Introduction to Perturbation Theory in Quantum Mechanics  book cover
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Introduction to Perturbation Theory in Quantum Mechanics




ISBN 9780849318771
Published September 19, 2000 by CRC Press
288 Pages

 
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Book Description

Perturbation theory is a powerful tool for solving a wide variety of problems in applied mathematics, a tool particularly useful in quantum mechanics and chemistry. Although most books on these subjects include a section offering an overview of perturbation theory, few, if any, take a practical approach that addresses its actual implementation

Introduction to Perturbation Theory in Quantum Mechanics does. It collects into a single source most of the techniques for applying the theory to the solution of particular problems. Concentrating on problems that allow exact analytical solutions of the perturbation equations, the book resorts to numerical results only when necessary to illustrate and complement important features of the theory. The author also compares different methods by applying them to the same models so that readers clearly understand why one technique may be preferred over another.

Demonstrating the application of similar techniques in quantum and classical mechanics, Introduction to Perturbation Theory in Quantum Mechanics reveals the underlying mathematics in seemingly different problems. It includes many illustrative examples that facilitate the understanding of theoretical concepts, and provides a source of ideas for many original research projects.

Table of Contents

PERTURBATION THEORY IN QUANTUM MECHANICS
Bound States
Equations of Motion
Examples
PERTURBATION THEORY IN THE COORDINATE REPRESENTATION
The Method of Dalgarno and Stewart
Logarithmic Perturbation Theory
The Method of Fernandez and Castro
PERTURBATION THEORIES WITHOUT WAVEFUNCTION
Hypervirial and Hellmann-Feynman Theorems
The Method of Swenson and Danforth
Moment Method
Perturbation Theory in Operator Form
SIMPLE ATOMIC AND MOLECULAR SYSTEMS
The Stark Effect in Hydrogen
The Zeeman Effect in Hydrogen
The Hydrogen Molecular Ion
The Delta Molecular Ion
THE SCHRODINGER EQUATION ON BOUNDED DOMAINS
One-Dimensional Box Models
Spherical-Box Models
Perturbed Rigid Rotors
CONVERGENCE OF THE PERTURBATION SERIES
Convergence Properties of Power Series
Radius of Convergence of the Perturbation Expansions
Divergent Perturbation Series
Improving the Convergence Properties of the Perturbation Series
POLYNOMIAL APPROXIMATIONS
One-Dimensional Models
Central-Field Models
Vibration-Rotational Spectra of Diatomic Molecules
Large-N Expansion
Improved Perturbation Series
Born-Oppenheimer Perturbation Theory
PERTURBATION THEORY FOR SCATTERING STATES IN ONE DIMENSION
On the Solutions of Second-Order Differential Equations
The One-Dimensional Schrödinger Equation with a Finite Interaction Region
The Born Approximation
An Exactly Solvable Model: The Square Barrier
Nontrivial Simple Models
Perturbation Theory for Resonance Tunneling
PERTURBATION THEORY IN CLASSICAL MECHANICS
Dimensionless Classical Equations
Polynomial Approximation
Canonical Transformations in Operator Form
The Evolution Operator
Secular Perturbation Theory
Canonical Perturbation Theory
The Hypervirial Hellmann-Feynman Method (HHFM)
Central Forces
MAPLE PROGRAMS
APPENDICES
Laplacian in Curvilinear Coordinates
Ordinary Differential Equations with Constant Coefficients
Canonical Transformations
REFERENCES

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Author(s)

Biography

Francisco M. Fernandez

Reviews

"Professor Fernandez [has been] active in the field of quantum mechanical perturbation theory for many years. … [T]he book has been prepared and written with-and to-the author's undisguised delight, offering a specific blend of the theory and computations. … The book informs the reader, in an admirably compact and still unbelievably digestible text, about virtually all the basic concepts … ."
- Zentralblatt MATH, Vol. 1048


"… summarizes in detail a wide variety of techniques used in perturbation theory that were developed for particular problems. To make the methods easily comparable, they are applied to the same models. Some of these models have exact solutions, so that improvement in the perturbation series and their limitations become clear…author intentionally avoids problems that require numerical computation and tries to keep the mathematics as simple as possible in order to concentrate the reader's attention on the ideas underlying each method…offers an invaluable source of ideas for theoreticians applying perturbation methods to various problems in chemistry and physics…it can also be used for teaching undergraduate and graduate courses…"
-Journal of the American Chemical Society, Vol. 123, No. 26

"…a very substantial presentation of perturbation theory in quantum mechanics, much of which is based on the publication of the author and his coworkers…the book is clearly an important reference work in the field, and the author has completed a laudable task."
-Mathematical Reviews, Issue 2002