Introduction to Quantum Mechanics: With Applications to Chemistry PDF
Preview Introduction to Quantum Mechanics: With Applications to Chemistry
INTRODUCTION TO QUANTUM MECHANICS With Applications to Chemistry BY LINUS PAULING, PH.D., Sc.D. Professor of Chemistry, California Institute of Technolouy AND E. BRIGHT WILSON, JR., PH.D. AS~lOciate Professor of Chemistry, Harvard Uni~ersity MeGRAW-HILL BOOK COMPANY, INCl. NEW YORK AND LONDON 1935 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 bereproduced in any form without permission of the publishers. XVII THE MAPLJiJ PRESS COMPANY, YORK, PA. 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 iv PREFACE learn something of these subjects as hestudies 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 in the discussion di pertin1option 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. U. Condon and H. P. Robertson for their own introduction to quantum mechanics. The constant advice of Professor R. C. Tolman is gratefully acknowledged, as well as the aid of Professor P. M. Morse, Dr. L. E. Sutton, Dr. G. W. Wheland, Dr. L. 0. Brockway, Dr. J. Sherman, Dr. S. Weinbaum, Mrs. Emily Buckingham Wilson, and Mrs. Ava Helen Pauling. LINUS PAULING. E. BRIGHT WILSON, JR. PASADENA, I^AMF., CAMBRIDGE MASS., July, 193J5. CONTENTS PREFACE CHAPTER I SURVEY OF CLASSICAL MECHANICS SECTION 1. Newton's Equations of Motion in the Lagntngian Form ..... 2 lIab.. TGehneerTahlriezee-ddiCmoeonrsdiionnaatlesI.so.tro.pi.c.Ha.r.mo.ni.c.Os\c.illAato.r.. .. . 46 Ic. The Invarian.ce.of.t.he.Eq.ua.ti.on.s.of.Mo.ti.on.i.n .the.L.ag.rai.b- gian Form 7 Id. An Example: T.he.I.so.tr.opi.c.Ha.r.mo.ni.c.Os.ci.ll.ato.r.in.P.ol.ar Coordinates 9 le. The Conservation of Angular Momentum . ........ 11 2. The Equations of Motion in t.he.H.am.il.to.ni.an.F.or.m....... 14 2a. Generalized Momenta 14 ....... 26. The Hamiltonian Function and Equations ...... 16 2c. The Hamiltonian Fu.nct.io.n .and.t.he.En.er.gy...... 16 2d. A General Example ...... 17 3. The Emission and Abso.rp.ti.on.o.f R.ad.ia.ti.on..... ....... 21 4. Summary of Chapter I 23 CHAPTER II THE OLD QUANTUM THEORY ........... 5. The Origin of the Old Quantum Theory 25 ................ 5a. The Postulates of Bohr 26 ...... 56. The Wilson-Sommerfeld Rules of Quantization 28 5c. Selection Rules. The Corresp.o.nd.en.ce.P.ri.nc.ipl.e.......... 29 6. The Quantization of Simple Systems 30 ...... 6a. The Harmonic Oscillator. Degenerate States 30 .................. 66. The Rigid Rotator 31 6c. The Oscillating and Ro.ta.ti.ng.D.ia.to.mi.c.Mo.le.cu.le.......... 32 6d. The Particle ina Box 33 ........... 6e. Diffraction by.a.Cr.ys.ta.l L.at.ti.ce............ 34 7. The Hydrogen Atom 36 .......... 7a. Solution of the Equations of Motion 36 76. Application of theQuantu.m.Ru.les....Th.e.En.er.gy.L.ev.el.s... 39 7c. Description of the Orbi.ts................ 43 7d. Spatial Quantization 45 ............ 8. The Decline of the Old Quantum Theory 47 v CONTENTS vi 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. WaveFunctions. Discreteand ContinuousSetsofCharac- teristic Energy Values 58 9d. The Complex Conjugate Wave Function ty*(x,t).... 63 10. The Physical Interpretation of the WaveFunctions 63 10a. Sk*(x,t)V(x,t) as a Probability Distribution Function. . 63 106. Stationary States 64 lOc. Further Physical Interpretation. Average Values of Dynamical Quantities 65 11. The Harmonic Oscillatorin Wave Mechanics 67 . . 11a. Solution of the Wave Equation . . 67 116. The Wave Functions for the Harmonic Oscillator andtheir Physical Interpretation ... . 73 lie. 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 ina Box 95 15. The Three-dimensional Harmonic Oscillator in Cartesian Coordi- nates 100 16. Curvilinear Coordinates 103 17. 