net Mathetnatical Methods in Chetnistry and Physics www.net Mathematical Methods in Chemistry and Physics Michael E. Starzak State University of New York at Binghamton Binghamton, New York Springer Science+Business Media, LLC www.net Library of Congress Cataloging in Publication Data Starzak, Michael E. Mathematical methods in chemistry and physics / Michael E. Includes bibliographical references and index.
ISBN 978-1-4899-2084-3 ISBN 978-1-4899-2082-9 (eBook) DOI 10.2454-dcI9 CIP © 1989 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1989. Softcover reprint of the hardcover 1st edition 1989 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher www.net Preface 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 eigenvec- tors 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 effec- tive 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. Bra-ket notation is introduced and used almost exclusively in subsequent chapters. Chapter 2 introduces function space vectors. To illustrate the strong paralleis between such spaces and N-dimensional vector spaces, the concepts of Chapter 1, e., 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 v www.net vi PreCace and their properties. Chapter 4 introduces practical techniques for the solution of matrix algebra and ca1culus 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 introdu~ed 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 familia~ physical concepts.
The final three chapters introduce more advanced applications of matriees 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. I wish to thank my wife Anndrea and my daughters Jocelyn and Alissa for their support throughout this project and Alissa for converting my pencil sketches into professional line drawings.
I am grateful to the students whose comments and suggestions aided me in determining the most effective way to present the material. I also wish to thank my readers in advance for their suggestions for improvement. Starzak Binghamton, New York www. The Scalar Product.
Scalar Product Applications. Other Vector Combinations. Orthogonality and Biorthogonality. Linear Independence and Dependence.
Orthogonalization of Coordinates. The Function as a Vector. Function Scalar Products and Orthogonality. Orthogonalization of Basis Functions.
Generation of Special Functions. Function Resolution in a Set of Basis Functions. Matrix Equations and Inverses. Rotation of Co ordinate Systems.
Eigenvalues and the Characteristic Polynomial .net viii Contents 3. Properties of the Charaeteristic Polynomial. Alternate Teehniques for Eigenvalue and Eigenveetor Determination. Similarity Transforms and Projections.
The Similarity Transform. Generalized Charaeteristie Equations. Matrix Deeomposition Using Eigenveetors. The Lagrange-Sylvester Formula.
Matrix Funetions and Equations. Diagonalization of Tridiagonal Matriees. Other Tridiagonal Matrices. Asymmetrie Tridiagonal Matriees.
Vibrations and Normal Modes. Equations of Motion for a Diatomie Moleeule. Normal Modes for Nontranslating Systems. Normal Modes Using Projeetion Operators.
Normal Modes for Heteroatomic Systems. A Homogeneous One-Dimensional Crystal. Cyclie Boundary Conditions. Heteroatomie Linear Crystals.
Normal Modes for Moleeules in Two Dimensions. Properties of Matrix Solutions of Kinetie Equations. Kineties with Degenerate Eigenvalues. The Master Equation.
Symmetrization of the Master Equation. The Wegseheider Conditions and Cyclic Reaetions. Graph Theory in Kinetics. Graphs for Kinetics.
Mean First Passage Times. Evaluation of Mean First Passage Times .net Contents ix 7. The Wind-Tree Model. Statistical Mechanics of Linear Polymers.
Polymers with Nearest-Neighbor Interactions. Other One-Dimensional Systems. Two-Dimensional Systems. Non-Nearest-Neighbor Interactions.
Reduction of Matrix Order. The Kinetic Ising Model. Hybrid Atomic Orbitals. Matrix Quantum Mechanics.
Hückel Molecular Orbitals for Linear Molecules. Hückel Theory for Cyclic Moleeules. Degenerate Molecular Orbitals for Cyclic Moleeules. The Pauli Spin Matrices.
Lowering and Raising Operators. Driven Systems and Fluctuations. Singlet-Singlet Kinetics. Multilevel Driven Photochemical Systems.
