1 INTRODUCTION TO QUANTUM MECIIANICS ー″んノ盟ん c 住〃 0 れ s C ん e 襯なケ BY LINUS PAULING, PH. D. , Sc. D. Professor 可 C ん e 襯 t , C 記ゾ 0 れ 1 れ就 e 0 / Tec ん叩 4 AND . BRIGHT WILSON, 3 , PH. D. ス 880C 観 e ~ 0 / を 880r 可 C ん e 襯け , ″ a の ard U れを ~ e ア 8 リ INTERNATIONAL STUDENT EDITION MCGRAW-HILL BOOK COMPANY, INC. NEW YORK ST. LOUIS SAN FRANCISCO LONDON MEXICO PANAMA SYDNEY TORONTO KöGAKUSHA COMPANY, LTD. TOKYO

XIV-50] 刃Ⅳ刃 GY 0 ACT AT ー 0 ル 0 c 〃刃、ー c 刃ユ CTIO ル 415 of the CouIomb and single exchange integrals being taken from the simple Heitler-London-Sugiura treatment of the hydrogen molecule or estimated from the empirical potential function for this molecule. These approximate treatments led to values in the neighborhood of 10 to 15 kcal for the activation energy ・ C001idge and James1 have recently pointed out that the approxi- mate agreement with experiment depends 0 Ⅱ the cancellation of large errors ar1Sing from approximations. The similar discussion of the activation energies Of a number Of more complicated reactions has been given by Eyring and collaborators. 2 1 A. S. COOLIDGE and Ⅱ . M. JAMES, 工 C ん e . 2 んい . 2 , 811 ( 1934 ). 2 Ⅱ . EYRING, 工襯 . C ん e . Soc. 53 , 2537 ( 1931 ) ; G. 刊 . KIMBALL and Ⅱ . EYRING, 記 . 54 , 3876 ( 1932 ) ; A. SHERMAN and Ⅱ . EYRJNG, 記 . 54 , 2661 ( 1932 ) ; R. S. BEAR and Ⅱ . EYRING, 記 . 56 , 2020 ( 1934 ) ; Ⅱ . EYRING, A. SHERMAN, and G. E. KIMBALL, 工 C ん e 襯 . 2 ん . 1 , 586 ( 1933 ) ; A. SHERMAN, C. SUN, and Ⅱ . EYRING, . 3 , 49 ( 1935 ).

