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Perturbation methods in quantum mechanics Pearson, Hans Lennart 1951-12-31

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he* Of 1 PERTURBATION METHODS IN QUANTUM MECHANICS by ' Hans Lennart Pearson A THESIS SUBMITTED IN PARTIAL FULFILMENT OF ' THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of MATHEMATICS We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF ARTS. Members "of the Department of Mathematics THE UNIVERSITY OF BRITISH COLUMBIA April, 1951 V ABSTRACT The solutions of the radial part of the Schrodinger equation for the hydrogen atom, which may be written (in atomic units) as f . L d_ r 2 d_ + i ( i+l) . 2l ^ ( r ) - E f ( r ) \ r 2 dr dr r 2 r J are well known in the standard case when the boundary conditions require that the wave function should vanish for infinite r . The eigenfunctions in this case are expressible in terms of Laguerre polynomials and the eigenvalues of the energy are E n = - i - ( n = l , 2 . . . ) n^ The problem of determining the eigenvalues when the boundary conditions require that should vanish for a finite r , say r 0 , is not as amenable to solution, and i t is only recently that several methods have been suggested for dealing'with this case. The method to be discussed here is due to Michels, de Boer, and B i j l . De Boer, considering the ground state alone, succeeds through the use of a perturbation method in finding the change in the eigen values for different r Q . In so doing, he makes an approximation, which a priori is not justified. In the present thesis, it is shown both qualitatively and quantitatively that the approximation is justified for the values of r 0 used. The logical extension of the method to states other than the ground state is made for two particular cases, and from the results of these two investiga tions, conclusions are drawn regarding the general applicability of de Boer's method. INTRODUCTION One of the most interesting facets of Quantum Mechanics to \ a mathematician is the set of methods developed therein for find ing approximate solutions to differential equations, the exact solutions of which cannot be found* These "perturbation" methods have had to be developed by the physicist since the number of pro blems for which the corresponding SchrSdinger equation is capable of exact solution is relatively small. The standard, problems in the last mentioned class lead to a study of the Legendre, Hermite, Laguerre, and Bessel functions — and as these names indicate, the required mathematics was developed long before the advent of' Quan tum. Mechanics. But for many other problems (for example, the pro blem of determining the interaction between an atom and a radia tion field) a new mathematical technique has to be developed. The usual approach for many of these perturbed eigenvalue problems is the following [3, pp. 149 et. seq.] The Schrodinger equation for the stationary state is = Wf where H - - V . It is assumed that H may be written as the sum of two parts for one of which, say HQ, the solution of the Schro'dinger equa tion i s known. Let the other part, H1 , be small enough to be regarded as a perturbation on H 0 . Let the eigenfunctio ns and eigenvalues to H Q be u n and E n respectively. That i s , H = H 0 + H ' and H Qu n = E n u n * T a n d w a r e t n e n expanded i n power series i n terms of a parameter X as follows: V " ¥ o + ^ 1 + ^ 2 + w = w + \WT + x2w0 + ... O 1 2 Substituting these values into the wave equation, ( H + X H ' H T J T + + . . . ) = • ( w o + xwx +...)( tyo + X)]/^ + . . where H T has been r eplaced by X H " , where X w i l l f i n a l l y be replaced by 1 . Equating c o e f f i c i e n t s of equal powers of X on each side of t h i s equation leads to a system of equations giving successively higher orders of the perturbation. H0fo - % Vo H o ¥ l + H T ¥ o - w o ¥ l + w l To H 0)jr 2 + H» Y l = W 0 ? 