Vectors are employed extensively in physics. Conceptually we all know what a vector is: a vector is a quantity which has a length (magnitude or norm) as well as a direction. This chapter begins with a summary of the properties of three-dimensional vectors. It focuses on these vectors and their mathematical representations as operators and matrices which provides the intuition for the various concepts and operations of vectors. The chapter examines how it is possible to generate probability distributions in terms of vectors and operators in order to gain an insight into the mathematical framework for the formulation of quantum mechanics. It summarises the properties of three-dimensional vectors in §6.1, §6.2, §6.3 and §6.4. The discussion is organised in a way that can be directly generalised to real and complex vectors in higher as well as lower dimensions.
Vectors are employed extensively in physics. Conceptually we all know what a vector is: a vector is a quantity which has a length (magnitude or norm) as well as a direction. This chapter begins with a summary of the properties of three-dimensional vectors. It focuses on these vectors and their mathematical representations as operators and matrices which provides the intuition for the various concepts and operations of vectors. The chapter examines how it is possible to generate probability distributions in terms of vectors and operators in order to gain an insight into the mathematical framework for the formulation of quantum mechanics. It summarises the properties of three-dimensional vectors in §6.1, §6.2, §6.3 and §6.4. The discussion is organised in a way that can be directly generalised to real and complex vectors in higher as well as lower dimensions.
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 In this chapter we present the mathematical framework of describing wave functions as vectors in a Hilbert space and observables as Hermitian operators on such a vector space. We describe the mathematical formulation of these ideas for finite and infinite-dimensional cases via the Dirac notation, and discuss the example of the Hilbert space of photon polarization. Important classes of linear operators on Hilbert spaces are discussed, including projectors, Hermitian and self-adjoint operators, as well as unitary operators. As an important centre piece for the mathematical language of quantum mechanics we present the spectral theorem. We discuss the subtleties of this framework in infinite dimensions, giving a brief overview of the distributional aspects of quantum mechanics and the conceptually important problem of finding self-adjoint extensions via the Weyl-von Neumann theorem
An examination of the postulates of quantum mechanics reveals a number of fundamental mathematical constructs that form its theoretical underpinnings. Many of the results that are summarized in this Chapter will only become clear after reading the rest of the book and a re-reading may be in order. The dynamical variables of classical mechanics are superseded by the quantum degree of freedom. An exhaustive and complete description of the indeterminate degree of freedom is given by its state function, which is an element of a state space that, in general, is an infinite-dimensional linear vector space. The properties of the indeterminate degree of freedom are extracted from its state vector by the linear action of operators representing experimentally observable quantities. Repeated applications of the operators on the state function yield the average value of the operator for the state [Baaquie (2013e)]. The conceptual framework of quantum mechanics is discussed in Section 2.1. The concepts of degree of freedom, state space and operators are briefly reviewed in Sections 2.3–2.5. Three distinct formulations of quantum mechanics emerge from the superstructure of quantum mechanics and these are briefly summarized in Sections 2.7–2.9. The Copenhagen quantum postulate The Copenhagen interpretation of quantum mechanics, pioneered by Niels Bohr and Werner Heisenberg, provides a conceptual framework for the interpretation of the mathematical constructs of quantum mechanics and is the standard interpretation that is followed by the majority of practicing physicists [Stapp (1963), Dirac (1999)].
We summarize below some of the postulates and definitions basic to our formalism, and present some important results based on these postulates. The formalism is purely mathematical in nature with very little physics input, but it provides the structure within which the physical concepts that will be discussed in the later chapters will be framed. State vectors It is important to realize that the Quantum Theory is a linear theory in which the physical state of a system is described by a vector in a complex, linear vector space. This vector may represent a free particle or a particle bound in an atom or a particle interacting with other particles or with external fields. It is much like a vector in ordinary three-dimensional space, following many of the same rules, except that it describes a very complicated physical system. We will be elaborating further on this in the following. The mathematical structure of a quantum mechanical system will be presented in terms of the notations developed by Dirac. A physical state in this notation is described by a “ket” vector, |〉, designated variously as |α〉 or |ψ〉 or a ket with other appropriate symbols depending on the specific problem at hand. The kets can be complex. Their complex conjugates, |〉 * , are designated by 〈| which are called “bra” vectors. Thus, corresponding to every ket vector there is a bra vector. These vectors are abstract quantities whose physical interpretation is derived through their so-called “representatives” in the coordinate or momentum space or in a space appropriate to the problem under consideration.
Abstract This chapter introduces the concept of linear vector functions of vectors and the related dyadic notation, a concept that is particularly important in the study of rigid body motion and the covariant formulations of relativistic mechanics. Linear vector functions of vectors have a rich structure, with up to nine independent parameters needed to characterise them, and vector outputs that need not even have the same directions as the vector inputs. The subject of linear vector operators merits a chapter to itself, not only for its importance in analytical mechanics, but also because study of it will help the reader to master the operator formalism of quantum mechanics. The chapter begins with a definition of operators and matrices, and concludes with the Dirac Notation, used by quantum mechanics for complex vectors.
If mathematics is the queen of science, then physics might well be the king. It successes are obvious. However, as a science, physics may have failed in one regard, and that is to explain what its basic concepts – such as state vectors, wavefunctions, and transformation matrices – actually represent. When studying quantum mechanics, it is, effectively, hard to keep up the initial enthusiasm, and those who branch out to other fields – which is most of us – quickly end up going through the motions only: we regurgitate models and equations and know how to solve the standard problems, so as to pass the exam, but then forget about them as soon as possible. This paper explores a very intuitive sentiment about the issue: the wavefunction is a rather ‘flat’ mathematical object – it is two-dimensional, basically – so it can’t do the trick, perhaps. In contrast, Maxwell’s equations have real vectors in them, which is why a deeper or more intuitive understanding of electromagnetism comes relatively easily. Indeed, when everything is said and done, we are just human beings living in three-dimensional space, and that is why vector equations (or systems of vector equations), as a mathematical tool, make sense to us. This paper further explores this sentiment. It also offers a way out by, predictably, presenting yet another possible physical interpretation of the wavefunction. More importantly (for the reviewer of this paper, at least), this paper offers a sensible response to the mainstream view that three-dimensional physical interpretations of the wavefunction cannot make any sense because of the weird 720° symmetry of the wavefunction when describing spin-1/2 particles (fermions or – for all practical purposes – electrons). The author does so by analyzing (1) Dirac’s belt trick more in detail – and what it implies in terms of the interaction between the observer and the object – as well as (2) Feynman’s derivation of the transformation matrices for spin-1/2 two-state systems.