Introduction to Perturbation Theory in Quantum Mechanics

by Fernandez; Francisco M.

• ISBN13: 9780849318771
• ISBN10: 0849318777
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-09-19
• Publisher: CRC Press

Summary

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 implementationIntroduction 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.

Author Biography

Francisco M. Fernandez, Ph.D., is Professor at the University of La Plata, Buenos Aires, Argentina

Table of Contents

Perturbation Theory in Quantum Mechanics	1	(12)
Introduction	1	(1)
Bound States	1	(4)
The 2s + 1 Rule	4	(1)
Degenerate States	4	(1)
Equations of Motion	5	(3)
Time-Dependent Perturbation Theory	6	(1)
One-Particle Systems	7	(1)
Examples	8	(5)
Stationary States of the Anharmonic Oscillator	8	(1)
Harmonic Oscillator with a Time-Dependent Perturbation	9	(2)
Heisenberg Operators for Anharmonic Oscillators	11	(2)
Perturbation Theory in the Coordinate Representation	13	(14)
Introduction	13	(1)
The Method of Dalgarno and Stewart	13	(4)
The One-Dimensional Anharmonic Oscillator	14	(1)
The Zeeman Effect in Hydrogen	15	(2)
Logarithmic Perturbation Theory	17	(3)
The One-Dimensional Anharmonic Oscillator	18	(1)
The Zeeman Effect in Hydrogen	19	(1)
The Method of Fernandez and Castro	20	(7)
The One-Dimensional Anharmonic Oscillator	23	(4)
Perturbation Theories without Wavefunction	27	(34)
Introduction	27	(1)
Hypervirial and Hellmann-Feynman Theorems	27	(1)
The Method of Swenson and Danforth	28	(10)
One-Dimensional Models	28	(4)
Central-Field Models	32	(5)
More General Polynomial Perturbations	37	(1)
Moment Method	38	(18)
Exactly Solvable Cases	39	(2)
Perturbation Theory by the Moment Method	41	(1)
Nondegenerate Case	42	(3)
Degenerate Case	45	(5)
Relation to Other Methods: Modified Moment Method	50	(6)
Perturbation Theory in Operator Form	56	(5)
Illustrative Example: The Anharmonic Oscillator	59	(2)
Simple Atomic and Molecular Systems	61	(22)
Introduction	61	(1)
The Stark Effect in Hydrogen	61	(9)
Parabolic Coordinates	61	(3)
Spherical Coordinates	64	(6)
The Zeeman Effect in Hydrogen	70	(6)
The Hydrogen Molecular Ion	76	(4)
The Delta Molecular Ion	80	(3)
The Schrodinger Equation on Bounded Domains	83	(22)
Introduction	83	(1)
One-Dimensional Box Models	83	(6)
Straightforward Integration	84	(1)
The Method of Swenson and Danforth	85	(4)
Spherical-Box Models	89	(6)
The Method of Fernandez and Castro	90	(1)
The Method of Swenson and Danforth	91	(4)
Perturbed Rigid Rotors	95	(10)
Weak-Field Expansion by the Method of Fernandez and Castro	96	(2)
Weak-Field Expansion by the Method of Swenson and Danforth	98	(3)
Strong-Field Expansion	101	(4)
Convergence of the Perturbation Series	105	(32)
Introduction	105	(1)
Convergence Properties of Power Series	105	(4)
Straightforward Calculation of Singular Points from Power Series	106	(2)
Implicit Equations	108	(1)
Radius of Convergence of the Perturbation Expansions	109	(8)
Exactly Solvable Models	109	(2)
Simple Nontrivial Models	111	(6)
Divergent Perturbation Series	117	(3)
Anharmonic Oscillators	118	(2)
Improving the Convergence Properties of the Perturbation Series	120	(17)
The Effect of H0	120	(6)
Intelligent Algebraic Approximants	126	(11)
Polynomial Approximations	137	(36)
Introduction	137	(1)
One-Dimensional Models	137	(10)
Deep-Well Approximation	138	(6)
Weak Attractive Interactions	144	(3)
Central-Field Models	147	(3)
Vibration-Rotational Spectra of Diatomic Molecules	150	(3)
Large-N Expansion	153	(6)
Improved Perturbation Series	159	(5)
Shifted Large-N Expansion	161	(2)
Improved Shifted Large-N Expansion	163	(1)
Born-Oppenheimer Perturbation Theory	164	(9)
Perturbation Theory for Scattering States in One Dimension	173	(20)
Introduction	173	(1)
On the Solutions of Second-Order Differential Equations	173	(1)
The One-Dimensional Schrodinger Equation with a Finite Interaction Region	174	(2)
The Born Approximation	176	(2)
An Exactly Solvable Model: The Square Barrier	178	(1)
Nontrivial Simple Models	179	(6)
Accurate Nonperturbative Calculation	179	(1)
First Perturbation Method	180	(1)
Second Perturbation Method	181	(2)
Third Perturbation Method	183	(2)
Perturbation Theory for Resonance Tunneling	185	(8)
Perturbation Theory in Classical Mechanics	193	(36)
Introduction	193	(1)
Dimensionless Classical Equations	193	(1)
Polynomial Approximation	194	(6)
Odd Force	196	(1)
Period of the Motion	197	(2)
Removal of Secular Terms	199	(1)
Simple Pendulum	199	(1)
Canonical Transformations in Operator Form	200	(3)
Hamilton's Equations of Motion	200	(1)
General Poisson Brackets	200	(1)
Canonical Transformations	201	(2)
The Evolution Operator	203	(3)
Simple Examples	204	(2)
Secular Perturbation Theory	206	(3)
Simple Examples	207	(1)
Construction of Invariants by Perturbation Theory	208	(1)
Canonical Perturbation Theory	209	(4)
The Hypervirial Hellmann-Feynman Method (HHFM)	213	(10)
One-Dimensional Models with Polynomial Potential-Energy Functions	216	(1)
Radius of Convergence of the Canonical Perturbation Series	217	(3)
Nonpolynomial Potential-Energy Function	220	(3)
Central Forces	223	(6)
Perturbed Kepler Problem	224	(5)
Maple Programs	229	(16)
Programs for Chapter 1	229	(1)
Programs for Chapter 2	230	(3)
Programs for Chapter 3	233	(2)
Programs for Chapter 4	235	(2)
Programs for Chapter 5	237	(1)
Programs for Chapter 6	238	(1)
Programs for Chapter 8	239	(1)
Programs for Chapter 9	240	(4)
Programs for the Appendixes	244	(1)
A Laplacian in Curvilinear Coordinates	245	(4)
B Ordinary Differential Equations with Constant Coefficients	249	(2)
C Canonical Transformations	251	(4)
References	255	(12)
Index	267