436 GENERAL T ″ EO 必 0 を Q UANTUM MECHANICS [XV-54 On application Of this equation it is found that the 襯 0 襯 e れ襯 佖 / れ c 0 れ s for the harmonic oscillator have the same form (Hermite orthogonal functions) as the coordinate wave functions (Prob. 54 ー 1 ) , whereas those for the hydrogen atom are quite different. 1 P て 0b1 em 54 ー 1. EvaIuate the momentum wave functions for the harmonic oscillator. Show that the average value of 2 : for the nth state given by the equation is the same as given by the equation Problem 54 ー 2. hydrogen atom, 2 öェア EvaIuate the moment am wave function for the normal 2 ( 工切十切ー十 2 切 : ) It iS convenient tO change tO polar coordinates in momentum space as well as 1 Ⅱ coordinate space. The further developments of quantum mechanics, including the discu SSIOII Of maximal asurements C onsisting not 0f the accurate determination Of the values Of a 1 1 ⅡⅱⅡⅡ number Of independent dynamical functions but of the approximate meas- urement 0f a larger number, the use 0f the theory of groups, the formulation of a relativistically invariant theory, the quantiza- tion 0f the electromagnetic field, etc. , are beyond the scope of this book. GeneraI References 0 Ⅱ Quantum Mechanics Matrix mechanics : M. BORN and P. JORDAN: “ Elementare Quantenmechanik," Julius Springer, Berlin, 1930. Transformation theory and general quantum mechanics : P. A. M. DIRAc: “ Quantum Mechanics," Oxford University press, New York, 1935. 1 The hydrogen-atom momentum wave functions are discussed by B. Podolsky and L. PauIing, P ん . 記 . 34 , 109 ( 1929 ) , and bv 刊 . A. Hylleraas, . /. ん . 74 , 216 ( 1932 ).

INDEX 459 Free particle, Free rotation ⅲ molecules 280 , 290 FrenkeI, 工 , , 437 Frequency, 0f harmomc oscillator, 5 of resonance, 320 Friedrichs, 202 Fues, . , 274 F u nd am ental frequency, definition of, 290 0 factor for electron spin, 208 Geib, K. Ⅱ . , 414 GeneraI solution Of wave 57 GeneraI theory 0f q uantum mechan- ics, 416 ガ . GeneraIized coordinates, 6 Generalized forces, 7 Generalized momenta, definition of 14 GeneraIiz ed perturbation theory, 191 Generaliz ed velocities, 7 Generating function, for associated Laguerre polynomials, 131 for associated Legendre functions, 128 definition of, 77 for Laguerre polynomials, 129 for Legendre polynomials, 126 GentiIe, G. , 361 Ginsburg, N. , 246 Gordon, W. , 209 Goudsmit, S. , 207 , 208 , 213 , 221 , 227 , 237 , 246 , 257 , 258 , 313 Gropper, L., 405 Group, completed, of electrons, 125 definition of, 231 Group theory and molecular vibra- Hamiltonian equations, Half-quantum numbers, 199 Guillemin, V. , 247 , 332 , tions, 290 16 353 HamiItonian form of equations of motion, 14 HamiItonian function, definition of, 16 and the energy, 16 and the wave equation, 54 Hamiltonian operator, 54 Harmonic oscillator, average of が , 161 classical, 4 ⅲ cylindrical coordinates, 105 energy levels f0 r, 72 ⅲ 01d quantum theory, 30 perturbed, 160 selection intensities fO ら 306 three-dimensional, ⅲ Cartesian coordinates, 100 wave functions, properties of, 77 ln wave mechanics, 67 ガ Harmonic oscillators, coupled, 315 397 Harmomcs, surface, 126 Harteck, P. , 358 , 414 Hartree, D. R. , 201 , 224 , 250 , 254 , 255 Hartree, W. , 255 Hassé, Ⅱ . R. , 185 , 228 , 385 , 387 Heat, of activation, 412 Of dissociation, of hydrogen mole- cule, 349 , 352 of hydrogen molecule-ion, 336 Heat capacity, of gases, 408 of solids, 26 Heats of sublimation, 388 Heisenberg, W. , 48 , 112 , 209 , 210 , 226 , 318 , 416 , 417 , 428 , 432 , 437 Heisenberg uncertainty principle, 428 HeitIer, W. , 340 , 361 , 364 Helium , solid , equilibrium distance in, 362 Helium atom , 210 accurate treatments of, 22 excited states of, 225 ionization potential of, 221 normal state Of, by perturbation theory, 162

