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Algebraic Methods in Quantum Chemistry and Physics




ISBN 9780849382925
Published October 24, 1995 by CRC Press
280 Pages

 
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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

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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