From Spinors to Quantum Mechanics discusses group theory and its use in quantum mechanics. Chapters 1 to 4 offer an introduction to group theory, and it provides the reader with an exact and clear intuition of what a spinor is, showing that spinors are just a mathematically complete notation for group elements. Chapter 5 contains the first rigorous derivation of the Dirac equation from a simple set of assumptions. The remaining chapters will interest the advanced reader who is interested in the meaning of quantum mechanics. They propose a novel approach to the foundations of quantum mechanics, based on the idea that the meaning of the formalism is already provided by the mathematics.In the traditional approach to quantum mechanics as initiated by Heisenberg, one has to start from a number of experimental results and then derive a set of rules and calculations that reproduce the observed experimental results. In such an inductive approach the underlying assumptions are not given at the outset. The reader has to figure them out, and this has proven to be difficult. The book shows that a different, bottom-up approach to quantum mechanics is possible, which merits further investigation as it demonstrates that with the methods used, the reader can obtain the correct results in a context where one would hitherto not expect this to be possible.
From Spinors to Quantum Mechanics discusses group theory and its use in quantum mechanics. Chapters 1 to 4 offer an introduction to group theory, and it provides the reader with an exact and clear intuition of what a spinor is, showing that spinors are just a mathematically complete notation for group elements. Chapter 5 contains the first rigorous derivation of the Dirac equation from a simple set of assumptions. The remaining chapters will interest the advanced reader who is interested in the meaning of quantum mechanics. They propose a novel approach to the foundations of quantum mechanics, based on the idea that the meaning of the formalism is already provided by the mathematics.In the traditional approach to quantum mechanics as initiated by Heisenberg, one has to start from a number of experimental results and then derive a set of rules and calculations that reproduce the observed experimental results. In such an inductive approach the underlying assumptions are not given at the outset. The reader has to figure them out, and this has proven to be difficult. The book shows that a different, bottom-up approach to quantum mechanics is possible, which merits further investigation as it demonstrates that with the methods used, the reader can obtain the correct results in a context where one would hitherto not expect this to be possible.
The mathematical formalism of quantum theory in terms of vectors and operators in infinite-dimensional complex vector spaces is very abstract. The definitions of many mathematical quantities used do not seem to have an intuitive meaning, which makes it difficult to appreciate the mathematical formalism and understand quantum mechanics. This book provides intuition and motivation to the mathematics of quantum theory, introducing the mathematics in its simplest and familiar form, for instance, with three-dimensional vectors and operators, which can be readily understood. Feeling confident about and comfortable with the mathematics used helps readers appreciate and understand the concepts and formalism of quantum mechanics. This book is divided into four parts. Part I is a brief review of the general properties of classical and quantum systems. A general discussion of probability theory is also included which aims to help in understanding the probability theories relevant to quantum mechanics. Part II is a detailed study of the mathematics for quantum mechanics. Part III presents quantum mechanics in a series of postulates. Six groups of postulates are presented to describe orthodox quantum systems. Each statement of a postulate is supplemented with a detailed discussion. To make them easier to understand, the postulates for discrete observables are presented before those for continuous observables. Part IV presents several illustrative applications, which include harmonic and isotropic oscillators, charged particle in external magnetic fields and the Aharonov–Bohm effect. For easy reference, definitions, theorems, examples, comments, properties and results are labelled with section numbers. Various symbols and notations are adopted to distinguish different quantities explicitly and to avoid misrepresentation. Self-contained both mathematically and physically, the book is accessible to a wide readership, including astrophysicists, mathematicians and philosophers of science who are interested in the foundations of quantum mechanics.
Abstract This book provides the reader with a contemporary and comprehensive introduction to Quantum Mechanics. It is suitable for beginners as well as for more advanced university students. Quantum mechanics is presented in a pedagogical fashion, with a clear logical organization. The various concepts and methods are introduced first in elementary terms, and later developed into more precise formulations. Systematic studies of approximation methods and the discussion of a wide class of physical applications follow. Part I of the book, together with the opening sections of Part II, provide adequate material for an introductory course of one semester at most universities. The rest of the book might be used in an advanced course on Quantum Mechanics. The basic material is fairly standard, even though some discussions such as those on general systems with time-dependent Hamiltonians, on metastable systems, as well as the discussions in some of the Complement sections, may not be found in other textbooks. The book also contains many original observations or new ways of illustrating even well-known subjects. In fact, the authors wish to convey in this book the sense of wonder in the logical simplicity and at the same time the beauty of subtle and far-reaching consequences of Quantum Mechanics, to young physics students in particular. Problem sets are provided at the end of each chapter, to be solved either analytically or by numerical methods. The solutions to both types of problems are given as separate pdf files or as Mathematica notebooks (there are 88 of them), all together on a CD accompanying the textbook. The presence of such a collection of numerical analyses enriches the main text and is one of the characteristic features of the book. With the many interesting systems discussed, the book will also be a useful reference for researchers and teachers. It provides the reader with a unique, enjoyable and rather complete textbook of Quantum Mechanics, destined to set a new standard for many years to come.
Introduction to the Maths and Physics of Quantum Mechanics details the mathematics and physics that are needed to learn the principles of quantum mechanics. It provides an accessible treatment of how to use quantum mechanics and why it is so successful in explaining natural phenomena. This book clarifies various aspects of quantum physics such as ‘why quantum mechanics equations contain “I”, the imaginary number?’, ‘Is it possible to make a transition from classical mechanics to quantum physics without using postulates?’ and ‘What is the origin of the uncertainty principle?’. A significant proportion of discussion is dedicated to the issue of why the wave function must be complex to properly describe our “real” world. The book also addresses the different formulations of quantum mechanics. A relatively simple introductory treatment is given for the “standard” Heisenberg matrix formulation and Schrodinger wave-function formulation and Feynman path integrals and second quantization are then discussed. This book will appeal to first- and second-year university students in physics, mathematics, engineering and other sciences studying quantum mechanics who will find material and clarifications not easily found in other textbooks. It will also appeal to self-taught readers with a genuine interest in modern physics who are willing to examine the mathematics and physics in a simple but rigorous way. Key Features: Written in an engaging and approachable manner, with fully explained mathematics and physics concepts Suitable as a companion to all introductory quantum mechanics textbooks Accessible to a general audience
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.
This authoritative, advanced introduction provides a complete, modern perspective on quantum mechanics. It clarifies many common misconceptions regarding wave/particle duality and the correct interpretation of measurements. The author develops the text from the ground up, starting from the fundamentals and presenting information at an elementary level, avoiding unnecessarily detailed and complex derivations in favor of simple, clear explanations. He begins in the simplest context of a two-state system and shows why quantum mechanics is inevitable, and what its relationship is to classical mechanics. He also outlines the decoherence approach to interpreting quantum mechanics. Distinguishing features: Provides a thorough grounding in the principles and practice of quantum mechanics, including a core understanding of the behavior of atoms, molecules, solids, and light. Utilizes easy-to-follow examples and analogies to illustrate important concepts. Helps develop an intuitive sense for the field, by guiding the reader to understand how the correct formulas reduce to the non-relativistic ones. Includes numerous worked examples and problems for each chapter.
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.