68 T ″刃 SC 〃ÖD / G 刃 WA VE EQ UA ー 0 Ⅳ [lll-lla employed ⅲ applying the wave equation, but also because this system is 0f considerable importance in applications which we shall discuss later, such as the calculation 0f the vibrational energies 0f molecules. The more difficult problem 0f the three- dimensional oscillator was treated by the methods 0f classical mechanics in section 10 , while the simple one-dimensional case was discussed according t0 the 01d quantum theory ⅲ Section 6 佖 . The potential energy may be written, as before, ⅲ the form 2k2 レ 2 , ⅲ which ェ is the displacement 0f the particle 取ェ ) = 0f mass 襯 from its equilibrium position の = 0. lnsertion 0f this in the general wave equation for a one-dimensional system (Eq. 9 ー 8 ) gives the equation d2V 8 ド襯 十一万一 ( ルー 272 襯 2 ) = 0 , 2 or, introducing for convenience the quantities 入 and 伐 = 4 2 襯レ 0 / ん , d2V , 十いー。 ' ェ ' ル = 0. = 8 ル / ん 2 ( 11 ー 1 ) ( 11 ー 2 ) desire functions ( の which satisfy this equation throughout the region 0f values ー t0 十 for ェ , and which are acceptable wave functions, i. e., functions WhiCh are continuous, single— valued, and finite throughout the region. A straightforward method 0f solution which suggests itself is the use 0f a power- series expansion for 協 , the coemcients Of the successive powers of 工 being determined by substitution of the series for 協ⅲ the wave equation. There iS, however, a very useful procedure which we may make use of ⅲ this and succeeding problems, consisting 0f the determination 0f the form 0f ⅲ the regions 0f large positive or negative values 0f も and the subsequent dis- cussion, by the introduction 0f a factor ⅲ the form 0f a power series (which later reduces t0 a polynomial) , 0f the behavior 0f for は一 small. This procedure may be called the 0 れ 0 襯 襯ん od. 1 The first step is the asymptotic solution 0f the wave equation when はい s very large. For any value 0f the energy constant } 仏 , a value 0f は一 can be found such that for it and all larger values 1 A. SOMMERFELD, “ Wave Mechanics, ” p ・ 11.

VII-27cl OT 〃刃 R APP ん OX W. 4 ア霽刃 7 ' 〃 ODS 201 drawback is the necessity of a knowledge of contour integration, but the labor involved ⅲ obtaining -the energy levels is often considerably less than other methods require. 27C. NumericaI lntegration. —There exist well-developed meth- odsl for the numerical integration of total differential equations which can be applied quite rapidly by a practiced investigator. The problem is not quite so simple when it is desired to find characteristic values such as the energy levels of the wave equa- tion, but the method is still practicable. Hartree,2 whose method of treating complex atoms is dis- cussed ⅲ Chapter IX, utilizes the following procedure. For some assumed value of 幵 ' , the wave equation is integrated numerically, starting with a trial function which satisfies the boundary conditions at one end of the range of the independent variable and carrying the solution into the middle of the range Another solution is then computed for this same value of ' , starting with a function which satisfies the boundary conditions at the other end of the range of . For arbitrary values of W these two solutions will not ⅲ general join smoothly when they meet for some intermediate value of ェ . ー「 is then changed by a small amount and the process repeated. After several trials a value of ル is found such that the right-hand and left- hand solutions 」 oi Ⅱ together smoothly (). e. , with the same slope) , giving a single wave function satisfying all the boundary conditions. This method is a quantitative application of the qualitative ideas discussed in Section 9C. The process Of numerical integra- tion consists 0f starting with a given value and slope for at a point and then calculating the value 0f at a near-by point 召 d2V by the use 0f values of the slope and curvature at , the latter 2 being obtained from the wave equation. This procedure is useful only for total differential equations ⅲ one independent variable, but there are many problems involving several independent @ariables which can be separated into total 1 . P. ADAMS, “ Smithsonian MathematicaI FormuIae," Chap. X, The Smithsonian lnstitution, Washington, 1922 ; 刊 . T. WHITTAKER and G. ROBINSON, “ CalcuIus of Observations," Chap. XIV, Blackie and n. , Ltd. London, 1929. 2 D. R. HARTREE, proc. Ca 襯レ記 ge 2 厖 I. Soc. 24 , 105 ( 1928 ) ; Mem. 4 ー 0 ん r 尸ん Soc. 77 , 91 ( 1932 ー 1933 ). 4