Ⅱ -8 ] TIIE D 刃 C ん / Ⅳ刃 0 T 〃 0 D QUÄNTUM T ″刃 0 必 47 mitted, as shown ⅲ Figure 7 ー 4. Values 士た for 襯 always cor- respond tO orbits lying ⅲ the 工リ plane. lt can be shown by the methods 0f classical electromagnetic theory that the motion 0f an electron with charge ー e and mass ? 〃 0 in an orbit With angular mornentum - ー - rise tO magnetic 2 んん e field corresponding t0 a magnetic dipole 0f magnitude 27 2 oc oriented in the same direction as the angtllar vector. The component of magnetic moment in the direction 0f the 名 axis The energy of magnetic interaction of the atom with IS ~ 4T ~ OC a magnetic field 0f strength ″ parallel t0 the-z axis is 4g7 〃 OC lt was this interaction energy which was considered tO give rise to the Z 襯佖れ effect (the splitting 0f spectral lines by a magnetic field) and the phenomenon 0f 、 paramagnetism. lt is 110W known that this explanation is only partially satisfactory, inasmuch as the magnetic moment associated with the spin 0f the electron' discussed ⅲ Chapter Vlll, also makes an important contribution. is called a お 0 ん r 襯 09 れ 0 れ . The magnetic moment ProbIem 7 ー 1. Calculate the frequencies 盟 ld wave lengths 0f the first five members Of the Balmer series for the isotopic hydrogen atom whose mass is approximately 2.0136 on the atomic weight scale, and compare with those for ordinary hydrogen. Problem 7 ー 2. Quantize the system consisting Of tWO neutral particles of masses equal t0 those Of the electron and proton held together by gravita- tional attraction, obtaining expressions for the axes Of the orbits and the energy levels. 8. THE DECLINE OF THE OLD QUANTUM THEORY The historical development 0f atomic and molecular mechanics up t0 the present may be summarized by the following divisi011 intO periods (which, 0f course, are not SO sharply demarcated as indicated) : 1913 一 1920. 1920 ー 1925. 192 The origin and extensive application of the old quantum theory 0f the atom. The decline of the 01d quantum theory. The origin of the new quantum mechanics and its application tO physical problems.

48 1927 ー ア″刃 0 ん〃 QU Ⅳし一″刃 0 [ Ⅱ The application Of the new quantum mechanics to chemical problems. The present time may well be also the first part of the era of the development of a more fundamental quantum mechanics, includ- ing the theory 0f relativity and of the electromagnetic field, and dealing with the mechanics 0f the atomic nucleus as well as of the extranuclear structure. The decline of the 01d quantum theory began with the introduc- tion of half-integral values for quantum numbers ⅲ place of integral values for certain systems, in order to obtain agreement with experiment. lt was discovered that the pure rotation spectra 0f the hydrogen halide molecules are not ⅲ accordance with Equation 6 ー 9 with K = 0 , 1 , 2 , but instead require Similarly, half-integral values of the osc illa- tional quantum number ⅲ Equation 6 ー 11 were found to be required ⅲ order to account for the observed isotope displace- ments for diatomic molecules. Half-integral values for the azimuthal quantum number ん were 引 so indicated by observations on both polarization and penetration of the atom core by a valence electron. Still more serious were cases in which agree— ment with the observed energy levels could not be obtained by the methods of the 01d quantum theory by any such subterfuge or arbitrary procedure (such as the normal state of the helium atom, excited states of the helium atom, the normal state of the hydrogen molecule ion, etc. ) , and cases where the methods 0f the old quantum theory led to definite qualitative disagreement with experiment (the influence of a magnetic field 0 Ⅱ the dielectric constant 0f a gas, etc. ). Moreover, the failure of the old quan- tum theory t0 provide a method of calculating transition probabil- ities and the intensities Of spectral lines was recognized and more clearly as a fundamental flaw. Closely related to this was the lack 0f a treatment of the phenomenon of the disper- sion 0f light, a problem which attracted a great amount of attention. This dissatisfaction with the 01d quantum theory culminated ⅲ the formulation by Heisenberg1 ⅲ 1925 of his quantum mechanics, as a method of treatment of ato systems leading t0 values 0f the intensities as well as freq ncies of spectral lines. The quantum mechanics of Heisen erg was rapidly 1 ・ W ・ . HEISENBERG, Z. /. 尸んい . 33 , 879 ( 1925 ).

