INTRODUCTION TO QUANTUM MECHANICS With Applications to Chemistry BY LINUS PAULING, PH. Professor of Chemistry, California Institute of Technolouy AND E. BRIGHT WILSON, JR. AS~lOciate Professor of Chemistry, Harvard Uni~ersity MeGRAW-HILL BOOK COMPANY, INCl.
NEW YORK AND LONDON 1935 www.com COPYRIGHT, 1935, BY THE MCGRAW-HILL BOOK COMPANY, INC. PRINTED IN THE UNITED STATES OF AMERICA All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers. XVII THE MAPLJiJ PRESS COMPANY, YORK, PA.com PREFACE In writing this book we have attempted to produce a textbook of practical quantum mechanics for the chemist, the experi mental physicist, and the beginning student of theoretical physics.
The book is not intended to provide a critical discus sion of quantum mechanics, nor even to present a thorough survey of the subject. We hope that it does give a lucid and easily understandable introduction to a limited portion of quantum-mechanical theory; namely, that portion usually suggested by the name "wave mechanics," consisting of the discussion of the Schrodinger wave equation and the problems which can be treated by means of it. The effort has been made to provide for the reader a means of equipping himself with a practical grasp of this subject, so that he can apply quantum mechanics to most of the chemical and physical problems which may confront him. The book is particularly designed for study by men without extensive previous experience with advanced mathematics, such as chemists interested in the subject because of its chemical applications.
We have assumed on the part of the reader, in addition to elementary mathematics through the calculus, only some knowledge of complex quantities, ordinary differential equations, and the technique of partial differentiation. It may be desirable that a book written for the reader not adept at mathematics be richer in equations than one intended for the mathematician; for the mathematician can follow a sketchy derivation with ease, whereas if the less adept reader is to be led safely through the usually straightforward but sometimes rather complicated derivations of quantum mechanics a firm guiding hand must be kept on him. Quantum mechanics is essentially mathematical in character, and an understanding of the subject without a thorough knowledge of the mathematical methods involved and the results of their application cannot be obtained. The student not thoroughly trained in the theory of partial differential equations and orthogonal functions must iii www.com iv PREFACE learn something of these subjects as he studies quantum mechan- ics.
In order that he may do so, and that he may follow the discussions given without danger of being deflected from the course of the argument by inability to carry through some minor step, we have avoided the temptation to condense the various discussions into shorter and perhaps more elegant forms. After introductory chapters on classical mechanics and the old quantum theory, we have introduced the Schroding^r wave equation and its physical interpretation on a postulatory basis, and have then given in great detail the solution of the wave equation for important systems (harmonic oscillator, hydrogen atom) and the discussion of the wave functions and their proper- ties,omitting none of the mathematical steps except the most l^JT!^?577 A, similarly detailed treatment has been given 1 in the discussion di pert in option Shruor^, the variation method, the structure of simple molecules, and, in general, 'iu --, important section of the book. In order to limit the size of the book, we have omitted from discussion such advanced topics as transformation theory and general quantum mechanics (aside from brief mention in the last chapter), the Dirac theory of the electron, quantization of the electromagnetic field, etc. We have also omitted several subjects which are ordinarily considered as part of elementary quantum mechanics, but which are of minor importance to the chemist, such as the Zeeman effect and magnetic interactions in general, the dispersion of light and allied phenomena, and most of the theory of aperiodic processes.
The authors are severally indebted to Professor A. Sommerfeld and Professors E. Robertson for their own introduction to quantum mechanics. The constant advice of Professor R.
Tolman is gratefully acknowledged, as well as the aid of Professor P. Emily Buckingham Wilson, and Mrs. Ava Helen Pauling. BRIGHT WILSON, JR.com CONTENTS PREFACE CHAPTER I SURVEY OF CLASSICAL MECHANICS SECTION 1.
Newton's Equations of Motion in the Lagntngian Form. The Three-dimensional Isotropic Harmonic Oscillator. The Invariance of the Equations of Motion in the Lagraib- gian Form. An Example: The Isotropic Harmonic Oscillator in Polar.
Coordinates 9 The Conservation of Angular Momentum le. The Equations of Motion in the Hamiltonian Form 14 2a. 14 The Hamiltonian Function and Equations 26. The Hamiltonian Function and the Energy 16 2d.