V-20al THE AGU 刃ん刃君 0 ん Y Ⅳ OM ー A ln Appendix VI it is shown that 0 for ′ l, に l(z 泚に② = (l 十回 ) ! 2 ()1 十 1 ) (l 一回 ) ! Using this result, we obtain the constant necessary to normalize the part 0f the wave function which depends on The final form for O( の is 129 十 1 = I. ( 19 ー 11 ) for l' ()1 十 1 ) (l 一回 ) ! 戸祠 ( cos の . (l 十回 ) い 2 ProbIem 19 ー 1. Prove that the definition of the Legendre polynomials Po ② is equivalent tO that of Equation 19 ー 1. ProbIem 19 ー 2. Derive the following relations involving the associated I ・ egend re functions : ー 22 声 p 同一 1 ② p ② 十 1 ) 十 1 ) ( 十り ( I 十一司十 1 ) ()1 十 1 ) (l ーレ司 ) ( I ーー司十 1 ) ()1 十 1 ) ( 19 ー 12 ) 1 ( 22 ( 1g13 ) 1 ゴ② , ( 1g14 ) ー 22 ) p 同 + 1 ( 2 ) p ② , ( 1g15 ) and ( I + 同 ) (l ー同十 1 ) 司② . ~ 十 1 十 1 ) 十 1 ) 20. T LAGUERRE POLYNOMIALS AND ASSOCIATED LAGUERRE FUNCTIONS 20a. The Laguerre Polynomia1s. —The Laguerre polynomials of a variable p, within the limits 0 ( p ( , may be defined by means 0f the generating function Lr(p) 卍に② ( 1g16 ) r=O To find the differential equation satisfied by these polynomials Lr(p) , we follow the now familiar procedure of differentiating the ( 20 ー 1 )

204 THE ん / T ー 0 METIIOD ⅣⅡ -27e lower and therefore a better value of 刃 than the first set. This process may then be repeated until the best set 転 is obtained and the best value of E. This procedure may be modified by the use 0f unequal intervals, and it can be applied tO problems ⅱ 1 tWO or more dimensions, but the diffculty becomes much greater ⅲ the case Of two dimensions. ProbIem 27 ー 3. Using the method of difference equations with an interval 佖 = , obtain an upper limit t0 the lowest energy ル 0 and an approximation t0 0 for the harmonic oscillator, with wave equation 2 十 ( 入ーの ' ル = 0 (see Eq. 11 ー 1 ). 27e. An Approximate Second-order Perturbation Treatment. The equation for the second-order perturbation energy (Eq. 25 ー 3 ) IS with and 十珱 : , ル 0 ール 0 ( 27 ー 46 ) sum may rearranged in such a manner as t0 permit an approximate value t0 be easily found. ル 0 ル 0 ' 十一」 it becomes ル 2 ル 0 ル 2 ( ルー W?) Now we can replacel 〃 ' 〃気 by ( 〃 ' ' ) 。 On multiplying by ( 〃気 ) 2 , obtaining ″ ? い。 ly verified by 1 To prove this, we note that 丑′協 multiplication by 協 2 * and integration). Hence ( 2 厖 丑気〃気 differs from this 0 Ⅱ ly by the term with ~ = ん , ( 気 ) 2 ・ The sum

334 Ⅱ 420 T ″刃 STR UCTURE OF SIMPLE 盟 0 刃 CU. ん刃 S in which we have made use Of the relation 4e2 ど rAB(E2 and have multiplied through by ル′ , given by 2 AB 2 2 The quantity 4 ( 42 ー 17 ) is the energy Of the electron ⅲ the field 0f the two nuclei' the mutual energy 0f the two nuclei being added t0 this tO give the t0tal energy Ⅱを lt is seen that on replacing ( ミ , の by the product function 三 ( 9 Ⅱ (n 沖 ( の ( 42 ー 18 ) , の this equation is separablel int0 the three differential equations ( 42 ー 19 ) ー襯 2 の dp2 十入デ 2 = 0 , ( 42 ー 20 ) ー H 2 an d ⅲ which 2 2 2 襯 0 / ′ ん 2 = 0 , ( 42 ー 21 ) 十 ー 2 十 2D ミー 十 1 ( 42 ー 22 ) and ( 42 ー 23 ) 00 The range of the variable ミ is from 1 t0 , and 0f from ー 1 t0 constant are confocal ellipsoids Of revolu- 十 1. The surfaces ミ tion, with the nuclei at the foci, and the surfaces constant are confocal hyperboloids. The parameters 襯 , 入 , and must æssume characteristic values ⅲ order that the equations PO e acceptable solutions. The familiar equation possesses such The subsequent procedure solutions for 襯 = 0 , 士 1 , 士 2 , of solution consists ⅲ finding the relation which must exist 1 The equation is also separable for the case that the two nuclei have different charges.