228 71 〃刃 SPINNING ELEC 7 0 [VIII-29e field (assumed to lie along the 名 axis) is ( 21 十名 2 ) , 21 and 22 being the 名 coordinates of the two electrons relative to the nucleus. The argument of Section 27C suggests that the variation function be of the form = 0 { 1 十 ( 21 十名 2 ゾ@1 , 41 , 21 , 昀 , 22 ) } , ( 29 ー 11 ) in which 0 is an approximate wave function for zero field. Variation functions of this form ()r approximating (t) have been discussed by Hassé, Atanasoff, and SIater and Kirkwood, 1 whose results are given ⅲ TabIe 29 ー 3. TABLE 29ー3. ー-ー・ VARIATION FUNCTIONS OR THE CALCULATION OF THE POLARIZABILITY 0 を THE 、・ ORMAL HELIUM 、 TOM Experimental value : 夜 = 0.205 ・ 10 ー 24 cm3 1 ヨー r2 00 Variation function e-Z'• { 1 十ス ( 21 十 2 ) } . ー 2 ' を { 1 十は ( 21e ー 2 " 十名 2e ー 2 " つ } 3. @1r2 ) 0 ・ 255e ー幻 8 { 1 十 ( 名 le ー 2 " ア 1 十 22e ー 2 " つ } 4. e ー 2 ' 8 { 1 十 ( 21 十司十 B ( 2 産 1 十名 2 ) } . ー 2 % { 1 十 ( 21 十名 2 ) 十 terms to quartic) } ー 2 ′ 8 ( 1 十の { 1 十 ( 名 1 十 32 ) 十 B ( 名産 1 十 2r2 ) } 7. e ー 2 1 十十ド十 ( 箚十名 2 ) } . 8. e ー 2 ' ・ { 1 十社十ド十い十 Bs)(Z1 十 22 ) 十夜 ( 21 ー 22 ) } 9. e ー 2 ' 8 { 1 十 CI 十 C2 卩十 C3S 十 C4S2 十 C5 2 十い十 Bs)(21 十 22 ) 十夜 ( 名 1 名 2 ) 十 1 十 22 ) } 10. e ー 2 % ( 1 十 cru 十ド ) { 1 十ス ( 21 十名 2 ) 十 B ( 1 十名 2 2 ) } 11. A non-algebraic function. 1 Ⅱ = Ⅱ s き , A = Atanasoff, SK = Slater and Kirkwood. Of these functions, 1 , 2 , 4 , and 5 are based 0 Ⅱ the simple screening-constant function 2 of Table 29 ー 1 ; these give 10W values Of 伐 , the experimental value (from indices of refraction extrapolated tO large wave length Of light and from dielectric 1 Ⅱ . R. HASSE, Proc. Ca 襯い記 ge Phil. Soc. 26 , 2 ( 1930 ) , 27 , 66 ( 1931 ) ; 工 V. ATANASOFF, P んい . 側 . 36 , 1232 ( 1930 ) ; 工 C. SLATER and 工 G. KIRK- WOOD, P んい . e 37 , 682 ( 1931 ). R. eferences 1 . 0.1 測 ・ 10 ー 24cm3 .164 SK .222 SK . 182 . 183 .201 . 127 . 182 . 194 .231 . 210

freatment for different functions / ( , 『 2 ) are given ⅲ Table 47 ー 1. lt is seen that the coeffacient 0f 一 e2 / 6 approaches a valuel only slightly larger than 6.499 ; this can be accepted as very close to the correct value. So far we have considered 0 Ⅱ ly dipole-dipole interactions. Margenau2 has applied the approximate second-order perturba- tion method of Section 27e t0 the three terms 0f Equation 47 ー 3 , 386 ー 6 .14 ー 6 . 462 ー 6 . 469 ー 6 . 482 ー 6 . 49 ー 6 . 4 間 ー 6 .498 盟 / SC 刃んん A ル EO US 」ア P. ん / CA770 Ⅳ S [XIV 47a obtaining the expression 6C205 究 6 135e2 鴫 1416e209 10 ( 47 ー 7 ) lt is seen that the higher-order terms become important at small distances. TABLE 47ー1. —VARIATION TREATMENT OF VAN DER 、 VAALS INTERACTION OF ′ 0 HYDROGEN ATOMS Variation function 術“ ( 1 ) 術 ( 2 ) 1 十 - ー ( 工 1 工 2 十″ 2 2 3 .f(rl, 2 ) 2. ー 2/@1 十 2 ). 3. 十お十 2 ) . 4. 十 B r 2. 5. 十 B@1 十 r2 ) 十 C 廬 r2. 6. 可朝 = 0.325 ) . 7. 十 B 廬 r2 十 C ド Polynomial to ら r2 ・ 10. PoIynomial to バ . 9. PoIynomialt to 務 . 8. 十 B 産 2 十 C 務十 D 戸 r3 11. E ール 0 ー 6 . 49 3 ー 6 .49899 ー 6 .4984 ー 6.00e2 / ん 6 2 122 ゾ@1, ) R. eference * SK PB PB SK PB PB PB * Ⅱ = Ⅱ sé , SK = Slater and Kirkwood, PB = PauIing and Beach. coeffcient# have been calculated by Pauling and Beach , あ c. . 2 Ⅱ . MARGENAU, P んい . 記側 . 38 , 747 ( 1931 ). More accurate values of the The source of the error has been pointed out by Pauling and Boach, あ c. た by Eisenschitz and London), being larger than the upper limit 8 given above. him for the coefficient, 24 % 8 = 8.68 , must be in error ()s first pointed ou も was made by S. C. Wang, んい . み 28 , 663 ( 1 7 ). The value found bY coefficient [Z. /. P んい . 60 , 491 ( 1930 ) ]. The first attack on this problem tion theory by R. Eisenschitz and F. London gave the value 6.47 for this 1 A straightforward but approximate application Of second-order perturba- ↑ The 001Y Ⅱ omial contains all terms of degree 2 or 1 s in ロ and 2 or le in 「 2 ・

