Introduction to quantum mechanics : with applications to chemistry

475 Pages·1935·17.593 MB·English

by  Linus Pauling, E Bright Wilson

INTRODUCTION
MECHANICS
QUANTUM
With Applications to Chemistry
LINUS-pAULfG, PBPM SC.D
ProfessorofChemistry, CaliforniaInstituteofTefyn&oftT
E. BRIGHT WILSON, JR., PH.D.
AssociateProfessorofChemistry,Harvard University
INTERNATIONAL STUDENT EDITION
MCGRAW-HILL BOOK COMPAN^IN '
NEW YORK AND LONDON
KOGAKUSHA COMPANY, LTD.
TOKYO.
INTRODUCTION TO QUANTUM MECHANICS
INTERNATIONAL STUDENT EDITION
Exclusive rights by Kogakusha Co., Ltd. for manufacture and export
from Japan. This book cannot be re-exported from the country to
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COPYRIGHT, 1935, BY THE
MCGRAW-HILL BOOK COMPANY, INC.
fill rights reserved. This book, or
parts Ike;'<;/, ir.ay rtct be reproduced
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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 presen^ia thorough
survey of the subject. We hope that it does gb^ela, lucid and
easily understandable introduction to a limjAdVpori*<m >
quantum-mechanical theorv; n*w*iv. *hat i^WJ<-- Via1^
suggested by the nnm- ( ,-..-/. ,., 1 <..,-,,, r^a^tinta of the
al&eussion ofthe Schrodinger wave equalkM . l?fce" problems
which ran 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
ofthe 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
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