# Algebraic Methods in Quantum Chemistry and Physics

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## Book Description

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.

## Table of Contents

ELEMENTARY INTRODUCTION TO LIE ALGEBRAS AND OPERATOR METHODS

Vector Spaces

Lie Algebras

Superoperators

Canonical Transformations

Operator Differential Equations

The Campbell-Baker-Hausdorff Formula

Basis Set for a Lie Algebra

SOME PRACTICAL APPLICATIONS OF FINITE-DIMENSIONAL LIE ALGEBRAS

Definition, Examples, and Some Applications of Finite-Dimensional Lie Algebras

Regular or Adjoint Matrix Representation

Eigenvalues of Superoperators

Faithful Matrix Representation

Disentangling Exponential Operators

THE QUANTUM-MECHANICAL HARMONIC OSCILLATOR

Eigenvalues, Eigenvectors, and Matrix Elements

Coherent States

The Coordinate Representation

Modeling Quantum-Mechanical Systems with Bosonic Algebra

MATRIX ELEMENTS OF EXPONENTIAL OPERATORS IN THE HARMONIC OSCILLATOR BASIS SET

Matrix Elements of Exponential Operators

Franck-Condon Factors

THREE-DIMENSIONAL LIE ALGEBRAS AND SOME OF THEIR REALIZATIONS IN QUANTUM MECHANICS

Eigenvalues and Matrix Elements

Angular Momentum and Bosonic Algebras

Second-Order Differential Operators

Exactly Solvable Models with Central Potentials

The Method of Canonical Transformations

Examples in Quantum Mechanics

Selection Rules

PERTURBATION THEORY AND VARIATIONAL METHOD

Perturbation Theory for Stationary States

The Vibration-Rotational Spectrum of a Diatomic Molecule

Perturbation Theory in Operator Form

Perturbation Theory and Canonical Transformations

Lie Algebras and the Variational Method

NUMERICAL INTEGRATION OF THE TIME-INDEPENDENT SCHRÖDINGER EQUATION

Approximate Difference Equation

The Propagation Matrix Method

An Exactly Solvable Problem

Propagation on a Grid

Perturbative Solutions

Exponential Solution

Product of Exponentials

EQUATIONS OF MOTION IN QUANTUM MECHANICS

Schrödinger, Heisenberg, and Intermediate Pictures

Approximate Methods

The Density Operator

Finite-Dimensional Lie Algebras and Observables

BILINEAR OSCILLATORS

General Bilinear Oscillator for One Degree of Freedom

Exactly Solvable Example

Transition Probabilities for a General Bilinear Oscillator

Solution to the Schrödinger Equation in the Coordinate Representation

Pseudo-Nonlinear Hamiltonians

Fokker-Planck Equation

Bilinear Approximation to Arbitrary Potential Energy Functions

PARAMETERIZATION OF THE TIME-EVOLUTION OPERATOR

The Magnus Expansion and Perturbation Theory

Simple Bilinear Hamiltonians

State Space of Finite Dimension

Product of Exponential Operators

SEMICLASSICAL EXPANSIONS IN STATISTICAL MECHANICS

The Canonical Ensemble

The Wigner-Kirkwood Expansion

The Harmonic Oscillator

The Euler-MacLaurin Summation Formula

The Poisson Summation Formula

NOTE: Introduction at the beginning of each chapter

## Author(s)

### Biography

Fernandez\, Francisco M.; Castro\, E.A.

## Reviews

"In short, this book is a nice introduction to the use of operator methods in quantum mechanics and chemistry and can also serve as a reference source because of the numerous problems solved in it."

-Ivaïlo Mladenov, Bulgarian Academy of Sciences, Sofia