Ⅸ -30d ] SLA TER'S TREA イ刃Ⅳ T 0 CO カ刃 X ス TOMS From this we see that 241 where / 0 つ is a function of alone. The integral of the first term in 〃 thus reduces to ・エ ( 可 ( 朝 P ( の臨 工磅 ( Ⅳ泚盻 ( ル ) Ⅳ , ( 30 ー 13 ) ⅲ which 2 ( の is used as a symbol for ( のⅲ which electron has replaced を as a result of the permutation 2. Because of the orthogonality of the u's, this is zero unless 2町@) = ( の except perhaps for equal tO the one value を . ln addition, since ( 3g14 ) two pairs must match. The factor containing these unmatched Similar treatment of the term ) 0 shows that all but perhaps the members 0f that pair have the same value of 襯 this integral will vanish, unless all the u's but one pair match and 2舛@) have the same quantum number 襯 z. We thus see that the factor 工4き@ゾ( ) P ⑦ d 叮 will be zero unless ( の and fun C tions iS J ( の ( の一 ( の記行 . lt can be shownl that 1 ( ん十回 ) ! 務再た 2 戸司 (cos ) に l(cos ら ) e ー % ) ん , ( 3g15 ) ( 3g16 ) ⅲ which is the smaller of and , and 島 is the greater. 2 (cos ) is an associated Legendre function, discussed in Section 196. Using this expansion we obtain for the 伊 part of 2T 2 す the above integral を ( お丐ー一物。 ) が叫 " ト ) 記記ゎ which 襯 s associated with (@) , 襯 : with ( の , 襯ー with ( の , and Pmf with ( の . This vanishes unless 襯ー襯ー十襯 = 0 and 2 襯 : ー襯 ~ ー襯 = 0 ; i. e. , unless ~ 襯 ~ 十 pmf = 襯 ~ 十襯 :. 1 For a proof of this see J. Ⅱ . Jeans, " Electricity and Magnetism," 5th *. Equations 152 and 196 , Cambridge University press, 1927.

83 lll-llc) 社ス MO Ⅳー C 08C / んん 0 / Ⅳル刃 MECIIÄ NICS A. 刊 . RVARK and Ⅱ . C. tJREY.•“ Atoms, Molecules and Quanta, ” McGraw- HiII Book Company, lnc. , New York, 1930. N. F. MOTT : “ An Outline of Wave Mechanics, ” Cambridge University Press, Cambridge, 1930. J. FRENKEL.•“ Wave Mechanics, ” Oxford University Press, 1933 ー 1934. K. K. DARROW : Elementary Notions of Quantum Mechanics, 記側 . M . 尸んい . 6 , 23 ( 1934 ). . C. KLMBLE : General PrincipIes of Quantum Mechanics, Part I, e 既 外ー od. P んい . 1 , 157 ( 1929 ). . C. KEMBLE: "Fundamenta1 Principles of Quantum Mechanics, ” Mc- Graw-HiII Book Company, lnc. , 1937. E. C. KEMBLE and E. L. HILI,: GeneraI Principles of Quantum Mechanies, Part Ⅱ , e 紗 . od. 2 んい . 2 , 1 ( 1930 ). S. DUSHMAN : "Elements of Quantum Mechanics," John WiIey & Sons, lnc. , ] 938.