2 + W l ? l + ¥ 2 T o etc. From the f i r s t of these i i t follows that i s one of the u n's. Solving the second of t h i s series of equations with "\JfQ replaced by u m w i l l give the f i r s t order solution. The solu tions to higher order i n H " are found from the succeeding equa tions. The above perturbation method has proved s a t i s f a c t o r y es p e c i a l l y since the perturbing terms ( i . e . , the i n t e r a c t i o n terms) are, i n practice, very small. But there is another class of pro blems for which t h i s d i r e c t method does not wrk. In t h i s l a t t e r type of problem, the perturbation due to a confinement of the qua turn mechanical system i s to be found. The re s u l t i n g changes i n the eigenvalues would appear as s h i f t s i n the spectral l i n e s of atoms under pressure or i n a crystal and the resulting changes in the eigenfunctions might show up, for example, In the rate of radioactivity of an atom under pressure. Mathematically, these problems can be formulated in the following way: find the eigenfunctions and eigenvalues of H "\JT= E "\JT where AJf must vanish at the ends of an interval, the interval being smaller than the usual range of the indepen dent variable. For example, for the problem discussed in the body of the thesis, the solution of Laguerre's equation vhi ch vanishes at 0 and r Q , where r 0 < 0 0 , is required. It is possible that this latter perturbation problem can be reduced to one of the f i r s t type mentioned but so far this idea has not been completely worked out. Meanwhile, however, several other attempts have been made to solve bounded eigenvalue pro blems. The best known of these, and the most general, is the graphical method of Sommerfield [4], i t is limited in i t s ac curacy and moreover gives no information about the eigenfunotions. A second method, which w i l l be discussed below is due to Michels, de Boer, and B i j l [ 2 ] . A third method, due to Auluck and Kothari [ l ] , makes use of asymptotic series. 1 . THE GROUND STATE 1 . 1 . An Outline of the De Boer Method The Schrfidinger equation for a hydrogen atom, in terms of spherical polar coordinates, i s Y " p*p r\ ft AX - 4 r 2 ^ = M M. (l) 2 where n - £- s/n Q JL 4. ! — J L , smQ je sm3G <D<p2 \i is the reduced mass and H is the energy i n ergs'. By a separation of variables, the radial part of the wave equation for T|f(p) f i s obtained. This is the only equation that need by used here, since the new boundary condition of this problem can involve a change ohly i n the radial part of the wave function. Equation (2) may be written in dimensionless form by introducing two d i - mensionless variables r and E with r = ap, E = H/H0 where a = H * V ft2 ' 0 2(i This substitution leads to the equation The method of solution given by de Boer [2] w i l l now be outlined. To solve equation (3), Sommerfeld's polynomial method is used. Let (4) 2 and t h e n (3) g i v e s t h e e q u a t i o n f d2 2d M+t) ,2] r , _ n (5) f o r d e t e r m i n i n g t h e p o l y n o m i a l f ( r ) . W r i t i n g f ( r ) = ^ _ c s r l e a d s t o t h e r e c u r s i o n f o r m u l a c s (scs-i) - SU*')} = Cs„, [ 2 '2] (6) The s m a l l e s t v a l u e of s t h a t o c c u r s must t h e r e f o r e s a t i s f y s ( s - 1) = ^ ( i + l ) ; i . e . , e i t h e r s = ~l or s = 1 . From t h e f i r s t boundary c o n d i t i o n , t h a t p f ( r ) must be f i n i t e a t r = 0 , t h e v a l u e s = j£ + 1 must be chosen. I n t h e un bounded problem, the second boundary c o n d i t i o n i s t h a t the wave f u n c t i o n s h o u l d v a n i s h f o r i n f i n i t e r and t h i s i s s a t i s f i e d 1 i f