This book is not meant to contain a detailed analysis of the foundations of quantum mechanics. However, we felt an introduction to the general theory was required in order to make the presentation self-sufficient and clear the following chapters from possible ambiguities. We shall, wherever possible, avoid the intricacies of the mathematical description of infinite-dimensional spaces, as well as the conceptual subtleties related to the interpretation of the formalism (although the approach taken will necessarily betray aspects of the author's stand on these matters), and content ourselves with a ‘pragmatic’ introduction to the theory, as the chapter's title suggests. Throughout the chapter, we will even occasionally refer to a hypothetical ‘user’ of the theory as an artefact to elucidate relationships between the formalism and its operational interpretation. Notions related to quantum non-locality and the information content of quantum states, such as entropies and the characterisation of quantum correlations (also known as entanglement), will be the subject of the second half of the chapter, as they are very relevant to the material covered in the book.
This book is not meant to contain a detailed analysis of the foundations of quantum mechanics. However, we felt an introduction to the general theory was required in order to make the presentation self-sufficient and clear the following chapters from possible ambiguities. We shall, wherever possible, avoid the intricacies of the mathematical description of infinite-dimensional spaces, as well as the conceptual subtleties related to the interpretation of the formalism (although the approach taken will necessarily betray aspects of the author's stand on these matters), and content ourselves with a ‘pragmatic’ introduction to the theory, as the chapter's title suggests. Throughout the chapter, we will even occasionally refer to a hypothetical ‘user’ of the theory as an artefact to elucidate relationships between the formalism and its operational interpretation. Notions related to quantum non-locality and the information content of quantum states, such as entropies and the characterisation of quantum correlations (also known as entanglement), will be the subject of the second half of the chapter, as they are very relevant to the material covered in the book.
This chapter sketches out quantum mechanics with the added purpose of clarifying further notation and adopted conventions. It presents a with a 'pragmatic' introduction to the theory, and then refers to a hypothetical "user" of the theory as an artefact to elucidate relationships between the formalism and its operational interpretation. Notions related to quantum non-locality and the information content of quantum states, such as entropies and the characterisation of quantum correlations (also known as entanglement), is the subject of the second half of the chapter. Quantum entanglement, which the founding fathers of the theory used to discuss as a "spooky action at distance", has by now acquired a well-established role as a resource for quantum communication, computation and metrology. The chapter also reviews very basic results in this area. A positive but not completely positive map that serves a key role in the characterisation of quantum entanglement is transposition, in any given basis.
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.
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.
In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation. Those readers already familiar with the subject can safely jump to the next chapter. An attempt has been made to make the presentation in this chapter as elementary as possible, and yet present the basic facts that will be needed later. There are many beautiful and important topics which will not be touched upon such as self-adjointness of Schrödinger operators, the general mathematical structure of quantum mechanics and the like. These topics are well described in other works, e.g.. Much of the following can be done in a Euclidean space of arbitrary dimension, but in this chapter the dimension of the Euclidean space is taken to be three-which is the physical case-unless otherwise stated. We do this to avoid confusion and, occasionally, complications that arise in the computation of mathematical constants. The interested reader can easily generalize what is done here to the R d , d > 3 case. Likewise, in the next chapters we mostly consider N particles, with spatial coordinates in R 3 , so that the total spatial dimension is 3 N . A Brief Review of the Connection Between Classical and Quantum Mechanics Considering the range of validity of quantum mechanics, it is not surprising that its formulation is more complicated and abstract than classical mechanics. Nevertheless, classical mechanics is a basic ingredient for quantum mechanics. One still talks about position, momentum and energy which are notions from Newtonian mechanics.
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.
Abstract The purpose of this book is to present an interpretation of quantum mechanics. But what are we talking about when we speak of “quantum mechanics?” Are we talking about a single theory or about many related theories? And what do we mean by “interpretation” here? In this chapter, I will address these questions in an introductory way. Of course, I do not intend to take part in the extensive discussions about what a scientific theory is, what is meant by interpreting a theory, in what situation we encounter equivalent theories, and many other questions related to or derived from these. My aim here is to present some preliminary ideas in a way that I hope will be sufficient to avoid misunderstandings in the following chapters. But, above all, I will state my position as to why, after so many years and so much literature on the subject, the interpretation of quantum mechanics deserves further discussion.