The Emission and Absorption of Radiation. Summary of Chapter I. 23 CHAPTER II THE OLD QUANTUM THEORY 5. The Origin of the Old Quantum Theory.
The Postulates of Bohr. The Wilson-Sommerfeld Rules of Quantization. The Correspondence Principle. The Quantization of Simple Systems.
The Harmonic Oscillator. The Rigid Rotator. The Oscillating and Rotating Diatomic Molecule. The Particle in a Box.
Diffraction by a Crystal Lattice. The Hydrogen Atom. Solution of the Equations of Motion .Application of the Quantum Rules. The Energy Levels .Description of the Orbits.
The Decline of the Old Quantum Theory .com vi CONTENTS SECTION PAGE CHAPTER III THE SCHRODINGER WAVE EQUATION WITH THE HARMONIC OSCILLATOR AS AN EXAMPLE 9. The Schrodinger Wave Equation 50 9a. The Wave Equation Including the Time 53 96. The Amplitude Equation 56 9c.
Discrete and Continuous Sets of Charac- teristic Energy Values 58 9d. The Complex Conjugate Wave Function ty*(x,t). The Physical Interpretation of the Wave Functions 63 10a. Sk* (x, t)V(x, t) as a Probability Distribution Function.
Stationary States 64 lOc. Further Physical Interpretation. Average Values of Dynamical Quantities 65 11. The Harmonic Oscillator in Wave Mechanics.
Solution of the Wave Equation. The Wave Functions for the Harmonic Oscillator and their Physical Interpretation. Mathematical Properties of the Harmonic Oscillator Wave Functions 77 CHAPTER IV THE WAVE EQUATION FOR A SYSTEM OF POINT PARTICLES IN THREE DIMENSIONS 12. The Wave Equation for a System of Point Particles 84 12a.
The Wave Equation Including the Time 85 126. The Amplitude Equation 86 12c. The Complex Conjugate Wave Function ty*(xi ZAT, t) 88 12d The Physical Interpretation of the Wave Functions. 88 13 The Free Particle 90 14.
The Particle in a Box 95 15. The Three-dimensional Harmonic Oscillator in Cartesian Coordi- nates 100 16. The Three-dimensional Harmonic Oscillator in Cylindrical Coordi- nates 105 CHAPTER V THE HYDROGEN ATOM 18. The Solution of the Wave Equation by the Polynomial Method and the Determination of the Energy Levels 113 18a.
The Separation of the Wave Equation. The Transla- tional Motion 113 186. The Solution of the *> Equation 117 www.com CONTENTS Vit SUCTION PAGE 18c. The Solution of the & Equation 118 18d.
The Solution of the r Equation 121 18c. The Energy Levels 124 19. Legendre Functions and Surface Harmonics 125 19a. The Legendre Functions or Legendre Polynomials 126 196.
The Associated Legendre Functions 127 20. The Laguerre Polynomials and Associated Laguerre Functions. The Laguerre Polynomials 20a. The Associated Laguerre Polynomials and Functions.
The Wave Functions for the Hydrogen Atom 132 21a. Hydrogen-like Wave Functions 132 216. The Normal State of the Hydrogen Atom 139 21c. Discussion of the Hydrogen-like Radial Wave Functions.
Discussion of the Dependence of the Wave Functions on the Angles t? and <p 146 CHAPTER VI PERTURBATION THEORY 22. Expansions in Series of Orthogonal Functions 151 23\xFirst-order Perturbation Theory for a Non-degenerate Level. A Simple Example: T>/> P^^tujj^ejjJH^mjinnin QsillRf. 160 ?jfo An Exjunplp- Tfor Normal FHium Atn 162 24.
J$tst-order Perturbation Theory for a Degenerate Level 165 24a. An Example: Application of a Perturbation to a Hydrogen Atom 172 25. Second-order Perturbation Theory 176 25a. An Example: The Stark Effect of the Plane Rotator .177 CHAPTER VII THE VARIATION METHOD AND OTHER APPROXIMATE METHODS 26.
The Variation Method 180 26a. The Variational Integral and its Pmpgrtjes 180 266. An Example fThe Normal State of Helium Atom. Application of the Variation Method to Other States.
Linear Variation Functions 186 26e. A More General Variation Method 189 27. Other Approximate Methods 191 27a. A Generalized Perturbation Theory 191 276.
