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Boundary perturbations of some eigenvalue problems Julius, Robert Stanely 1956

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BOUNDARY PERTURBATIONS OF SOME EIGENVALUE PROBLEMS by ROBERT STANLEY JULIUS A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the DEPARTMENT of MATHEMATICS We accept t h i s t h e s i s as conforming to the standard required from candidates f o r the degree of MASTER OF ARTS. Members of the DEPARTMENT OF MATHEMATICS. THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1956. ABSTRACT In treating so-called "bounded quantum mechanical problems" one i s generally l e d to d i f f i c u l t eigenvalue problems. For example, the problem of finding the wave equation f o r a hydrogen atom confined to a sphere of f i n i t e radius leads one to f i n d i n g solutions of Laguerre's equation which vanish f o r given f i n i t e values of the independent v a r i a b l e . In t h i s thesis we discuss three such problems i n some d e t a i l . Since 1937 a number of papers on these problems have appeared; we describe here the p r i n c i p a l methods of these papers, and improve one of them. We then develop a new method which i s s u f f i c i e n t l y general to t r e a t each problem, and which can be established rigorously i n each case. This method provides us with asymptotic expressions f o r the perturbed eigenvalues. ACKNOWLEDGEMENT The author wishes to thank Dr. T.E. Hull f o r h i s invaluable help i n the preparation of t h i s t h e s i s . TABLE OF CONTENTS I . INTRODUCTION I I I . HISTORICAL SURVEY 3 1. General Methods 4 2. The Harmonic O s c i l l a t o r 11 3. The R i g i d Rotator 18 4. The Hydrogen Atom 24 I I I . FURTHER RESULTS 32 1. General Procedure 32 2. Spec ia l Cases 43 i The Harmonic O s c i l l a t o r 43 i i The R i g i d Rotator 44 i i i The Hydrogen Atom 45 BIBLIOGRAPHY 47 I. INTRODUCTION Since 1937 a number of p h y s i c i s t s have been interested i n the solutions to bounded quantum mechanical problems such as the l i n e a r harmonic o s c i l l a t o r , the r i g i d rotator and the hydrogen atom. Even though such s i t u a t i o n s do not ar i s e i n r e a l i t y , enclosing a hydrogen atom i n a sphere of f i n i t e radius might be expected to give, f o r example, an approximation to the eff e c t of pressure on spectral l i n e s . A r t i f i c i a l l y r e s t r i c t i n g the quantum mechanical system does not a l t e r the d i f f e r e n t i a l equation to be solved, but affe c t s only the boundary conditions. To i l l u s t r a t e , l e t us consider a p a r t i c u l a r case, the l i n e a r harmonic o s c i l l a t o r . The d i f f e r e n t i a l equation to be solved i s where we use the notation adopted by Auluck and Kothari ( 1 ) . The f i r s t few eigenfunctions and eigenvalues of the unbounded problem are "a 8 * " X , A = I U*=-4* X\x - | ) ( X + ») , X r 2 u , = i * * x x U " J3)(* + J3), 3 - 2 -It i s c l e a r that these "unperturbed" solutions can also be used as perturbed solutions i n some s p e c i a l circumstances. For example, U 2 given above, which i s the second excited state i n the unbounded problem, can also be interpreted as the ground state when the walls of the enclosure are at x 0 = + 1 ; i t follows then, that f o r t h i s p a r t i c u l a r enclosure, X =» 2 i s the lowest eigenvalue. S i m i l a r l y , can be thought of as the ground state when the walls of the enclosure are at x 0 = + J3 - Jo" , or as the second excited state when they are at x Q = + J3 + J5 then i n the former case \ = 4 i s the smallest eigenvalue, and i n the l a t t e r i t i s the second eigenvalue. We are l e d , therefore, to consider the following graph. - 3 -Other points can be obtained as we did the four shown; s t i l l others can.be obtained by i n t e r p o l a t i o n methods, or by using tables g i v i n g zeros of other solutions of Hermite's equation. While some of the papers on the subject are concerned with the r e s u l t s of these numerical c a l c u l a t i o n s , most of the papers are devoted to f i n d i n g the asymptotic behavior of X near the ends of the curves. Langer, Cherry, E r d e l y i and others have developed a theory concerning the asymptotic behavior, f o r large values of a parameter, of the solutions to i n i t i a l value problems. Some of the papers which we w i l l discuss, and t h i s t h e s i s , constitute an attempt to i n i t i a t e an analogous theory f