Ⅱ】ア″刃 D 刃 C. ん 7 Ⅳ刃 OF ア〃刃 0 D QUANTUM ア″ EO 49 developed by Heisenberg, Born, and Jordan1 by the introduction of matrix methods. ln the meantime Schrödinger had inde- pendently discovered and developed his wave mechanics,2 stimulated by the earlier attribution of a wave character to the electron by de BrogIie3 ⅲ 1924. The mathematical identity of matrix mechanics was shown Schrödinger4 and by Eckart. 5 The further development of the quantum mechanics was rapid, especially because of the con- tributions 0f Dirac, who formulated6 a relativistic theory of the electron and contributed to the generalization of the quantum mechanics (Chap. XV). GeneraI References on the 01d Quantum Theory A. SOMMERFELD: "Atomic Structure and Spectral Lines," E. p. Dutton & Co. , lnc. , New York, 1923. A. . RUARK and H. C. UREY: "Atoms, MoIecules and Quanta, ” McGraw- HiIl Book Company, lnc. , New York, 1930. 1 M. BORN and P. JORDAN, 記 . 34 , 858 ( 1925 ) ; M. BO#N, W. HEISENBERG, and P. JORDAN, 防記 . 35 , 557 ( 1926 ). 2 . SCHRöDINGER, れれ . d. 2 んい . 79 , 361 , 489 ; 80 , 437 ; 81 , 109 ( 1926 ). 3 L. DE BROGLIE, Thesis, Paris, 1924 ; れれ . de 〃んい . ( 10 ) 3 , 22 ( 1925 ). 4 刊 . SCHRöDINGER, ユれれ . d. んい . 79 , 734 ( 1926 ). 5 C. ECKART, P ん ys. 側 . 28 , 711 ( 1926 ). 6 P. A. M. DIRAC, Proc. 0 鱶 Soc. A 113 , 621 ; 114 , 243 ( 1927 ) ; 117 , 610 ( 1928 ).

XIII-46fJ S. ん T 刃ん ' 8 刃は 7 ' 」一刃 0 COMPLEX 0 ん刃 CU. ん刃 s 379 6 0 and d C 0 6 d C Ⅱ Ⅱ we neglect all exchange integrals of 〃 except the single exchange integrals between adj acent atoms, which we call ( 面 ) ] , and all exchange integrals occurring ⅲ△ , the secular equation is found by the rules of Section 46e to be Q 十夜ール Y2Q 十ー V2Q 十 2 夜ー与ル Q 十夜ール The solutions of this are ル = Q 十 2 伐 and ル = Q ー 2 of which the former represents the normal state, 伐 being negative ⅲ sign. The energy for a single structure (I or Ⅱ ) is = Q 十伐 ; hence the so れ 0 れ ce between the two structures stabilizes the system by the amount 住 . Extensive approximate calculations Of resonance energies f01 molecules, especially the aromatic carbon compounds, havc been made, and explanations of several previously puzzling phenomena have been developed. 1 Empirical evidence has alSO been advanced tO show the existence Of resonance among several valence—bond structures in many simple and complex 1 ) 101ecules.2 lt must be emphasized, as was done ⅲ section 41 , that the use 0f the term so れ 0 れ ce implies that a certain type of approximate treatment is being used. ln this case the treatment is based on the valence-bond wave functions described above, a procedure which is closely related to the systematization of molecule formation developed by chemists over a long period of years, and the introduction of the conception of resonance has per- mitted the valence-bond picture to be extended to include 1 . HÜCKEL, Z. /. p んい . 70 , 2 ( ( 1931 ) , etc. ; L. PAULING and G. w WHELAND, 工 C 厖襯 . んい . 1 , 362 ( 1933 ) ; L. PAULING and 工 SHERMAN, 碗 d. 1 , 679 ( 1933 ) ; 工 SHERMAN, 仂記 . 2 , 488 ( 1934 ) ; W. G. PENNEY, p oc. ひ 4. Soc. A146 , 223 ( 1934 ) ; G. W. WHELAND, 工 C ん e 襯 . p んい . 3 , 230 ( 1935 ). 2 L. PAULING, 工ユ . C ん e 襯 . Soc. 54 , 3570 ( 1932 ) ; proc. . cad. . 18 , 293 ( 1932 ) ; L. PAULING and 工 SHERMAN, 工 C 厖襯 . んい . 1 , 606 ( 1933 ) ; G. W ・ . WHELAND, 記 , 1 , 731 ( 1933 ) ; L. 0. BROCKWAY and L. PAULING, Proc. Ⅳ観 . cad. S . 19 , 860 ( 1933 ).