t h e power s e r i e s t e r m i n a t e s . I t f o l l o w s f r o m (6) t h a t i f t h e s e r i e s i s t o t e r m i n a t e , t h e n t h e h i g h e s t power n o f r t h a t o c c u r s must s a t i s f y 2" - 2 = 0 a i . e . n = a (7) Thus t h e e i g e n v a l u e s o f E , by ( 4 ) , are E = - 1/n 2 ( 8 ) However, t h e second boundary c o n d i t i o n i n the problem c o n s i d e r e d here i s t h a t the wave f u n c t i o n s h a l l v a n i s h a t a f i n i t e r e q u a l s a y , t o r Q . T h i s c o n d i t i o n i s t h a t 00 f ( r Q ) = I E b s r 0 s = 0 (9) 0 £+1 S 0 De Boer now proceeds t o t r e a t the ground s t a t e {][ = 0, n = 1) o n l y , by t a k i n g f o r the u n p e r t u r b e d case a = 1 , E = -1 ; and f o r the p e r t u r b e d c a s e , l e t t i n g 3. a = i + P - 1 + 2 z3 Substituting these values into (6) gives b 5 5 (s-i) = b 5-1 5 - 2 -(3 wh ich, neglecting ji against 1 , leads to The wave function can then be written as with b g = -pb^ . Then from (9) the value of p can be obtained as I3 = 2bU (10) (11) (12) (13) where b^ i s independent of p A set of values of A E ^  2 p are given: »S / 3 * to3 A E in e-V. 5 a 0 5 3 .45 .0927 6 a 0 6 .727 .0196 7 a D 7 .13^3 .00375 8 a c S .0257 .00069 ft is the radius of the f i r s t Bohr orbit. This completes the resume of [ 2 ] . 1.2 Uniform Convergence of the Series Upon study of this paper, the question arises whether the approximation used in arriving at the expression (11) for the coefficients b_ i s valid. This approximation t a c i t l y assumes S that for s large, the contribution of the sth term i s so small 4. s-1 that the error introduced by replacing (1 + f$ ) by 1 is negligible. This assumption w i l l now be justified, f i r s t l y , by a qualitative approach which w i l l lead to the conclusion that for ^ "small enough" the approximation can indeed be made; secondly, by a more quantitative approach whereby an ac tual range for the true value of w i l l be obtained and an upper bound for the error introduced by the approximation cal culated. Let the exact series corresponding to the approximate series appearing in (12) be denoted by 2. p C where r 5 ~ si (*-,)! Then ^ s ^ - (s-2-/3)a-3-/g) «-pX-fi)( 2r \ Si 2r , O-lS) ( 2* \ 2 . ( 2 - / 8 X 1 - / 8 } f2r \3 1 To indicate the dependence of the series inside the bracket on p , c a l l i t S(y3) . The corresponding approximate series is q,n) _ j _ pf L _ i _ (2t)*+ -iLterf + o(O) - 2ll, r -t- -t M l 3, f If i t can now be shown that S(/3) converges uniformly with res pect to fi , then the qualitative conclusion w i l l follow, since Sip) being uniformly convergent means that | Sh(p) - SN (/?) | £ for a l l n > fixed N(e ), where N is independent of /3 . Thus N can be chosen so that "the remainder of the S({3) series is as small as is desired for a l l J3 , and may therefore be neglected i n the calcula tions. This leaves a f i n i t e number of terms in the S{j3) series, '5. each of which approaches the corresponding term i n S(0) as (3 -*• 0 . Therefore, f o r small enough, the sum; of the f i r s t N terms of S((3) w i l l be close enough to the sum of the f i r s t N terms of S(0) to j u s t i f y using S(0) i n the computations. S (yS) converges uniformly by the Weierstrass M t e s t , since i f 2r < M and 0 < ft. < 1, then M 1 ' (.S-2-/3)(s-1-it)~ (1-/3)1 2f \ y A / ~ ( S I ) S I The seri e s whose sth term i s -: : — i s a convergent series ( s - l ) s . of positive terms; hence S(y?) converges uniformly. This completes the q u a l i t a t i v e argument. 