Chapter 1. Vectors and operators

Oliver Davis Johns

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.

Alexander S. Holevo

K. Kong Wan

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.

Oliver Davis Johns

Abstract Linear vector functions of vectors, and the related dyadic notation, are important in the study of rigid body motion and the covariant formulations of relativistic mechanics. These functions 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. This chapter defines linear operators and discusses operators and matrices as well as special operators, dyadics, resolution of unity, complex vectors and operators, real and complex inner products, eigenvectors and eigenvalues, eigenvectors of real symmetric operator, eigenvectors of real anti-symmetric operator, normal operators, determinant and trace of normal operator, eigen-dyadic expansion of normal operator, functions of normal operators, exponential function, and Dirac notation.

Lev Deych