X-36a] T 〃記 OT TION 0 20 ん 4T0 」一一 C 0 ん霽 CU. 刃 S 1 ö2V cos2 ö2 sin ー十 十 ötY sin2 ög2 sin2 C öx2 2 cos ö2V 8 ユル 1 sin 277 協 = 0. ( 36 ー 1 ) 十 sin2 öxöp The angles x and do not occur ⅲ this equation, although derivatives with respect to them do. They are therefore cyclic coordinates (Sec. 17 ) , and we know that they enter the wave function ⅲ the following manner : = 0(B)eiMpeiKx ( 3 2 ) ⅲ which 〕ー and K have the integral values 0 , 士 1 , 土 2 , Substitution of this expression in the wave equ ation c onfirms this, yielding as the equation ⅲ dO sin dtY Sin ーー cos2 十 十一 K2 sin2 C 8 24 COS KM ー ル O = 0. ( 3 3 ) Sin 2 M2 Sin2 We see that 穆 = 0 and = are singular points for this equation (Sec. 17 ). lt is convenient to eliminate the trigonometric functions by the change of variables ェ = 1 2 ( 1 0( の = ( の , at the same time introducing the abbreviation 8 ル —K2 cos ) , the result being dT { 十 K ( 2 ェー 1 ) } 2 ( 36 ー 4 ) ( 3 5 ) T = 0. ( 3 6 ) 4 ェ ( 1 ー The singular points, which are regular points, have now been shifted to the points 0 and 1 of も so that the indicial equation must be obtained at each of these points. Making the sub- stitution ( の = G ( の , we find by the procedure of Section 17 that s equals 121K ーし while the substitution 1 ーの一十 x ー

IX-32c] 7 ' 〃刃 M 刃″ OD 0 ア〃霽 S 刃ん - CO Ⅳ S / ST 刃Ⅳ 7 ' / 刃ん D 255 sions for the single-electron wave functions which fit these results fairly accurately. Such functions are 0f course easier t0 use than numerical data. The most serious drawback to Hartree's method is probably the neglect 0f interchange effects, i. e. , the use 0f a simple product- 40 50 20 0 0 C み々 e 〆 r “た 0 ・ 4 び 6 0 ・ 8 3 ・ 0 40 FIG. 32 ー 1. ーー The electron distribution function の for the normal rubidium atom, as calculated: l, by Hartree's method 0f the self-consistent field ; by the screening-constant method; and lll, bY the Thomas-Fermi statistical method. type wave function instead 0f a properly antisymmetric one. This error is partially eliminated by the procedure 0f Hartree and BIack described above, but, although in that way the energy corresponding tO a gwen set 0f functions ( ん ) is properly calcu- lated, the functions ( た ) themselves are not the best obtainable because of the lack of antisymmetry 0f Fockl has considered this question and has given equations which may be numerically solved by methods similar t0 Hartree's' but which include inter- change. SO far Ⅱ 0 applications have been made 0f these' but 2 several computations are in progress. Figures 32 ー 1 , from Hartree, shows the electron distribution function for Rb + calculated bY this method' together with those gwen by 0ther methods for comparison ・ 1 V. FOCK, Z. /. ん . 61 , 126 ( 1930 ). , See D. . HARTREE and ・ W. HARTREE' P 「 . . S . A 1 , 9 ( 1935 ).