o r boundary value problems. The main purpose of t h i s t h e s i s i s to develope a single procedure which can be applied to each of several classes of problems. I t s a p p l i c a t i o n to each class can be rigor o u s l y j u s t i f i e d i n each case, and the problems mentioned above appear as special cases. Before doing t h i s we w i l l describe the more important r e s u l t s obtained to date, and indicate how one of them can be improved. I I . HISTORICAL SURVEY The r e s u l t s we are going to discuss could be c l a s s i f i e d either according to the method used, or according t o the problem f o r which they were developed. We have a r b i t r a r i l y - 4 -chosen the l a t t e r c l a s s i f i c a t i o n . 1. General Methods. We begin by discussing a method due to P r o e l i c h (Zt); i t i s the only one which was developed independently of any p a r t i c u l a r a p p l i c a t i o n . We also include i n t h i s s e c t i o n a br i e f description of two numerical methods t y p i c a l of those which could be applied to the problems being considered. In 1938 Proelich (£) developed the method outlined below. Consider the two equations 1) u" * I gu) + X)u * 0 2) y" + (goo + X + A \ ) y s 0 , and the two sets of boundary conditions u(0) = 0 , y(0) = 0 or u'(0) = 0 , y'(0) = 0 and u(b) = 0 , y(b - € ) - 0 . Multiplying equation (1) by ( - y(x)) , equation (2) by u(x) , and adding, we get j x |y'(x)uoo- u'umxjj. + A * uu)y(x) = 0. Integrating t h i s from x = 0 to x = b - e and solving f o r £ X we obtain ( 3 ) a. -y;<)Y'(b-° • J UU)Y("fc)clt 0 Equation (3) i s an exact expression f o r , but, of course, we do not know y(t) . I f , however, € i s "small", then i t i s reasonable to expect that y(t) 2z u ( t ) ; then i f b i s an ordinary point of the d i f f e r e n t i a l equation, we w i l l have y* (b - 6 ) u' (b - € ). I f t h i s i s the case then - u(b-e)u'(b-e) - — X I • J u * ( t ) d t e It follows, then, that t\ ^ € U'(b)) x as e - 0 . J u 4(t)dt o (By f (x) ^ g(x) as x X Q - we s h a l l always mean l i m ££*) - 1 .) x*x 0 S ( X > The f i r s t numerical method which we w i l l discuss was developed primarily by Rauscher (£) i n 1949• The method involves matrices, and was o r i g i n a l l y designed f o r use with aeronautical v i b r a t i o n problems. In these s i t u a t i o n s , the matrix elements are determined experimentally. The method can, however, be adapted to eigenvalue problems of the type under consideration. Convert the system M y" + 9<*>y = -Ay , Ay(o)+ By'(o) = y(b) = 0} into (5) y<x> = - A j G(x,t) y i t l d t t - 6 -where G(x, t ) . i s the Green's Function associated with (6) yM + tjtxiy s o , Ay(o) • 6y'u)* ytb) = o. Choose n points 0 < x - L < X 2 < « . « < x n < b . Let us now approximate to y(x) i n the following manner: (7) Y ( x ) = XI Si^^j , J = » , V w K « r « y . 5 y U j ) . i~\ The g-;(x) are the polynomials of lowest possible degree which s a t i s f y i ) the boundary conditions of equation (4), i i ) g j ( x k ) = 5 j k . I f we now put X - - ^  , equation (5) becomes n b r\ (8) YL { U(V>3i(t><tt]/j = uLv'yj . L e t t i n g x • x^ j. « 1, 2, ..., n, (7) and (8) y i e l d the matrix equation A Y = oo Y b where A » (A<j) , A ^ = [ G(x t |t) o^UJdt , and Y~ ^  V* I f we now substitute Yv. into (7) and take X i , «= - — ,K=1,2,.., n, we get approximations to the f i r s t n eigenvalues and corresponding eigenfunctions of the o r i g i n a l equation (4). In general, a considerable amount of work i s entailed i n obtaining the above approximations. G (x, t ) i s of the form + U . ( X \ U x l i ) - U x ( X ) U , ( t ) 2 W where u-^  and U2 are l i n e a r l y independent solutions of equation (6). We take the ( + ) when 0 < x < t and the (-) when t < x < b . The c o e f f i c i e n t s a^, a 2 , p t t must be determined so that G(x, t ) s a t i s f i e s the boundary conditions of the problem. G (XJL, t ) w i l l have to be tabulated f o r 0 < t < h , i = 1, 2, ... n and the matrix elements J ^ ( X L . t ^ l t j d t 0 evaluated numerically. It i s only aft e r the above has been completed that one can begin solving the matrix equation by any of several straightforward methods. The following i s perhaps a more e f f i c i e n t numerical method of solving the type of equation with which we are dealing. The method was developed i n 1950 by Milne (7). - 8 -We are given U" • (31x1 + \\ u = 0 , u(o) = u l b ) * 0 , which we replace by o + 9<„v-- 0 . Divide the i n t e r v a l (0, b) into (n + 1) subintervals of length h = ~- ( . Let V ( i h , jh) = Vi;j- , and l e t (9) V ( x , 0 c Ku KU )cosJH t . K The i n i t i a l conditions may be chosen a r b i t r a r i l y , and we w i l l l a t e r choose them i n a convenient fashion. Mow make the following approximations: Our p a r t i a l d i f f e r e n t i a l equation now becomes d o ) viji+, • Vt|i., = • Vc,^ + ^ 9 - V C J where g^ = g(ih) • The boundary conditions are now (11) V«j = V n + I > i = 0 . We now choose the i n i t i a l conditions to be such that i n the difference equation approximation they become - 9 -(12) »/,eS 0, 1*1 and V l l = . = 1 , 1 . 