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starzak m mathematical methods chemistry physics 1989

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Textbook in PDF format Imposingly thick text derived from a one-semester course intended to acquaint advanced undergraduate (and beginning graduate) students with the concepts and methods of linear mathematics. Though physics is referred to in the title, the book is in almost every organizational and notational respect. Mathematics is the language of the physical sciences and is essential for a clear understanding of fundamental scientific concepts. The fortuna te fact that the same mathematical ideas appear in a number of distinct scientific areas prompted the format for this book. The mathematical framework for matrices and vectors with emphasis on eigenvalue-eigenvector concepts is introduced and applied to a number of distinct scientific areas. Each new application then reinforces the applications which preceded it. Most of the physical systems studied involve the eigenvalues and eigenvectors of specific matrices. Whenever possible, I have selected systems which are described by 2 x 2 or 3 x 3 matrices. Such systems can be solved completely and are used to demonstrate the different methods of solution. In addition, these matrices will often yield the same eigenvectors for different physical systems, to provide a sense of the common mathematical basis of all the problems. For example, an eigenvector with components (1, -1) might describe the motions of two atoms in a diatomic molecule or the orientations of two atomic orbitals in a molecular orbital. The matrices in both cases couple the system components in a parallel manner. Because I feel that 2 x 2, 3 x 3, or soluble N x N matrices are the most effective teaching tools, I have not included numerical techniques or computer algorithms. A student who develops a clear understanding of the basic physical systems presented in this book can easily extend this knowledge to more complicated systems which may require numericalor computer techniques. The book is divided into three sections. The first four chapters introduce the mathematics of vectors and matrices. In keeping with the book's format, simple examples illustrate the basic concepts. Chapter 1 intro duces finite-dimensional vectors and concepts such as orthogonality and linear independence. Braket notation is introduced and used almost exclusively in subsequent chapters. Chapter 2 introduces function space vectors. To illustrate the strong parallels between such spaces and N-dimensional vector spaces, the concepts of Chapter 1, e.g., orthogonality and linear independence, are developed for function space vectors. Chapter 3 introduces matrices, beginning with basic matrix operations and concluding with an introduction to eigenvalues and eigenvectors and their properties. Chapter 4 introduces practical techniques for the solution of matrix algebra and calculus problems. These include similarity transforms and projection operators. The chapter concludes with some finite difference techniques for determining eigenvalues and eigenvectors for N x N matrices. Chapters 5-8 apply the mathematics to the major areas of normal mode analysis, kinetics, statistieal mechanics, and quantum mechanies. The examples in the chapter demonstrate the paralleis between the one-dimensional systems often introduced in introductory courses and multidimensional matrix systems. For example, the single vibrational frequency of a one-dimensional harmonie oscillator intro duces a vibrating molecule where the vibrational frequencies are related to the eigenvalues of the matrix for the coupled system. In each chapter, the eigenvalues and eigenvectors for multieomponent coupled systems are related to familiar physical concepts. The final three chapters introduce more advanced applications of matrices and vectors. These include perturbation theory, direct products, and fluctuations. The final chapter introduces group theory with an emphasis on the nature of matrices and vectors in this discipline. The book grew from a course in matrix methods I developed for juniors, seniors, and graduate students. Although the book was originally intended for a one-semester course, it grew as I wrote it. The material can still be covered in a one-semester course, but I have arranged the topics so chapters can be eliminated without disturbing the flow of information. The material can then be covered at any pace desired. This material, with additional numerical and programming techniques for more complicated matrix systems, could provide the basis for a two-semester course. Since the book provides numerous examples in diverse areas of chemistry and physics, it can also be used as a supplemental text for courses in these areas. Each chapter concludes with problems to reinforce both the concepts and the basic ex am pies developed in the chapter. In all cases, the problems are directed to applications

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starzak m mathematical methods chemistry physics 1989