Equilibrium and Stationary-State Properties. Fluctuations about Equilibrium. Fluctuations during Reactions. The Kinetics of Single Channels in Membranes.
Other Techniques: Perturbation Theory and Direct Products. Development of Perturbation Theory. First-Order Perturbation Theory-Eigenvalues. First-Order Perturbation Theory-Eigenvectors.
Second-Order Perturbation Theory-Eigenvalues. Second-Order Perturbation Theory-Eigenvectors. Direct Sums and Products. A Two-Dimensional Coupled Oscillator System.
Introduction to Group Theory. Vectors and Symmetry Operations. Matrix Representations of Symmetry Operations. Group Operations and Tables.
Properties oflrreducible Representations. Applications of Group Theory. Generation of Molecular Orbitals. Normal Vibrational Modes.
Ligand Field Theory. Direct Products of Group Elements. Direct Products and Integrals. Vectors Vectors are used when both the magnitude and the direction of some physical quantity are required.
A force applied to an object on a frictionless table (a two-dimensional system) can be any magnitude and it may pull the object along any direction on the table (Figure 1. This force is represented by a line with length proportional to the magnitude of the force. This li ne lies along the direction in which the force is applied. The line normally begins from the object and terminates with an arrow (Figure 1.
If it acts on a rigid body, the force could be applied at any point on the body, i., the physical location of the vector is less important than its magnitude and direction. If the body is elastic, a force vector applied to different parts of the body may give a different response. In such ca ses, the vector cannot be separated from its location on the body. For a rigid body, two forces of different magnitude which act in exactly the same direction will produce a net force equal to the sum .of the two constituent forces: (1.1) To translate into a vector format, either vector is moved so it starts from the terminus of the second vector.
The resultant vector, F I ' is a single vector which starts from the origin and ends at the terminus of the second vector; it has the same direction as the original two vectors. This resultant vector is found by arranging vectors in head-to-tail fashion and connecting the first tail to the final head. This head-to-tail vector addition is valid even when the vectors have different directions. Two forces are oriented at a right angle in Figure 1.
The total force is found by transposing either vector to the head of the other (Figure 1. The resultant vector then connects the initial tail and final head. The force from the two vectors is equivalent to a single force directed horizon- tally. Its magnitude can be found geometrically since the transposed vector is perpendicular to the initial vector creating a right tri angle.
The resultant (hypotenuse) is (1.net 2 Chapter 1 • Vectors Figure 1. The force on an object on a table expressed as the magnitude of a directed line in the x-y plane. If the two veetors make an angle () rat her than 90°, the law of eosines ean be used: (1.3) Veetors ean also be used to loeate positions in spaee. Under sueh cireumstanees, the veetor represents the distanee and direetion from one point in spaee to another.
Although most veetors in spaee involve three dimensions, two- dimensional systems ean illustrate the eoneept. The veetor r in Figure 1.4 represents the motion from the origin to a point j2 distant at a +45° angle. The veetor s represents a motion from the origin of j2 at a -45° angle. The addition of these two veetors by eonneeting the head of r to the tail of s is like the addition of forees.
In this ease, the first veetor ehanges the loeation in spaee from the tail to the head of the veetor. The head of the first veetor then serves as the origin for the seeond veetor. The head of this veetor is the final spatial loeation. The veetors are arranged in sequenee to determine the final position.
The order of the veetors is not important in this ease. The subtraetion of two veetors requires only a ehange in the direetion of the seeond veetor.4 ) involves the translation to the head of r as its first step. The position of the + s veetor is shown as a dashed Une. The subtraetion is performed by reversing the direetion of the s veetor as shown.
The resultant then connects the initial tail to the head of the negated veetor. Any number of veetors ean be added in this fashion to produee a net resultant. For example, there is no reason that the veetors of Figure 1.4 be loeated at some origin.