1 Using equations (10j^  (11) and (12), we can now generate the matrix V. . i = 0, 1, n+1, j = 0, 1, ... n . The submatrix 1±j, i = 1, 2, ... n , j = 0, 1, ... (n -1 ) , i s triangular and non-singular. We are able, then, to solve f o r the n constants A k, k - 0, 1, 2, ... (n-1) fromthe set of equations (13) L A i v i i ? 0 i--o where A n i s defined to be unity. The s o l u t i o n f o r the A k i s e a s i l y achieved since the co-e f f i c i e n t matrix i s t r i a n g u l a r . We now re-write equation (9) as (14) V i j ' X^UKicos/Jkjh . K: I Multiplying both sides of (14) by Aj and summing from j = 0 to j = n we obtain, with the aid of equation (13) (15) Lc»UHij^A J co*jr i l j k} = 0 , and i t can be shown that (15) implies that w J^AjCOsJTKjK r 0 . j = 0 - 1 0 -That i s , where u^, k = 1, 2, . . . n are the n roots of the c h a r a c t e r i s t i c equation cosnyu + A„_,cos(n-i)/« A,cos/« • A 0 = o . Having found the eigenvalues, the eigenfunctions are now easily-determined i n the following manner. (We now omit the subscript K from and u.^  .) Into the difference equation (10) put 8 COS j /A Since equation (10) becomes { 1 6 ) u i + l , U c o s ^ - h ^ U i - Ui., . The boundary conditions imply that u Q = o , we choose any non-zero Uj_, say U]_ « 1 , and solve f o r the one a f t e r the other. The f a c t that ^ n+± should turn out to be zero serves as a check on the calculations* Equation (16), with u. = u.^ , k = 1, 2, • •• n , together with the boundary conditions provides the f i r s t n eigenfunctions. - 11 -2. The Harmonic O s c i l l a t o r . In t h i s sec t ion we w i l l summarize the r e s u l t s of two papers deal ing with the l i n e a r harmonic o s c i l l a t o r . Both papers begin by considering the graphica l method out l ined i n the i n t r o d u c t i o n . The e a r l i e r paper, w r i t t e n by Chandrasekhar(2) i n 1943 employs /a method which i s e s s e n t i a l l y that invented i n 1937 by Miche l s , de Boer and B i j l ( 6 ) . We w i l l develop the method here, f o r the Harmonic o s c i l l a t o r , and also develop a modif icat ion to the technique, which cons iderably extends i t s u t i l i t y . The equation to be solved i s (17) U"4 { l i - i x * ) + X } u = 0 where , E - ( * + i ) t » w and The mass of the o s c i l l a t o r i s m, {ct/1 i s the p o t e n t i a l energy due to a displacement * , and u(x-j X) i s the wave funct ion of an o s c i l l a t o r with energy E . The general so lu t ion of equation (17) i s (18) We see that i f the "natural" condit ions that u(x , \) 0 as x + oo are imposed, then we .must take X equal to an odd p o s i t i v e integer and B = 0 , or X equal to an even - 12 -pos i t i v e integer or zero, and A = 0 . For the sake of c l a r i t y we w i l l consider only the odd solutions to the problem, and l a t e r state the r e s u l t s for the even ones. I f we i n s i s t that u(x Q, X) = 0 , where x 0 i s large, we expect that X w i l l be close to X 1 where X T = 2K - 1, K any p o s i t i v e integer. Let us put X = X 1 + A X , and make the approximation M - X — M - X T unless M • X f , i n which case M - X = - A X , substituting t h i s into equation (18) (with B = 0), we obtain f o r K = 1 (19) u ( x A ) s A * ^ { x - a | j ^ i x - } , and f o r K > 1 « » , > ) * A 4 - « * { » • f ' ^ - ' - ^ - v ) •(••V) „ » . . - 4 M f f i * - ' - « ) & " * E^ ,- v n;^;>.'