide to specific topics. Either way, it provides the clarity and depth needed to transform quantum theory from an abstract challenge into a coherent, intuitive framework.
Applying quantum mechanics to many-particle systems is an active area of research. The popularity of using quantum trajectories as a computational tool has exploded over the last decade, finally bringing this methodology to the level of practical application. This book explores this powerful tool to efficiently solve both static and time-dependent systems across a large area of quantum mechanics. Many of the pioneers who have either developed the subject from its inception or have improved on the subject contribute to this volume. They offer their insights on Lagrangian and Eulerian methods as well as various mixed quantum-classical techniques, among other topics--Provided by publisher.