Chapter 3 Abstract Linear Space of State Vectors

‎Louis Marchildon

Steven Herbert

Abstract Quantum mechanics is mathematically formulated in terms of a Hilbert space, which is a vector space with a defined inner product. Linear algebra is the study of vector spaces and the (linear) operations thereon, and so it follows that having a solid grasp of the basics of linear algebra is essential for studying quantum mechanics and, therefore, quantum computation. All the linear algebra prerequisites necessary to understand the theoretical results presented in the remainder of the book are detailed in this chapter. The chapter also acts as a nomenclature reference for the book, by providing all the notations that are used thereafter.

Tai L. Chow

Linear vector space is to quantum mechanics what calculus is to classical mechanics. In this chapter the essential ideas of linear vector spaces will be discussed. The reader is already familiar with vector calculus in three-dimensional Euclidean space E 3 (Chapter 1). We therefore present our discussion as a generalization of elementary vector calculus. The presentation will be, however, slightly abstract and more formal than the discussion of vectors in Chapter 1. Any reader who is not already familiar with this sort of discussion should be patient with the first few sections. You will then be amply repaid by finding the rest of this chapter relatively easy reading. Euclidean n - space E n In the study of vector analysis in E 3 , an ordered triple of numbers ( a 1 , a 2 , a 3 ) has two different geometric interpretations. It represents a point in space, with a 1 , a 2 , a 3 being its coordinates; it also represents a vector, with a 1 , a 2 , and a 3 being its components along the three coordinate axes (Fig. 5.1). This idea of using triples of numbers to locate points in three-dimensional space was first introduced in the mid-seventeenth century. By the latter part of the nineteenth century physicists and mathematicians began to use the quadruples of numbers ( a 1 , a 2 , a 3 , a 4 ) as points in four-dimensional space, quintuples ( a 1 , a 2 , a 3 , a 4 , a 5 ) as points in fivedimensional space etc. We now extend this to n -dimensional space E n , where n is a positive integer. Although our geometric visualization doesn't extend beyond three-dimensional space, we can extend many familiar ideas beyond three-dimensional space by working with analytic or numerical properties of points and vectors rather than their geometric properties.

Norman Goldstein

F. S. Levin

The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.