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 Solutionof the *>Equation 117 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. TheLegendre FunctionsorLegendre Polynomials 126 196. The Associated Legendre Functions 127 20. TheLaguerre Polynomials and Associated Laguerre Functions . .129 20a. TheLaguerre Polynomials 129 206. The Associated Laguerre Polynomials and Functions ... 131 21. The Wave Functions for the Hydrogen Atom 132 21a. Hydrogen-like Wave Functions 132 216. The Normal State of the Hydrogen Atom 139 21c. Discussionofthe Hydrogen-like Radial Wave Functions. . 142 2ld. Discussion of the Dependence of the Wave Functions onthe 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-degenerateLevel ... 156 23a. A Simple Example: T>/> P^^tujj^ejjJH^mjinnin QsillRf.nr . 160 ?jfo An Exjunplp- TforNormal FHium Atn 162 24. J$tst-order Perturbation Theory for a DegenerateLevel 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 ExamplefThe Normal State of Helium Atom 184 26c. Application of theVariation Methodthteo_OtherStates ...... 186 26d. 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. Numerical Integration 201 27d. Approximationbythe UseofDifferenceEquations .... 202 270. AnApproximateSecond-orderPerturbationTreatment 204 . . 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 TreatmentoftheNormalHelium Atom. . . 221 29d. Excited States of the Helium Atom 225 29e. The Polarizability of the NormalHelium Atom 226 CHAPTER IX MANY-ELECTRON ATOMS 30. Slater's Treatment of Complex Atoms 230 30a. Exchange Degeneracy 230 306. Spatial Degeneracy 233 30c. Factorization and Solution of the Secular Equation. . . 235 30d. Evaluation of Integrals ...* 239 30e. Empirical Evaluation of Integrals. Applications. . . . 244 31. Variation Treatments for Simple Atoms 246 31a. The Lithium Atom and Three-electron Ions . .... 247 316. Variation Treatments of Other Atoms. . . 249 32. 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 Methodsfor 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 3355d.. AA SMiomrpeleAcPcoutreanttiealTrFeuantcmteinotn.forThDeiaMtoormisce MFoulnecctuiloens . . . 227617 . . . 36. The Rotation of Polyatomic Molecules 275 36a. The Rotationof Symmetrical-top Molecules 275 36b. The Rotation of Unsymmetrical-top Molecules 280 CONTENTS ix SECTION PAGE 37. The Vibiationof Polyatomic Molecules 282 37a. Normal Coordinates in Classical Mechanics 282 376. Normal Coordinates in Quantum Mechanics 288 38. The Rotation of Moleculesin 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 .... 294 39a. A Simple Example .... 296 40. The Emission and Absorption of Radiation.V .... 299 40a. The Einstein Transition Probabilities .... 299 406. The Calculation of the Einstein Transition Probabilities by Perturbation Theory 302 40c. Selection Rules and Intensities forthe 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 IntensitiesfortheHydrogenAtom . .312 400. Evenand Odd Electronic States and their Selection Rules. 313 41. 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 VariationTreatments 331 42c. The Separation andSolutionoftheWave Equation.... 333 42d. Excited States ofthe Hydrogen Molecule-ion 340 43. The Hydrogen Molecule 340 43a. The Treatment of Heitlerand London. 340 436. Other Simple Variation Treatments. . . 345 43c. The Treatment of James and Coolidge 349 43d. ComparisonwithExperiment 351 43e. Excited States ofthe Hydrogen Molecule 353 43/. Oscillation and Rotation of the Molecule. Ortho and Para Hydrogen 355 44. The Helium Molecule-ion Hef and the Interactionof Two Normal Helium Atoms 358 44a. The Helium Molecule-ion Hef 358
The list of books you might like