1.3 An Upper Bound f o r the Error due to de Boer's Approximation In order to obtain the range i n which the true value of j3 l i e s and an upper bound to the error involved i n using the approximation for the d i f f e r e n t values of r , consider f i r s t of a l l bounds for the expression which i s the exact expression f o r f ( r ) corresponding to de Boer's value obtainable from (11). Now r + l4n£r - r u s ) Note that the r i g h t hand side of (15) i s de Boer's f ( r ) . 6. Mso r 4-2: i . ( s - , ) » ( . v > s - ' 2 5 = 2 ' K ft , ^ ( < - T ^ r ) 0 - - - (l-/9)(-/3) „ s ^ ' + ^ S ! (S-i)(i+/3) S R ' f 2 S - C i - / i) ,-*0 - / » ) f - / « ) Combining (15) and (16): -/3 Z SU5-0 ^(S-a-/3)(S-3-/3)~-- Q-/3K-/3) P 2 From e r t = 1 + rt + + ... i t follows that * 2 . <=>yt -rt -I 1.16) (17) (18) and then _ . r * _ e j ^ 1 r u ^ L . It is easily verified that the '. integral is convergent at the f* - r u - I j r lower limit. Putting J —5 q U - | (t) , 7. and noting that = 2(1 - 2/3 + 2 ^ ' - - - ) > 2(1 -2/0 (17) may then be written -pfte)* Exact Series *-/3f(tf^) *-pF(2-H/3) <2°) Rewriting, - c f ^ U ^ 1 ^ •« E x a c t S e r . e s ^ j i ^ U - j u ( 2 1 ) O O Therefore, the maximum percentage error that can arise w i l l be 2 I r \r\j | e - r u - i J2-1Ii U ^ I ^ 7 7 7 - ^ X t O O Z ( 2 2 ) d u u 2 Thus, i f an upper bound can be found for t h i s expression, i t w i l l certainly also be an upper bound for the actual percentage error that occurs. To this end, i t w i l l be necessary to get ap proximations to the area under the curve eYK) - ru -1 ^ U* It w i l l be shown f i r s t that this curve is always concave upwards for u ^ 0 . .a. Av u 2 ( r e r u - r } - ( e K U - r u - i } 2 u du " U* u £y eru{ r a U 2 - M - r u +6} - 2 r u - 6 (23) du 2~ U 4 - bn +_6__ f n y n _ l * n -5n+6 and, for n ^ 4 , Jr? is always positive since the two roots of n 2 - 5n + 6 occur at n = 2 and n = 3 . There fore, the second derivative being always positive for u posi tive, y(u) w i l l be concave upwards. ) FIG. 1 9. An upper bound f o r (22) can now be obtained i n the following manner. The denominator of (22) which i s the area under y(u) from u = 0 to u = 2 may be decreased by taking instead the s h a d e d area shown i n Figure 1. By elementary analytic geometry, the equations of the l i n e s through SA, AB, and BT are 6 K 3 r 2 y = {e r (r -2 )4 - r +2} u + ero-r) - 2 r - 3 (25) y = ^ U + q; (26) respectively. The u coordinate of A , the point of i n t e r  section of (24) and (25) i s = e r ( 3 - r ) - 2 r - 3 - £ * ( 2 } The u coordinate of B , the point of in t e r s e c t i o n of (25) and (26) i s 2 r u„ = ( 2 r - 5 ) + H e r(3-r)-^r - 9 e* r(r-») - H e r ( f - 2 ) - 3 r - 7 ( 2^) Thus the shaded area of Figure 1 can be written J ^ ' u + f-'Jdu +£U8{[er(r-2)+r-h2]u + er ( 3 - r)-2r-3} du Y(e ' "Cr-»)+r + l u + e - 0 - 2 r ) - f r - 3 | < | u (29) The numerator of ( 2 2 ) , which i s the area under y(u) from u = 2 - 4/* to u = 2 may be increased by taking instead the area of the trapezoid whose verti c e s are R , T , ( 2 , 0 ) , and (2 - 4^3 , 0 ) . This area i s 2f 10. Thus the value of 4 ^ 4 x 100 °l w i l l be an upper bound for (29) the error. A table of these values is given. ro terror < 5 7.8 6 2.36 7 .47 8 .136 It may be concluded that for r Q larger than 6 , de Boer's method gives quite accurate results for the change in the eigen values,. Note that in expression ( 3 0 ) , which is used in calcula ting an upper bound for the error in {3 , ^ i t s e l f appears. That this fact does not seriously prejudice^ the calculations w i l l be brought out more clearly in the following discussion, wherein a range for the true value of ^ w i l l be found. 1.4- Determination of the Range for ^ Solving ^ (S-2-/3XS-3-/9) Q-/8)f-/9) s-, K S _ ( 3 D which i s the exact f(r) set equal to 0 for r = r G , for ^ would give the true value of {3 . That i s , one would, like t o solve . 2 P \ \ + p + J>'s i ts-,)! (i + p)"' 1 2 r°.)-l f3 ' ! (s-.)