XII-42cJ T 〃〃 D 刃 . 」耄 0 ん ECU. ん刃イ 0 Ⅳ 339 We now consider the set of equations 42 ー 28 for different values 0f ~ as a set 0f simultaneous linear homogeneous equations in the unknown quantities の . ln order that the set may possess a Ⅱ 0 Ⅱ - trivial solution, the determinant formed by the coefhcients of the cz's must vanish. This gwes a determinantal equation involving 入 and from which we determine the relation between them. are interested ⅲ the normal state of the system, with 襯 = 0 and I even. The determinantal equation for this case is 1 2 0 0 3 4 0 21 り 1 0 1 一 1 21 0 臨 0 0 0 ー 20 ー ( 42 ー 29 ) 39 77 0 The only non-vanishing terms are ⅲ the principal diagonal and the immediately adjacent diagonals. As a rough approximation ()o the first degree ⅲ入 ) we can neglect the adjacent diagonals ; the roots of the equation are then = 当 X , = 1 IX ー 6 , 3 ? イ 7 入ー 20 , etc. We are interested ⅲ the first of these. ln order to obtain it more accurately, we solve the equation agam, including the first two non-diagonal terms, and replacing ⅲ the second diagonal term by ! X. This equation, 1 2 3 15 ・ 1 ・ 1 1 り編 has the solution l//3X 十 2135X2 十 8505X3 , ⅲ which powers of 入 higher than the third are neglected. Hyller- aas carried the procedure one step farther, obtaining l//3X 十イ 35X2 十チく 505X3 ー 0.000013 入 4 ー 0.0000028X5. This equation expresses the functional dependency of 0 Ⅱ入 for the normal state, as determined by the equation. The next step is tO introduce this in the ミ equation, eliminating 円 and then to solve this equation tO Obtain the characteristic

X-36bl 7 ' 〃刃ん 07 Ⅵ TION 0 20 ん TO 」耄 c 」 10 カ刃 CU. も刃 281 procedure which has been used with success ⅲ the interpretation of the spectra of molecules of this type. Let us write the wave equation symbolically as 〃協 = ル協 . ( 36 ー 15 ) lnasmuch as the known solutions of the wave equation for a symmetrical-top molecule form a complete set of orthogonal functions (discussed ⅲ the preceding section), we can expand the wave function ⅲ terms of them, wnting ( 3 16 ) the secular equation lmmediately factors into equations corre- between functions with the same values of and 」寿 so that we find that the only integrals which are not zero are those a solution 0f the unsymmetrical-top wave equation (Sec. 27 の , equation corresponding to the use of the series of Equation 36 ー 16 inertia Äo, お 0 ( = ス 0 ) , and Co. Ⅱ we now set up the secular wave functions for a hypothetical molecule with moments of in which we use the symbol V}KM to represent the symmetrical-top sponding tO variation functions of the typel ( 3 17 ) On substituting this expression in the wave equation 36 ー 15 we obtain the equation ( 3 18 ) simultaneous homogeneous linear equations in the coeff. cients : ド and integration, this equation leads to the following set of values Of these two quantum numbers. On multiplication by the argument from now 0 Ⅱ being understood to refer to definite ⅲ which for simplicity we have omitted the subscripts and 」 , 十み ( 3 19 ) fixed axis ⅲ space (see Sec. 52 , Chap. XV). the total angular momentum of the system and its component along a 1 The same result follows from the observation that and M correspond ⅲ which ん has the value 1 for = K and 0 otherwise, and 〃ん

194 THE VÄ TION 既刃〃 OD ⅣⅡ -27a which corresponds to ordinary first-order perturbation theory, inasmuch as, if ″ can be written as 〃 0 十〃′ , with then ル。 = 〃… has the value ル。 = ル十工の″ 'F 鼠の (@) , which is identical with the result of ordinary first-order per- turbation theory 0f Section 23 when p(c)d.c = . Equation 27 ー 9 is more general than the corresponding equation of first- order perturbation theory, since the functions 鼠の need not correspond t0 any unperturbed systern. On the other hand, it may not be SO reliable, in case that a poor choice of functions 鼠の is made ; the first step of ordinary perturbation theory is essentially a procedure for finding suitable zeroth-order functions. lt may happen that some of the non-diagonal terms are large and others small ; in this case neglect of the small terms leads to an equation such as 〃 11 ール〃 12 ″ 22 ール 0 0 0 ″ 33 ール 0 0 0 0 〃 44 ール which can be factored into the equations 〃 11 ール 0 0 0 01 れ 0 -4 ( 27 ー 10 ) lt is seen that this treatment is analogous to the first-order perturbation treatment for degenerate states as given ⅲ Section 24. The more general treatment now under discussion is espe- cially valuable ⅲ case that the unperturbed levels are not exactly equal, that is, ⅲ case 0f approximate degeneracy. A second approximation to the solution 0f Equation 27 ー 7 can be made ⅲ the following manner. Suppose that we are interested ⅲ the second energy level, for which the value 22 is found for the energy as a first approximation. 尾 introduce this expression for ル everywhere except ⅲ the term ″ 22 ール