-' , , '"- v ,x"4 Then, since. u(x Q, X) = u(xo, X* + A X) = o we can'solve equations (19) and (20) f o r A X ; t h i s gives ( 2 1 ) ZL ... Q * W A X ) T 1 tn-Ql vir> and - 13 -(22) ( A » V + * X s , K >l). I f X - 3 + A X , that i s , when k = 2, equation (22) reduces to (23) Un*i)! U*3t**) , Expressions (21) and (22), and the spe c i a l case (23) are v a l i d only f o r large X Q ; unfortunately, t h i s i s just when the series involved converge most slowly. Even though equation (22) implies that *X = 0 at the zeros of u(x, X 1 ) , the formula i s not v a l i d f o r x Q near these zeros. This i s ea s i l y seen. For example, equation (23) implies that aX i s p o s i t i v e or negative, according as x 0 i s greater than or l e s s than J3 . This, of course, i s inco r r e c t . The reason that equation (22) i s not v a l i d f o r x 0 near these zeros of u(x, X T), i s that the expression - 14 -(24) X • V ) » n - l - V » " - » - V » j r » " * ' «»• <*"+•).» which occurs i n equation (20) i s very sensitive to small changes i n \* i f x i s near a zero of u(x, \ f)» Let us, then,modify the method of Michels, de Boer and B i j l , bearing t h i s f a c t i n mind. In (24), l e t us replace \* by \* + A \ (which i s the exact value of \) and then expand i n powers of A \ , keeping only the f i r s t two terms of the polynomial. From (17) we then obtain where "•• 'x-i 1 and Hence, putting x = x Q and solving for & \ we obtain U = A'* AX = 2K-1 1-6X , K > » ) . - 15 -In the special case that X 1 * 3 ( i . e . k=2) we now f i n d that X.lTf - 1 ) ' U * 3 * A A ) . I t i s e a s i l y seen from the above expression that A X = 0 when x 0 = J3 , and that A X i s p o s i t i v e or negative according as x Q i s l e s s than or greater than J J ( i f x 0 i s close enough to J3 )• The corresponding r e s u l t s f o r the even s o l u t i o n to the problem are as follows. I f X - X» + AX where X » - 2k, k - 0, 1, 2, then the method of Michels, de Boer and B i j l gives and A>c ii^fl {\, V * A A , K > l ) , F(x.) ' where and - 16 -Xn.V '\UK4l>! L (2n)» y* nsic*t Our modification to this method gives F(x.) • G(Xo) J ' where K GU)= X^j, 2(2n-i-V)(ln - * - y)-(an-at - v)...(-V)x w. tvsi -fc=l The second paper that we shall summarize was written in 1945 by Auluck and Kothari(l). The authors are able to obtain approximate expressions for En in the two limiting cases where xQ < < Jn+ ± and x 0 > > J n+1 . (i) X C << Jn+T~ Under these circumstances the author develops two distinct approaches. The first of these is as follows, a) It can be shown that where I ** T 4) ) 4} and where - 17 -I t can also be shown that, f o r large K M K m i\xx) = 2"* Tf"'* ri2m+1) K"m"5 cos (JaK x - mn- OWK)). Putting K = i X + "5. and m = + , we have, upon i n s i s t i n g that the wave functions vanish at x = + x Q , b) A second method of approximation i n the case that x Q < < Jn~+~I~ i s obtained i n the following manner. Replace x by x - x Q i n the d i f f e r e n t i a l equation; we obtain I n s i s t now, that u(0) * u(2x Q) = 0 , and look f o r a s o l u t i o n of the form where The authors shew that A j ^ i s n e g l i g i b l e unless k = n+1 , and that - 18 -( i i ) X 0 » / n T T The authors show that under these conditions V » * V * , « ( M t t t d ^ a l t . . . ) t i l , ^ and l , T i ^ o i * i i l # . . . ) c o i a 2 l Using the above expressions we f i n d that 3. The Rigid Rotator In t h i s section we apply the method of Michels, de Boer and B i j l to the r i g i d rotator; we also treat the - 19 -problem with our modification to the technique. Before doing t h i s , we include f i r s t those graphs of X v.s. x 0 arrived at by Sommerfeld and Hartmann(ll) i n 1940. The fu n c t i o n u(x, X) s a t i s f i e s the equation ( 2 5 ) «" •(&.•,}=&)«•• where m i s a po s i t i v e integer, and the changes of variable x = cos 9 and u(x, X) = s i n 9 • ^ G1 (9, X) have been made. *@* i s the wave function of the problem. It i s e a s i l y shown that the general s o l u t i o n of equation (25) i s given by U(X,X>= A( I + f XHMzrt.f X)-" (M,- » vxn) + B ( X 4. f CMxr.-'XXMan.r X ) " (<VX ) »M-»\\ (2r»+»)l ' J where = (m + k) (m + k - 1 ) . I t can also be shown that i f we i n s i s t that u(+ 1, X) = 0 , then we must put \ - % , where A or B i s zero according and k i s even or odd. By using the method shown i n the introduction, Sommerfeld and Hartmann were able to draw graphs s i m i l a r to those below, of X v.s. x 0 . X - 20 -For m 2 the X v.s. x Q graphs are s i m i l a r to the one f o r m = 1 , the only important difference being that 4 X I — r I , = 0 i f m ^ 2 ; i t i s only f o r m - 0 that c) X —-=— -*._«> a s x. -» i . It i s t h i s f a c t that Sommerfeld 6 *o .. • ° .. and Hartmann investigated i n d e t a i l i n (11). With the aid of the asymptotic expressions which we s h a l l derive f o r A X , t h e i r r e s u l t s w i l l appear as special cases i n part I I I . We w i l l now f i n d the Michels, de Boer, B