! C /?) Let the term inside the brackets in (32) be denoted by . (32) by ]> " 2 s" r P s S< ( s - i ) Now, what de Boer does is to replace ^ which is 7? >^ (see (15) ). Therefore, the value |3, • i l . •that he arrives at for /3 i s actually smaller than the true one. De Boer's w i l l be used as the lower limit of the range for To obtain a value of p larger than the true one, replace 5 by 2--iw^H^r (33) which i s < ^ (from (16) ) . But in solving for p from fi = r0 /5 , de Boer's has to be inserted in Z arid i t might be that although >^ (p) < ^ , 2 ( is not < ^ > , since |?ct is smaller than the true value. It w i l l next be shown that this d i f f i c u l t y can be avoided. / ^ Z ~ H I+/3 ^ £ l S ! ( s - .)! d+^) s-' ^ r ° J r , + « 2 g« r» 0 ^ ) 2 ' i s * Q - 4 ) Q - f ) a V 1 since 1 ~ > ' " A for s > 4 i f ft 14-/3 ' I +• f9o. ' I is small. where <T is the series involving fo in the brackets above. Now, from f3 ^ ~ ^ a (T = |^  , i t can be seen that the value of f3 obtained is actually too large, since (T is smaller than the corresponding exact series and f3 is the dominant term in the expansion of p ^ ~ ^ g . Thus for an upper limit to the range of fi , the smaller of the 2 values obtainable from the quadratic 12. Z 3 (frfp <r= rQ i . e . w i l l be used. Now r 2 t<r+r 0 ) and from (18), (19), and ( 2 0 ) . . = F(2-Uf:) > F(2-4/,j = / y * d u (35) But-.g^ -ru-i f e - w - r u - » , f e r v , - r u - > , f e " - y l du = I ijz dU - y so a lower l i m i t f o r e r u - r u - t j fa :— du U' can. be obtained by taking the lower bound found i n (29) for. i 2 e - r u - i , CP d u and subtracting from i t the upper bound of e K t ; - r u - t , [p du found i n ( 3 0 ) . Then the value of ^ used as ami upper l i m i t to the range of p i s calculated here by solving (34) with <f1 < cf i n place of <f , where <5~' i s { t } times the mentioned abovei. o The range of p for the d i f f e r e n t r„ 's i s tabulated below. 13. < True f3 < 5 3.45 xlO" 3 24.7 XlO" 3 6 .727 x lO"*3 1.27 x 10~ 3 7 1.38 x lO" 4 1.84 x 10~k 8 2.57 x 10~ 5 9.5 x lCf 5 In finding each bound, two main approximations have been made. In each case, the f i r s t approximation was to replace the exact series by a simpler one; i t would appear that this step cannot be avoided. The simpler series were then replaced by equivalent integrals, and the second approximations oc curred in the numerical evaluation of these integrals. To be sure the last approximations were not too crude a check was made; for example', when an upper limit was required, a lower limit was also calculated and the difference between the two limits was compared to the range ^ g s-Vo S -x 2L sics-i) Z. In each case the difference was only a small fraction of this range and thus the error introduced by the second approximation, over which there is some control, was correspondingly small. This completes the discussion of the ground state. It is natural to inquire further and see i f the method is applicable for states other than the ground state. First an s state with n different from 1 , and then a typical p state, w i l l be investigated. General conclusions concerning the applicability of the -method w i l l be drawn from the results of these investigations. 2. OTHER STATES 2.1 The s-State Consider, then, the s state with n = 2 . For the un- perturbed atom, a = 2 and E = - l / 2 ; therefore de Boer would take for the perturbed atom a = Z + (3 and Substituting these values into (6) gives for the recurrence r e l a t i o n i n t h i s case b s s ( s - i ) = b s - ,{ 2 "M-" 2 l = 2 b s -  S'2\y (36) which, neglecting ^ with respect to 2 , leads to b, = « b a = - i ' - ^ b s = s ! ( S f , ) ( s - 2 ) ( 3 7 ) ( S»3) The corresponding wave function i s For t h i s to vanish at r = r„, R must be (38) 2 'o (39) Two values of are given below r c Z9 5 10 .59 .072 It may be remarked immediately that the value .59 for r Q = 5 i s c e r t a i n l y not small compared to 2 ; and thus one would expect a large error to a r i s e from making the approxima t i o n that gives (37) . For r 0 = 10, however, the i s more reasonable. 