INDEX 465 Rotational levels, allowed, 392 R0tational quantu m number, 33 Rotational and vibrational energy 0f molecules, 405 Rotational wave functions, sym- metry of, 355 Rotator, plane , Stark effect of, 177 Rydberg constant, 41 Rutherford atom, 26 Russell-Saunders coupling, 238 RummeI, K. W. 358 Rumer, G. , 365 , 375 255 R. ubidium ion, 、 functions 292 , 312 , 420 , 432 , 437 Ruark, A. . , 49 , 54 , 83 , 92 , 258 , ⅲ 01d quantum theory, 31 rigid, 271 wave equation Of, 177 Schrödinger, 刊っ 49 ガ . Schaefer, A. , 293 Sachsse, Ⅱ . , 358 for, for even and Odd ・ wave functions, ecules, 266 , 309 SeIection rules, for diatomic m01- 171 SecuIar perturbation, definition of, ⅲ vibration problems, 286 and variation m eth od, 188 sum of roots of, 239 numerical solution of, 188 for molecules, factoring , 369 factoring, 173 definition of, 171 SecuIar equation, for atoms, 23 「 approximate, 204 Second-order perturbation theory, for helium , 185 Screemng constants, for atoms, 256 53 wave function including the time, wave equation , 50 equation in three dimensions, 84 ガ . 313 , 3 Selection rules, for the harmonic oscillator, 306 of old quantum .theory, 29 for surface—harmomc wave func— tions, 306 Self-consistent field, method of, 2 立 . and the vari ation metho d , 252 Separation, Of electronic and nuclear motion in molecules, 259 of hydrogen molecule-ion equa- tion , 333 of variables, 56 , With curvilinear coordinates , 105 0f wave equation, 113 Series of orthogonal functions, 151 SheIIs of electrons, completed, 234 in lithium atom, 248 Sherman, A. , 415 Sherman, , 365 , 379 ShortIey, G. Ⅱ . , 203 , 246 , 258 Single-electron wave functions, 254 use of, 231 Singlet states, 214 , 220 Singularity, definition of , 109 Size of the hydrogen atom, 140 Slater, 工 C. , 191 , 221 , 222 , 228 , 230 , 233 , 252 , 254 , 256 , 275 , 294 , 299 , 348 , 361 , 364 , 366 , 377 , 387 , 403 Slater's ak's and bk's, table of, 243 , 245 Smith, J. P. , 226 Smyth, C. P. , 412 Solution, of の equation, 117 Of secular equation for atom, 235 Of wave equation, approximate, 180 by expanslon in series , 191 SommerfeId, A. , 28 , 36 , 41 , 49 , 68 , 82 , 92 , 112 , 125 , 155 , 207 , 209 , for molecules. 354 , 237 , 314 Spectroscopic nomenclature, Spatial quantiz ation, 45 Spatial degeneracy, 233 403 , 437