i j l - t y p e approximations to A X ; l e t us derive them here f o r the even solutions, and l a t e r state the r e s u l t s f o r the odd ones. The procedure i s as follows. Put X = X* + a X , where X' • , and k * 1, 2, ..., (B - 0). As before, we make the approxai mat ions ^ 2 L _ i - X — ^21,-1 " ^ T u n l e s s L = R , i n which case - X = - a X . I f B = 1 , that i s , i f X» = (m+l)m we f i n d that and i f k > 1 we obtain (27) U U . X l ^ : A l l - x ' J ^ B O r t - A * C U ) ) where BU) - 11 and - 22 -since u(x 0, \) - 0 , solving (26) and (27) f o r A X we obtain (28) ! * i " + J (M1)t.t-yKMt,.rV)"-cMB-v) x*« 2 (in) I f o r k = 1 , and f o r k > 1 we obtain ( 2 9 ) sia c c Xo) Equations (28) and (29) are v a l i d f o r . x Q near unity. As we showed i n section 1, the method can be modified to provide an expression f o r A X which i s v a l i d near those zeros of u(x, \ T) which l i e between zero and unity. The r e s u l t of t h i s modification gives (30) A X * ) K>l, where I f m = 0 and k = 1 , that i s , \» = 0 , (28) reduces to & \ ± — L f - 2 3 -Let us now compare (29) and (30) when m • 0 and k • 2 , ( i . e . X' = 3 x 2 = 6). Equation (29) becomes (3D -61 *a > V ((2i>-i>un-t)-fcH(»-iX»-'H-fr)-(?»»-fe)xi«) whereas (3®) becomes (32) ^ , - J X : ' It i s e a s i l y seen that equations (31) and (32) are asymptotically equal as x Q -> 1 , but that (32) i s v a l i d f o r xQ near ~ -while (31) i s not. The corresponding r e s u l t s f o r the odd solutions to the problem are as follows. I f X = X T + A X , X T = , K = 1, 2, then i f k = 1 AX*= *a X? + 7 < X'H M>n.j V) • • ( M» - V) ^ »»+, 3' & u ^ T o i — * ' - 24 -For k > 1 , the approximation due to the method of Michels, de Boer and B i j l i s e(x.) F(x.) whereas our modification to the method gives E, F and G are defined below: 4 la •»•»)! * t>«0 4. The Hydrogen Atom. In 1937 Michels, de Boer and B i j l (6) developed the method which, i n sections (2) and (3) we have seen applied to the harmonic o s c i l l a t o r and r i g i d rotator r e s p e c t i v e l y . In t h i s section we outline the relevant portion of t h e i r paper (they also discuss a p h y s i c a l a p p l i c a t i o n of the s i t u a t i o n , - 25 -and do some numerical calculations) and give the r e s u l t s of our modification to t h e i r method. We w i l l then discuss a paper written i n 1946 by de Groot and ten Seldam(j>) and one written i n 1953 by Dingle (2.). We sta r t with the equation u'» • (I - £i!*L> +X)u* o where u(x) i s the wave fu n c t i o n density, I t i s e a s i l y shown that the solution which i s f i n i t e at the o r i g i n i s where X = - -—g » M- > 0 • Since u(x, X) w i l l diverge as x-*• 0 0 , unless the series terminates, the condition that l i m u(x, X) » 0 r e s t r i c t s ^ x-*» X to the set - ~ 2 > u - a p + k , k * l , 2, ... . I f the boundary conditions are u(0, X) = u ( x 0 , X) = 0 , the method of Michels, de Boer and B i j l gives - 26 -which i s the r e s u l t stated i n (6). In general we obtain A / * U = — .y^***') 2. n i » p t » * i ) i For K > 1 we obtain where and For p = 0 and K = 2 , that i s u-f - 2 , t h i s reduces to A > ^ ^ , ( > . ) which i s v a l i d f o r large x • - 27 -Our modification of the method gives where K-> j n ^ ^ ^ ^ For p = 0 and K = 2 the above reduces to A/* ta= 7 " 1 , ( X = ) , rt»3 which i s v a l i d f o r large x 0 and f o r x 0 near 2 . The second paper on the hydrogen atom which we summarize i s by de Groot and t e n Seldam (j>). The authors discuss the method of Michels, de Boer and B i j l , and reproduce t h e i r r e s u l t s except f o r a change i n notation. They also f i n d Michels, de Boer and B i j l - t y p e approximations to & \ when p = 0 and jj,f » 2 , and when p = 1 and H T = 2 (see the previous part of t h i s section). The authors note that the series-involved i n the Michels, de Boer and B i j l - t y p e approximations converge most slowly when they are most accurate; i t i s f o r "this reason that they f i n d an i n t e g r a l representation f o r these s e r i e s . - 28 -The authors show that sums of the form 00 Ox) '- Z can be expressed i n terms of the f u n c t i o n x A straightforward recurrence r e l a t i o n i s then developed connecting F i m + 1 (x) and F i m ( x ) , where Fi,(x) = E i l x ) - E i 10 E i ( x ) i s given by and has been previously tabulated. The authors then consider t he following cases. a) E < 0 , r 0 large b) E < 0 , r Q small c) E ± 0 d) E > 0 , E