2.2 An Upper Bound f o r the Error An approximation to the upper bound for the error involved i n t h i s case w i l l now be obtained i n a manner analagous to that used previously. The exact expression t h i s time for f (r) i s • r - i g + ^ r + (S + ft)*-> s i ($.,)! ' Now (40) < r - ( i + f ) r a + > ... C , R < A-2 S " ' s'. ( s - 0 ! This would be de Boer's f ( r ) . Also r-^ + t i r ' + f <s-3-P —<*-(*>P{«pr* >r-(a +^;r •+- ^  ( 2 + f 3 ) * - ' s! (s-i)Cs-2; . V 2 < r 2 s " 0-/3)'"* r 5 > r - ( ^ + t ) r + (2+/3>J"' S! (S-iKS-2) = r + r a + pj> s-(s r-o(s-2) { 2 T T ^ } (42) 1 6 . C o m b i n i n g (41) a n d ( 4 2 ) : ' f3^ S K S - O C S - 2 ) • ^ 2S-' ( S - 3 - / 3 ) - - - (l - / 3 ) ( - ^ ) ( - | - / 8 ) r S '5 S ! ( S ^ f Z K S - I ) ( S - 2 ) (43) F r o m t h e e x p o n e n t i a l s e r i e s , - t s-V s e r t - 4 t ' - r t - l t h e r e f o r e , s ! C s _ ( ) ( s - 2 ) = J d u j ~ ^ <h (44) S e t t i n g J duj ^ 3 dq = G tt) a n d n o t i n g t h a t 2 T £ = O-/»>(• + £)-' = (.-/?)(»-! + f - ) ' > \- if* s o G{2^jy) > G C i - I / J ) (43 ) m a y t h e n b e w r i t t e n [3 G 0-1/9) ^ /3 < 5 ( 2 ^ r ) <: E x a c t S e r i e s << pGo) R e w r i t i n g , /*jdu ^ dq ^ E x a c t Series S/sJ^J ^ ^ " " qq (46) T h u s , t h e m a x i m u m p e r c e n t a g e e r r o r t h a t c a n a r i s e w i l l b e i-is./» ° I l J u l ~ ? ^ - d o X 1 0 0 X (47) 17. To find an upper bound for this, which w i l l be an upper bound for the error due to de Boer's approximation, i t w i l l be necessary to obtain an upper bound for the numerator of (47) and a lower bound for the denominator; so here again, approximate values of the integrals appearing are required. These can be obtained in the following way. F I G 2. Consider f i r s t the finding of an upper bound for the numera tor of (47) . The ordinate at any point in the lower diagram is the area up to that point in the top'diagram. The area of the 1.8. 3 trapezoid formed by A, B, ( 1 , 0 ) , and (1 - 2 /*» °) which one would like to use for the upper bound, w i l l be increased by taking instead of the ordinate at A , the value obtained 3 by taking the area .of the trapezoid 0 , E, C, and (1 - g/3. °) in the top diagram, and instead of the ordinate at B , the sum of 2 such areas from the top diagram. This leads to the expression + - g - »J| { 4 d , for an upper bound for the numerator. Next, to get a lower bound for the denominator, one can obtain an ordinate smaller than that of B by taking the area under the lines 1 and 2 in the top diagram. This area i s I f i r h + -y } <Jq +J ([(r -J)e r+| J + ^  +-3]<j -( r - H ) e r - r 2 - 3 r - ^ ] d c , where q = e K + rg+3r+t + i* Letting this area equal R , and using the area under the lines 3 and 4 in the diagram, one obtains as a lower bound for the denominator the expression r U c r' O where u = _ e - f - r - 1 - R - e ' - f - r - i - f Evaluating (48) and (49) for r Q = 5 and r Q = 10 gives 19. ds upper bounds for (47) the values 110 percent, 50 percent respectively. These values are not very impressive, but then the approximations made in obtaining bounds for the integrals in (47) were very crude. It would require a great deal of com putation to make them more accurate, but at least, the way in which this can be done has been made clear by the above discussion. 