This chapter forms a natural divide half-way into the book. In the first half, we discussed the principles of discrete time (DT) classical mechanics (CM). In the second half we focus mainly on quantum principles. This chapter is a good place in which to take stock of what we have done, what we plan to do, and how the two halves of the book are related. The common theme is Lee's approach to DT mechanics, discussed in Chapter 10. As we implied previously, theorists whose work is relevant to us can be classified into two types: the applied mathematicians and the mathematical physicists, who may also be called fundamentalists. This division is based not on any value judgements but on the motivations and ambitions driving a theorist's work, which are generally easy to identify. The applied mathematicians fall into two categories. The first consists of those interested in finding better ways of understanding CM and, if necessary, finding ever better approximations to it. They explore DT CM with that in mind and are generally not remotely interested in quantum mechanics (QM). The second category consists of quantum theorists, such as Bender, who develop DT numerical simulation to approximate standard QM. Lattice gauge theorists also fall into the applied group, since their discretization of spacetime is regarded at all times as an approximation to the continuum.
This chapter forms a natural divide half-way into the book. In the first half, we discussed the principles of discrete time (DT) classical mechanics (CM). In the second half we focus mainly on quantum principles. This chapter is a good place in which to take stock of what we have done, what we plan to do, and how the two halves of the book are related. The common theme is Lee's approach to DT mechanics, discussed in Chapter 10. As we implied previously, theorists whose work is relevant to us can be classified into two types: the applied mathematicians and the mathematical physicists, who may also be called fundamentalists. This division is based not on any value judgements but on the motivations and ambitions driving a theorist's work, which are generally easy to identify. The applied mathematicians fall into two categories. The first consists of those interested in finding better ways of understanding CM and, if necessary, finding ever better approximations to it. They explore DT CM with that in mind and are generally not remotely interested in quantum mechanics (QM). The second category consists of quantum theorists, such as Bender, who develop DT numerical simulation to approximate standard QM. Lattice gauge theorists also fall into the applied group, since their discretization of spacetime is regarded at all times as an approximation to the continuum.
Scope of this book Quantum mechanics is a difficult subject, and this book is intended to help the reader overcome the main difficulties in the way to understanding it. The first part of the book, Chs. 2–16, contains a systematic presentation of the basic principles of quantum theory, along with a number of examples which illustrate how these principles apply to particular quantum systems. The applications are, for the most part, limited to toy models whose simple structure allows one to see what is going on without using complicated mathematics or lengthy formulas. The principles themselves, however, are formulated in such a way that they can be applied to (almost) any nonrelativistic quantum system. In the second part of the book, Chs. 17–25, these principles are applied to quantum measurements and various quantum paradoxes, subjects which give rise to serious conceptual problems when they are not treated in a fully consistent manner. The final chapters are of a somewhat different character. Chapter 26 on decoherence and the classical limit of quantum theory is a very sketchy introduction to these important topics along with some indication as to how the basic principles presented in the first part of the book can be used for understanding them. Chapter 27 on quantum theory and reality belongs to the interface between physics and philosophy and indicates why quantum theory is compatible with a real world whose existence is not dependent on what scientists think and believe, or the experiments they choose to carry out. The Bibliography contains references for those interested in further reading or in tracing the origin of some of the ideas presented in earlier chapters.
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.
The goal of this book is to present classical mechanics, quantum mechanics, and statistical mechanics in an almost completely algebraic setting, thereby introducing mathematicians, physicists, and engineers to the ideas relating classical and quantum mechanics with Lie algebras and Lie groups. The book emphasizes the closeness of classical and quantum mechanics, and the material is selected in a way to make this closeness as apparent as possible. Much of the material covered here is not part of standard textbook treatments of classical or quantum mechanics (or is only superficially treated there). For physics students who want to get a broader view of the subject, this book may therefore serve as a useful complement to standard treatments of quantum mechanics. Almost without exception, this book is about precise concepts and exact results in classical mechanics, quantum mechanics, and statistical mechanics. The structural properties of mechanics are discussed independent of computational techniques for obtaining quantitatively correct numbers from the assumptions made. The standard approximation machinery for calculating from first principles explicit thermodynamic properties of materials, or explicit cross sections for high energy experiments can be found in many textbooks and is not repeated here.
This book explains the often counterintuitive physics of quantum mechanics, unlocking this key area of physics for students by enabling them to work through detailed applications of general concepts and ideas. The book states general principles first in terms of wave mechanics and then in the standard abstract linear space formalism. It offers a detailed discussion of measurement theory — an essential feature of quantum mechanics — and emphasizes the art of numerical estimates. Along the way, the book provides a wealth of physical examples together with numerous problems, some easy, some challenging, but all of them selected because they are physically interesting. The book is an essential resource for advanced undergraduates and beginning graduate students in physics.
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 book studies the mathematical aspect of path integrals and Hamiltonians – which emerge from the formulation of quantum mechanics. The theoretical framework of quantum mechanics provides the mathematical tools for studying both quantum indeterminacy and classical randomness . Many problems arising in quantum mechanics as well as in vastly different fields such as finance and economics can be addressed by the mathematics of quantum mechanics, or quantum mathematics in short. All the topics and subjects in the various chapters have been specifically chosen to illustrate the structure of quantum mathematics, and are not tied to any specific discipline, be it quantum mechanics or stochastic systems. The book is divided into the following six parts, in accordance with the Chapter dependency flowchart given below. Part one addresses the Fundamental principles of path integrals and (Hamiltonian) operators and consists of five chapters. Chapter 2 is on the Mathematical structure of quantum mechanics and introduces the mathematical framework that emerges from the quantum principle. Chapters 3 to 6 discuss the mathematical pillars of quantum mathematics, starting from the Feynman path integral, summarizing Hamiltonian mechanics and introducing path integral quantization. Part two is on Stochastic processes . Stochastic systems are dissipative and are shown to be effectively modeled by the path integral. Chapter 7 is focused on the application of quantum mathematics to classical random systems and to stochastic processes.