not large e) E -> + oo The l a s t portion of the paper i s devoted to the physical importance of the problem. In t h i s connection the introduction to (10) may also be of i n t e r e s t . - ( x - 29 -I t w i l l be seen i n part III that the i n t e g r a l s F i m ( x ) obtained by de Groot and t e n Seldam are very s i m i l a r to those which we w i l l obtain i n a more natural and d i r e c t fashion. ' The t h i r d paper, written by Dingle (£) i n 1953 contains possibly the most complete, from a mathematical point of view, treatment of the hydrogen atom problem. The author s t a r t s with the r a d i a l equation (33) £ (TR) + 1 ^ + iiiL£ - * 1 £ ± : M M 0. and makes the following changes of variable These reduce (33) to the form (34) 5ll ( t « + l - i t t t . C l £ t 2 ) ) ^ * ) s 0 . ? % The author then provides the following forms of the s o l u t i o n to (34): a) a convergent series expansion which i s convenient only f o r small ©, b) an expression found by the Wentzel-Kramers-Brillouin method, which i s asymptotic to the solution, and which i s v a l i d f o r ^ near a zero of ^ - 4$ n + 4p(p+U), - 30 -c) an asymptotic expansion i n powers of f which i s v a l i d f o r large $ • Each of these solutions i s then used to provide an approximation f o r A n i n terms of ? • The convergent series i s used to provide e s s e n t i a l l y that approximation to A n found by Michels, de Boer and B i j l . The Wentzel-Kramers-Brillouin solution i s ( 3 5 ) c o s n ( n - p ) ( - i A ( S ^ e ^ ) t s i n n c n - P ) ( A ( ? , ^ 6 ( s l ) where A W feai -| T p l p + » ) - ¥ n y +• j x J " * -and { n x - p * l » * ^ } *• Then putting $ s S 0 and equating (35) to zero we get A H * £ t * A T T U - P > ~ T i r * Ln. • For the lowest state (p = 0 , n 0 8 8 l ) t h i s reduces to - ' 3 1 -A n ^ ± < * p - (1#*(J;-T-)± - Z c o s K W t j . - i ) ) , The asymptotic expansion of ( f R) f o r large ? i s given by the author as 1111211 c o s n C n - P ) * * ?sM | - ifill^^li....] • J j ( a p T a n n - P - i ) i s i n i n « i - p » w i * * , s - n J l • ' " • P ^ + ' l » . . . } . I f we now put $ = $ 0 , n = and equate the above to zero, we obtain A n ^ i t f l « m « - P ) i - ' - ^ Q T P ) ( n , - f - » ) / f , . . . # K - p - i m « . « r M | +> ( n, - ?) (n,+ p+«)/j#« In the case where p = 0 , n Q = 1 , t h i s reduces to I t i i . - * t=0 The author also discusses two other cases. He f i n d s an expression f o r A n when S0 i s such t h a t E 0 , and he uses t h i s to f i n d that the smallest: eigenvalue i s zero (when p = 0) i f the radius of the enclosure i s He also writes down Proelich*s(/t) i s near a zero of the unrest r i c t e d I I I . Further Results. In section 1 of t h i s part we w i l l develop an approximate expression f o r & X which i s sim i l a r to that of F r o e l i c h . We w i l l then prove that, unlike F r o e l i c h ' s , our approximation to A X i s v a l i d near singular as well as. ordinary.; points of the d i f f e r e n t i a l equation. In section 2 we w i l l give the asymptotic expressions f o r aX f o r 'the harmonic o s c i l l a t o r , the r i g i d r o t a t o r , and the hydrogen atom problems. 1. General Procedure The equation s a t i s f i e d by the so l u t i o n to the unrestricted problem i s of the form ( 3 6 ) u" + (91x1 f A )U - 0 where one condition on u i s that . u(0, X) = 0 or u T ( 0 , X) = 0 the other boundary condition i s u(x', X) = 0 , x 1 > 0 , where x f may or may not be a s i n g u l a r i t y of the equation; the o r i g i n (as i n the hydrogen atom problem) may be a singular point. A l l points x, 0 < x < x T are ordinary points of the equation. Let y(x, X + t X) be the s o l u t i o n to the r e s t r i c t e d problem, where y(xo, X + & X) = 0 , the condition on r . 1.8354 . o formula f o r use when $ 0 s o l u t i o n . y(x, X + A X) at the o r i g i n being the same as that on u(x, X) • Then y(x, X + A X) s a t i s f i e s the equation y M + ( q u ) + A + A A ) y = o , which may be rewritten as (37) y" + lgu> U ) y = -*Xy . By v a r i a t i o n of parameters i t i s easy to convert equation (37) into the i n t e g r a l equation x (3$) y(x) = uix)~ ~ j (vix)ult>-uux>vi*)J y ii\ dt o where u(x) i s the solution to the unr e s t r i c t e d problem, and where v(x) i s a so l u t i o n of equation (36) such that In (38) l e t us put x = x 0 ; since y ( x Q ) = 0 we can solve for A X to get . \ W U (Xo) (39) A A = . V(x,)J "uit)yit)dK " uix.i J\wy(-t>c|t 0 0 Equation (39) i s