Algebraic Methods in Quantum Chemistry and Physics

Francisco M. Fernandez, E.A. Castro

• Publisher: CRC Press
• Year: 2020
• eBook ISBN: 9781000722666
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

Algebraic Methods in Quantum Chemistry and Physics provides straightforward presentations of selected topics in theoretical chemistry and physics, including Lie algebras and their applications, harmonic oscillators, bilinear oscillators, perturbation theory, numerical solutions of the Schrödinger equation, and parameterizations of the time-evolution operator. The mathematical tools described in this book are presented in a manner that clearly illustrates their application to problems arising in theoretical chemistry and physics. The application techniques are carefully explained with step-by-step instructions that are easy to follow, and the results are organized to facilitate both manual and numerical calculations. Algebraic Methods in Quantum Chemistry and Physics demonstrates how to obtain useful analytical results with elementary algebra and calculus and an understanding of basic quantum chemistry and physics.

chapter one

Elementary introduction to Lie algebras and operator methods

1.1 Introduction

Most textbooks on quantum mechanics exhibit extensive use of commutators between linear operators either to derive suitable analytic expressions or to simplify the calculation.¹ Moreover, it is often possible to construct sets of linear operators closed under commutation that are particularly useful when one is able to express the observables of the physical model in terms of those operators. With some additional properties such sets become Lie algebras of utmost relevance in theoretical physics and chemistry.

The main purpose of this chapter is to provide a simple general introduction to some basic mathematics necessary for a subsequent discussion of Lie algebras, operator methods, and their applications to theoretical physics and chemistry. We do not attempt a formal approach to mathematical physics and suppose the reader to be familiar with the main ideas and concepts relevant to those fields. As pointed out in the general introduction, even our treatment of the main subject of this book, the Lie algebras, is not as rigorous and detailed as those appearing in more specialized books.² Instead, we try to provide some useful mathematical tools to d...

Table of contents

• Cover
• Series Page
• Title Page
• Copyright Page
• Introduction
• Acknowledgments
• Table of Contents
• Chapter One Elementary Introduction to Lie Algebras and Operator Methods
• Chapter Two Some Practical Applications of Finite-Dimensional Lie Algebras
• Chapter Three The Quantum-Mechanical Harmonic Oscillator
• Chapter Four Matrix Elements of Exponential Operators in the Harmonic Oscillator Basis Set
• Chapter Five Three-Dimensional Lie Algebras and Some of Their Realizations in Quantum Mechanics
• Chapter Six Perturbation Theory and Variational Method
• Chapter Seven Numerical Integration of the Time-Independent Schrödinger Equation
• Chapter Eight Equations of Motion in Quantum Mechanics
• Chapter Nine Bilinear Oscillators
• Chapter Ten Parametrization of the Time-Evolution Operator
• Chapter Eleven Semiclassical Expansions in Statistical Mechanics
• APPENDIX A Functions of Operators and Matrices
• APPENDIX B The Liouville Transformation
• REFERENCES
• INDEX