2.3 The p State Finally, consider the application of de Boer's method to a p state, for example the state with n = .2 . Here de Boer would take for the perturbed atom a = 2 + p and E ~ -1/4 + l/4 Z3 , just as in the previous case. From (6), the recurrence relation this time is bs(scs-i) -2} _ ? . S - 3 - / 9 - c- Ds-i 2 + /3 (50) which, neglecting with respect to 2 leads to b 2 = l S ' [ S(S - / ) - 2 ] [ ( S-i ) ( 5 - 2 ) - 2 ] - - -[3(a)-2j -p (S-3)\ (51) The corresponding wave function is w i +h bs=-/3bs The value of p is obtained as before, by demanding that this vanish for r = r Q . 20. The values for r Q = 5 and 10 are given. ro f3 5 10 .39 .041 As in the previous case, the value of p for r g = 5 is not small compared to 2 ; and therefore, de Boer's approxima tion is again relatively crude. The exact f(r) in th is case is 2 ^ g '"*(s-3-f >(s-»i-/» - - - <-/s) rs Y + 2i (?+/9)«-2{ 5 ( s . l ) - ? j { ^ . l ) ( s _ g ) _ g } _ - { 3 r ? ? - ? } ( 5 4 ) It is not as straightforward a matter this time to obtain good bounds for (54) as in the previous cases, but a set of bounds very similar to the ones found in (43) can be obtained by doing the following. f , 2 „ ( S - 3 X S - H) f-/3) KS \5H) > r + Z 2*-»[s($ - . ) - 2 } { < S - 0 ( 5 - 2 ) - 2 } - {3(?)-2) (55) The right hand side of (55) would be de Boer's f ( r ) . But (55) in turn is equal to — 2 ( S - 3 ) ? ( - /3) r s  which is >r 2 + f />(s-2) = r 2 - / 3 s • (s - i X s -e) (56) _ 2 1 . A I S O ( 5 4 ) 4 r * + 5 (p + yj)*"* ( S - 3-/3X5-*/-/3)-- ( - / s ) ^ S! ( s - D ! = Y*- si ( s r-,)(S-2) [ P 2+^ ] s-? ^ r 2 - / ? 5 s i ( s r - o ( s- 2 ) fH^H (57) Combining ( 5 5 ) , ( 5 6 ) and ( 5 7 ) , -/3 ^  S ! (S-i){S-2) ^ J* Boer Ser.es ^ £ x acf i er .es - f ? ^  S H*-»)(W)C 2^] ( 5 8 ) and using G(t) as defined before, this leads to - (i G (2 ) < Exact Series < - (3 G(l - ) ( 5 9 ) Rewriting -,r.ur^ -?-r'-' < E x a c t S e r i e s ^ Y J o J u | - ^ ^ ( 6 0 ) Thus the maximum percentage error that can arise w i l l be The numerator of ( 6 1 ) is at least half as large as the de nominator, so not very good results w i l l be obtained by using 22. "this, and for this reason no values have been calculated. The. reason that this expression for the maximum percentage error i s less satisfactory than those found previously, i s that, in this case, the de Boer series does not lead immediately to an integral. In fact, it is of such an awkward form that a further approximation has to be made before a series express ible as an integral can be obtained. However, (56) is not the best possible next approximation to the de Boer series. Only computational hazards l i e in the way of getting a better app roximation to the maximum percentage error with which to re place (61). 23. CONCLUSION In conclusion, i t may be said that de Boer's method leads to reasonable values for the change in the eigenvalues for quite small r when the ground state is considered. But the above statement is not true for other states. For it was seen in the last two cases investigated that for r = 5 , the value of p found did not justify using de Boer's method for a f i r s t approxi mation to the change i n the energy. But when a larger value of r was taken (here, r = 1 0 ) a value of p was obtained that ac tually was small compared to 2 . It may be inferred, from the discussion of the last two cases, that, in general, increasing either i or n w i l l lead to increasing the lower bound to the range of values of r for which reasonable results are obtained. BIBLIOGRAPHY Auluck, F . C . and Kothari, D.S., Quantum Mechanics of  a Linear Harmonic O s c i l l a t o r . Proceedings of the Cambridge Philosophical Society, 41, 175 (1946). Michels, A., de Boer, J. B i j l , A., Uniform Compression  of a Hydrogen Atom. Physica 4, 991 (1938). S c h i f f , L.I., Quantum Mechanics. McGraw-Hill, 1949. Sommerfeld, A. and Hartmann, H., The Bounded Rigid  Rotator. Annalen der Physik, 37, 333 (1940). 

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