exact, but, of course, we don't know y ( t ) . However, i f x 0 i s close enough to x 1 , u(t) should be a good approximation to y(t) (0 < t < x Q ) . Let us, then, define the following approximation to A X : (40) - x = w  Vix 0 ) J uMOt i t t - u ( X o ) j**v<t)uU)alt • " * . We would now l i k e to show two things; that A X a X. as x 0 -*• x 1 , and that, i n the asymptotic approximation, we x may neglect the term u(x Q) • \ v(t) u(t) dt i n the denominator of ( 4 0 ) . I t w i l l s u f f i c e to show that i ) \ u<*)y<*)dt * \ V i t i J t 0 o /•X. » Xp i i ) V l X o i j uM-udt - U(X«) I v ( * )w ( t ) t J t ^ V(x 0) J u v K ) d t « ) v i x 0 > j u«t ) yct)4t - u i x . ) ( v K ) y ( * ) d t V l X t ) JXV(i)<At . i i i . o w i - - . | p To simplify the following- discussion, l e t us now consider only the harmonic o s c i l l a t o r , and indicate l a t e r any changes necessary f o r the treatment of the other problems. Then i n equation (36) and u(x, X) i s of the form 7 35 -n where P«(x) '- 21 a t 3^ ; we may assume that a n > 0 . k = 0 Let us f i r s t prove i ) , i i ) and i i i ) i n the case that x T - + oo We proceed as follows. y(x, X ) i s of the form K>n*i where GlnUl r £ b K X* and b k ^ 0 when k >, n + 1 i f X Q i s large enough ( i t can be seen from equation ( 1 8 ) that i f X Q i s large enough, A X < 2 ) . Now, to show ( i ) using the Lebescjue bounded con-vergence theorem we need and t h i s i s true f o r each f i x e d x , since any so l u t i o n of the o r i g i n a l d i f f e r e n t i a l equation i s , f o r each f i x e d x , an analytic function of x 0 . To show that \u(x) y(x)\ i s bounded by an integrable function, we introduce the func t i o n y * ( x ) . - o , x , n . - 36 -.Now, i t i s e a s i l y shown that |y*(x)| i s bounded f o r a l l p o s i t i v e x , independently of x 0 , i f x 0 i s large enough. We proceed to do t h i s as follows. oo and therefore £ <•**•" $ I I M X.K . Now, i f 0 < x < x 0 , then IbKxK I_b«xefc (42) Equations (41) and (42) together imply that | y * u ) | « 2 i A U " i x X £ ibK»x* Kro f o r a l l p o s i t i v e x ; that is?, f o r some f i n i t e M which does not depend on x 0 i f x 0 i s large enough, |Y*U)| 4 M } x>, o . - 37 -We have shown, then, that |u(x) y*(x)| < M |u(x)| where )u(x)| i s integrable from zero to i n f i n i t y . Lebescjue's qonvergence theorem states that j u u ) y * ( x ) d x ^ j u M x j d x as x.-»°°; O p c l e a r l y , then, Xe f To prove ( i i ) we need only prove that <•)) w l(x) d V(x*M u l ) x - > t « ° cis x--» while x. x'^oo I w u , ) ) v^x)uix)dx I < The f i r s t of these statements i s e a s i l y proved. Let a be some number larger than the largest zero of u(x). For a < X Q we can write v ( x Q ) i n the form V ( X . ) = U(X.> \ • J U»IX) Since (x) U<X> = A * i^PnCX) i t i s easy to show that •» xS" V(x () ^ os x,-» - 38 -Therefore, since ) u M x ) d x ? 0 VU.l ) U v ( X ) d x -> * 0 0 OS X E -S>oo. 0 To prove the second statement we note that sinee i Kv VU) ^ ^ 2 — os *-»«> \ vtx)ucxjdx - \vuji i(x>elx + \v<x)u(x)dx 9 <* v X „ ( v t x ) m x ) d x t \ ^ i x - ) dx as *.->*>. Now, u<x.) ( 5^ *, d x - » o as and since .«< < 0 0 j j v i x ) U ( x ) d x j 9 I if* MIX.) I V(xJM(X)dx = 0 . X . - » « o ' C l e a r l y , then, we have proved ( i i ) . The proof of ( i i i ) i s s i m i l a r t o that of ( i i ) . We have already shown by ( i ) that - 39 -j u i x)yu> o ( x * ( u * ( X ) o l x a s x 0 -> oo tand i n proving ( i i ) we showed that f x * V ( X . ) \ U l ( X ) o l x "* t • * *• ~* 0 0 y therefore V i x e ) I uix)y(x) c l x -* * oo « s * . -» «*> . Also i n the course of proving ( i ) we showed that where M does not depend on x Q i f x Q i s large enough. Therefore ly(x)| $ M ' l u i x ) ! ( « N< x s x0 J and so W(x0) j vix)yix)olx| $ |uiXo) Jv(x)y(xjolx | * | u<x.> J vuj y m d x | x, X , $ j u<v«) j v ( x ) y < x ) e U | +• M'|u<x.>| J | U(x) v<xj| c4 I t i s cle a r that uu.) j v u ) y u)oix -> o o s x. -> <*> , - 40 -and M'uu.) ] | V(x)u(x)|olx -*o qs x and t h i s was a l l that remained to be done t o complete the proof of ( i i i ) . J We have shown, then, that and also that I f we now put v i x ) * u(x) (* , J UMt) ' i t i s easy to show that W = 1 ; i f we also choose A such that 1 ) Ul(X)fllX = I j our approximate expression f o r A \ reduces to the simple form ) uMx) where, of course, a i s larger than the largest zero of u(x) . - 41 -It i s cl e a r now, that steps ( i ) , ( i i ) and ( i i i ) are also e a s i l y proved when x 1 (where u(x f) = 0) i s an ordinary point of the d i f f e r e n t i a l equation. In t h i s case we f i n d that A \ ^ as x, -» x 1 and that as x„-»x* i f u i s nomalized so that « \ u l(x)olx = \, 0 In t h i s case, u(a) ^ 0 , and no zero.', of u separates a and x 1 ; a may be greater than or l e s s than x 1 . The proofs of the previous r e s u l t s may e a s i l y be adapted t o the other problems i n which we are interested. The proofs are v i r t u a l l y unchanged i f we allow u(x) to be of the form ?