1st Edition

Algebraic Methods in Quantum Chemistry and Physics

By Francisco M. Fernandez, E.A. Castro Copyright 1995

    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.

    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

    Biography

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

    "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