80 T 刃 SC 社ÖD ーⅣ G 霽記 ⅲ ity, the polynomials 〃。 (E) the Hermite polynomials. The func tions Ⅳス / 刃 EQ UATION [ I Ⅱ -1 な introduced ⅲ Section 110 are = ( 11 ー 20 ) are called the ″襯 e 0 れん 0 れ al / c 0 れ s , ・ they are, as we have seen, the wave functions for the harmonic oscillator. The 工 k(c)dc = 1 , i. 。 . , which normalizes value of Ⅳ纏 which makes , is Ⅳれ 2 れれ ! ( 11 ー 21 ) The functions are mutually orthogonal if the integral 0 v er configuration space 0f the product of any two of them vanishes : メ襯 . ( 11 ー 22 ) TO prove the orthogonality of the functions and to evaluate the normalization constant given ⅲ Equation 11 ー 21 , it is convenient tO consider tWO generating func tions : and Using these, we obtain the relations e い一 ( を一い , 2 ー (t ーま 十 STe¯E2dE ーを , ー十 2 畦十 2 畴ード d ミ 十 e e ード e ー ( ←←の記 十 2 22S2 ド 2 れ 十・ Considering coefrients of ⅲ the two equal series expansions, 工 - 。 (k ) 〃の。→ ' 破、 " i 。訂。 , 。。 , and hasthe we see that value 2 。 ! Ⅳ for = ⅲ consequence Of which the functiong

242 ーⅣ Y - 霽ん刃 C ア 0 Ⅳ ATOMS [IX-30d This completes the proof of the theorem that 〃。。 = 0 unless ) 襯員 s the same for 協れ and 協禰 Of the non-vanishing elements 〃。 0 Ⅱ ly certain 0f the diagonal ones need tO be evaluated ⅲ order tO calculate the energy levelS' as we have seen ⅲ the last section. Because Of the orthogonality of the u's, Equation 30 ー 13 vanishes unless 2 = 1 (the identity operation) when a diagonal element 〃。。 is being considered. Since the u'S are alSO normalized, thiS expression reduces tO ( 3g17 ) a relation which defines the quantities ム . ) 工⑦ ( の 0 ( の 0 ( の d , , = ム , , ( 0 、 contributes the terms interchange Of を and の respectively. The first ch0ice Of 3g15 to = 1 and = ( ) , the identity operation and the Similarly, the orthogonality 0f the u's restricts P ⅲ Equation イけ > イ while the second yields ) 工⑦可 ( の色 0 ( の 0 ( の 0 , け > イ イロ > を The integral Kii vanishes unless the spins 0f ( の and ( の are parallel, i. e., unless 〃 The functions ム reduce tO integrals over the radial part 0f = 瓢ゾ ( 広 i)d ム We shall not evaluate these further. ( 3g20 ) The functions and Kii may be evaluated by using the expansion for 1 〃新 given ⅲ Equation 3g16. For 諸 the 窪 記窪 , which vanishes part 0f the integraJ has the form unless = 0. The double sum ⅲ the expansion 3g16 thus reduces tO a single sum over ん , which can be written ( 3g21 ) ⅲ which 襯 ~ and が′襯 : are the quantum numbers previously represented bY and ゎ respectively. and ん are given by