•' U ( X > - A - J t ^ P r t U ) > K>o ,rv \>o* J our r e s u l t s , then, are v a l i d f o r the bounded hydrogen atom. The r i g i d rotator problem i s d i f f e r e n t i n two ways. The equation to be solved i s of the form 42 -where h(x) f 1 , but i s continuous i n the open i n t e r v a l under consideration; and the range of the independent variable i s from zero to the s i n g u l a r i t y at x = 1 ( i . e . © For the r i g i d rotator The solutions of the unrestricted problems are of the form where P n ( x ) Is the nth Legendre polynomial. It i s e a s i l y shown that i f we define A X. to be *x - wm*.> , V(x„) Jv»tx)uMx)dx - Utx.)j MX) V(x)M(x)otx o o then 6> X A X as x 0 -> x f . (Both x 0 and x* must not be greater than unity.) We can also f i n d that Feu J uMx) where, i n t h i s case x1 j hlx)uMx)dx = I. 43 -2. Special Cases In t h i s section we apply the r e s u l t s developed above to f i n d the asymptotic formulas f o r t X f o r the harmonic o s c i l l a t o r , the r i g i d r o t a t o r , and the hydrogen atom problems. The d i f f e r e n t i a l equations f o r the three problems are found i n S c h i f f (£, Ch. 4) ; i n the d e r i v a t i o n of the asymptotic formulas, the necessary summations and d e f i n i t e i n t e g r a l s are to be found i n S c h i f f (£, Ch. 4) and i n Whittaker and Watson (12, Ch. 15). i . The Harmonic O s c i l l a t o r The d i f f e r e n t i a l equation f o r the wave function i s 2m ol-r* x C T * U = E U and the following changes of variable convert the equation into the form * ( ( t - i » M • * ) « « < » . The eigenfunctions of the unbounded problems are 5i: h where H n ( t ) i s the nth Hermite polynomial. - 44 -I f we i n s i s t that u(+ x Q , X ) = 0 , i t i s easily-shown that where X » n + A X . i i . The R i g i d Rotator The wave function s a t i s f i e s the equation -L d ( sine £ & \ + ( \ . j £ \ ^ - 0 sine ote V X e * T V* si**-* ' u * I f we put x • cos 9 and u(x) - s i n 9 • (9) t h i s becomes s*ls* • ( JL + *x )u - o. The s o l u t i o n to the unrestricted problem i s uu) = U - x * ) * P f ( x ) = U - x 1 ) 2 ^ ' j ^ U ) where Pn( x) i s the nth Legendre polynomial, and where \- n(n +1) > n ^  m>, o ^ Upon i n s i s t i n g that u(+ X Q , X ) • 0 , we can e a s i l y f i n d t h a t , f o r m • 0 - 45 -where X • n(n+l) + a X • For m >, 1 we f i n d that ft 5 f M W U n - l K ) ! where r - Sdfi 0 r n" m** 1 t whichever i s an integer, and 2 2 where X = n(n+l) + A X . We note here that <} A -I ~ o< r»» s o, which i s i n agreement with the graphs given i n section 3 , part I I . i i i . The Hydrogen Atom The d i f f e r e n t i a l equation f o r the r a d i a l part of the wave function R i s and the changes of variable - 46 -convert t h i s into the equation 2pi + U - £i*i!> + X)IA = o. The solutionsto the unbounded problemr are where X - —ir and L i s an associated Laguerre polynomial, n* I n s i s t i n g that u(xo, X ) = 0 , we f i n d that \ 2 X0 JL * a w = a s x„ -> <*> n*"*3 in-p-i)Kntpj! where X = - — 9 + A X . For the ground state where x\ = 1 and p = 0 t h i s reduces to BIBLIOGRAPHY (1) . F.C. Auluck and D.S. Kothari, Proc. Camb. P h i l . S o c , 41 (1945) , 175 . (2) . S. Chandrasekhar, Astrophys. J., SI (1943) , 263. (2.). R.B. Dingle, Proc. Camb. P h i l . S o c , A£ (1953) , 103 (AJ. H. Fr o e l i c h , Phys. Rev., j& (1938) , 945-(£). S.R. de Groot and C.A. ten Seldam, Physica, 12 (1946) , 669. ( 6 ) . A. Michels, J. De Boer and A. B i j l , Physica, 1± (1937) , .981. ( J ) . W.E. Milne, J. Research U.S. Bur. Stand., (1950) , 245-( 8 ) . M. Rauscher, J. Aero. Sc., 16 (1949) , 345-(£). L.I. S c h i f f , Quantum Mechanics (McGraw-Hill, 1949) , Ch . 4 . (1O0. C.A. t e n Seldam and S.R. de Groot, Physica, IB (1952) , 8"91. (11) . A. Sommerfeld and H. Hartmann, Ann. £ . Phys. (5) 37 (1940) , 333 . (12) . E.T. Whittaker and G.N. Watson, Modern Analysis (Cambridge, 1915) , Ch. 15. 

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