The Wentzel-Kramers-Brillouin Method 198 27c. Approximation by the Use of Difference Equations. An Approximate Second-order Perturbation Treatment .com via CONTENTS SUCTION PAQB CHAPTER VIII THE SPINNING ELECTRON AND THE PAULI EXCLUSION PRINCIPLE, WITH A DISCUSSION OF THE HELIUM ATOM 28. The Spinning Electron 207 29.
The Helium Atom. The Pauli Exclusion Principle 210 29a. The Configurations Is2s and Is2p 210 296. The Consideration of Electron Spin.
The Pauli Exclusion Principle 214 29c. The Accurate Treatment of the Normal Helium Atom. Excited States of the Helium Atom 225 29e. The Polarizability of the Normal Helium Atom 226 CHAPTER IX MANY-ELECTRON ATOMS 30.
Slater's Treatment of Complex Atoms 230 30a. Factorization and Solution of the Secular Equation. Evaluation of Integrals. Empirical Evaluation of Integrals.
Variation Treatments for Simple Atoms 246 31 a. The Lithium Atom and Three-electron Ions. Variation Treatments of Other Atoms. The Method of the Self-consistent Field 250 32a.
Principle of the Method 250 326. Relation of the Self-consistent Field Method to the Varia- tion Principle 252 32c. Results of the Self-consistent Field Method 254 33. Other Methods for Many-electron Atoms 256 33a.
Semi-empirical Sets of Screening Constants 256 336. The Thomas-Fermi Statistical Atom 257 CHAPTER X THE ROTATION AND VIBRATION OF MOLECULES 34. The Separation of Electronic and Nuclear Motion 259 35. The Rotation and Vibration of Diatomic Molecules 263 35a.
The Separation of Variables and Solution of the Angular Equations 264 356. The Nature of the Electronic Energy Function 266 35. A Simple Potential Function for Diatomic Molecules. A More Accurate Treatment.
The Morse Function. The Rotation of Polyatomic Molecules 275 36a. The Rotation of Symmetrical-top Molecules 275 36b. The Rotation of Unsymmetrical-top Molecules 280 www.com CONTENTS ix SECTION PAGE 37.The Vibiation of Polyatomic Molecules 282 37a.
Normal Coordinates in Classical Mechanics 282 376. Normal Coordinates in Quantum Mechanics 288 38. The Rotation of Molecules in Crystals 290 CHAPTER XI PERTURBATION THEORY INVOLVING THE TIME, THE EMISSION AND ABSORPTION OF RADIATION, AND THE RESONANCE PHENOMENON 39. The Treatment of a Time-dependent Perturbation by the Methoa of Variation of Constants.
The Emission and Absorption of Radiation. The Einstein Transition Probabilities. The Calculation of the Einstein Transition Probabilities by Perturbation Theory 302 40c. Selection Rules and Intensities for the Harmonic Oscillator 306 40d.
Selection Rules and Intensities for Surface-harmonic Wave Functions 306 40e. Selection Rules and Intensities for the Diatomic Moleculev- The Franck-Condon Principle 309 40/. Selection Rules and Intensities for the Hydrogen Atom. Even and Odd Electronic States and their Selection Rules.
The Resonance Phenomenon 314 41a. Resonance in Classical Mechanics 315 416. Resonance in Quantum Mechanics 318 41c. A Further Discussion of Resonance 322 CHAPTER XII THE STRUCTURE OF SIMPLE MOLECULES 42.
The Hydrogen Molecule-ion 327 42a. A Very Simple Discussion 327 426. Other Simple Variation Treatments 331 42c. The Separation and Solution of the Wave Equation.
Excited States of the Hydrogen Molecule-ion 340 43. The Hydrogen Molecule 340 43a. The Treatment of Heitler and London. Other Simple Variation Treatments.
The Treatment of James and Coolidge 349 43d. Comparison with Experiment 351 43e. Excited States of the Hydrogen Molecule 353 43/. Oscillation and Rotation of the Molecule.
Ortho and Para Hydrogen 355 44. The Helium Molecule-ion Hef and the Interaction of Two Normal Helium Atoms 358 44a. The Helium Molecule-ion Hef 358 www.com INTRODUCTION TO QUANTUM MECHANICS CHAPTER I SURVEY OF CLASSICAL MECHANICS The subject of quantum mechanics constitutes the most recent step in the very old search for the general laws_goyrning the motion of matter.