CHAPTER II THE OLD QUANTUM THEORY 5. THE ORIGIN OF THE OLD QUANTUM THEORY The 01d quantum theory was born ⅲ 1900 , when Max P1anck1 announced his theoretical derivation of the distribution law for black-body radiation which he had previously formulated from empirical considerations. He showed that the results of experl- ment 0 Ⅱ the distribution of energy with frequency of radiation ⅲ equilibrium with matter at a given temperature can be accounted for by postulating that the vibrating particles of matter (considered to act as harmonic oscillators) do not emit or absorb light continuously but instead only ⅲ discrete quanti- ties 0f magnitude ん proportional to the frequency 材 of the light. The constant of proportionality, ん , is a new constant of nature ; it is called れ c ん ' s co れ s 厄 and has the magnitude 6.547 x 10 ー 27 erg sec. lts dimensions (energy X time) are those of the 01d dynamical quantity called ac 0 れ , ・ they are such that the product 0f ん and frequency ′ (with dimensions sec—l) has the dimensions 0f energy. The dimensions of ん are also those of angular momen- tum, and we shall see later that just as ん′ is a れ 0 れル襯 of radiant energy 0f frequency so is ん / 2T a natural unit or quantum of angular Ⅱ 101 Ⅱ e Ⅱ tu Ⅱ 1. The development of the quantum theory was at first slow. lt was not until 1905 that Einstein2 suggested that the quantity Of radiant energy ん was sent out in the process Of emission Of light not ⅲ all directions but instead unidirectionally, like a particle. The name I ん田れ襯 or をん 0 んれ is applied to such a portion 0f radiant energy. Einstein also discussed the photo- electric effect, the fundamental processes of photochemistry, and the heat capacities of solid bodies ⅲ terms of ・ the quantum theory. 市 e Ⅱ light falls on a metal plate, electrons are emitted from it. The maxnnum speed of these photoelectrons, however, 1 M. PLANCK, スれル d. ん . ( 4 ) 4 , 553 ( 1g1 ). 2 A. EINSTEIN, スれれ . d. んい . ( 4 ) 17 , 132 ( 1 5 ). 25

= 襯ん , ア霽 0 ん D QUANTUM THEORY 襯 = 士 1 , 士 2 , 襯ん -7b ( 7 ー 14 ) Hence the component 0f angular momentum 0f the orbit along the 名 axis can assume only the quantized values which are integral multiples 0f ん / 2T. The quantum number is called the 襯 - netic 0 れ襯襯 6 催 , because it serves t0 distinguish the various slightly separated levels int0 which the field-free energy levels are split upon the application 0f a magnetic field t0 the atom. This quantum number is closely conne cted with the orientation 0f the old-quantum-theory orbit ⅲ space' a question discussed ⅲ Section 7d. The second integral is easily discussed by the introduction 0f the angle x and its conjugate momentum Px = 2 , the t0tal angular momentum 0f the system , bY means 0f the relation' given ⅲ Equation 1-41 , Section pxdx = d 十ル d . ln this way we obtain the equation $pxdx = んん , ( 7 ー 15 ) ( 7 ー 16 ) ⅲ which px is a constant 0f the motion and ん is the sum 0f が and 襯 . This integrates at once t0 = たん , 2 or た = 1 , 2 , ( 7 ー 17 ) Hence the total angular momentum 0f the orbit was restricted by the 01d quantum theory t0 values which are integral mul- tiples 0f the quantum unit 0f angular momentum ん / 2 The quantum number ん is called the 0 襯ん記 9 社 0 れ襯忸わな . To evaluate the first integral it is convenient t0 transform it ⅲ the following way, involving the introduction 0f the angle X and the variable れ = 1 を with the use 0f Equation 7 ー 6 : 1 du 2 dr 2 dx = ・ ル d = 市 r2 dx 2 dx From Equation 7 ー 10 we find 0 Ⅱ differentiation dx. を COS (X ー xo) dx ( 7 一 18 ) ( 7 ー 19 )

CHAPTER XV GENERAL THEORY 0 QUANTUM MECHANICS The branch of quantum mechanics to which we have devoted our attention ⅲ the preceding chapters, based 0 Ⅱ the Schrödinger wave equation, can be applied ⅲ the discussion 0f most questions WhiCh arise in physics chemistry. It is sometimes conven— ient, however, to use somewhat different mathematical methods ・ and, moreover, it has been found that a thoroughly satisfactory general theory 0f quantum mechanics and its physical inter- pretation require that a considerable extension 0f the simple theory be made. ln the following sections we shall give a brief discussion 0f matrix mechanics (Sec. 51 ) , the properties of angular momentum (Sec. 52 ) , the uncertainty principle (Sec. 53 ) , and transformation theory (Sec. 54 ). 51. MATRIX MECHANICS ln the first paper written 0 Ⅱ the quantum mechanicsl Heisen- berg formulated and successfully attacked the problem of calcu- lating values 0f the frequencies and intensities of the spectral lines which a system could emit or absorb ; that is, of the energy levels and the electric-moment integrals which we have been discussing. He did use wave functions wave equations, however, but instead developed a formal mathematical method for calculating values of these quantities. The mathematical method is one with which most chemists and physicists are not familiar ()r were not, ten years ago), some of the operations involved being surprisingly different from those of ordinary algebra. Heisenberg invented the new type of algebra as he needed it ; it was immediately pointed out by Born and Jordan,2 however, that in his new quantum mechanics Heisenberg was 1 W ・ . HEISENBERG, 2. /. p んい . 33 , 879 ( 1925 ). M. BORN and P. JORDAN, . 34 , 858 ( 1925 ). 6

IV PREFACE learn something Of these subjects as he studies quantum mechan- ics. ln order that he may d0 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 t0 condense the various discussions int0 shorter and perhaps more elegant forms. After introductory chapters on classical mechanics and the 01d quantum theory, we have introduced the Schrödinger wave equation and its physical interpretation on a postulatory basiS' and have then gwen in great detail the solution 0f the wave equation for important systems (harmonic oscillator' hydrogen atom) and the discussion 0f the wave functions and their proper- ties, omitting none 0f the mathematical steps except the most elementary. A similarly detailed treatment has been given in the discussion 0f perturbation theory, the variation method' the structure of simple molecules, and, in general' in every important section 0f the b00k. ln order to limit the size 0f the book, we have omitted from discussion St1Ch advanced tOPiCS as transformation theory general quantum mechanics (aside from brief mention in the last chapter) , the Dirac theory 0f the quantization of the electromagnetic field, etc. We have also omitted several subjects which are ordinarily considered as part 0f elementary quantum mechanics, but which are 0f minor importance t0 the chemist, such as the Zeeman effect and magnetic interactions general, the dispersion 0f light and allied phenomena, and rnost 0f the theory 0f aperiodic processes. The authors are severally indebted t0 Professor A. Sommerfeld and Professors . U. Condon and Ⅱ . P. Robertson for their own introduction tO quantum mechanics. The constant adVice of Professor R. C. ToIman is gratefully acknowledged, as well as the aid of Professor P. M. Morse, Dr. L. E. Suttcn, Dr. G. W. Wheland, Dr. L. O. Brockway, Dr. 工 Sherman, Dr. S. Weinbaum, Mrs. Emily Buckingham Wilson, and Mrs. Ava HeIen PauIing. LINUS PAULING. E. BRIGHT WILSON, R. PASADENA, CALIF. , CAMBRIDGE, MASS. , J リ , 1935.

XV-541 アスⅣ S 0 ん、一 0 Ⅳ社霽 0 437 工 v. NEUMANN : “ Mathematische GrundIagen der Quantenmechanik ' ' JuIi Springer, Berlin, 1932. Questions Of frhysical interpretation : T. HEISENBERG: "The Physical Principles of the Quantum Theory, University of Chicago Press, Chicago, 1930. GeneraI references : A. . RUARK and Ⅱ . C. UREY: “ Atoms, MoIecules and Quanta,' McGraw-Hill Book Company, lnc. , New York, 1930. 刊 . U. CONDON and P. M. MORSE: “ Quantum Mechanics," McGraw ・ HilI Book Company, lnc. , New York, 1929. A. SOMMERFELD : “ Wave Mechanics," Methuen & Company, Ltd. London, 1930. Ⅱ . WEYL: "The Theory of Groups and Quantum Mechanics," . P. Dutton & Co. , lnc. , New York, 1931. J. FRENKEL: “ Wave Mechanics, ” Oxford University Press, New York 1933.