UBC Theses and Dissertations

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UBC Theses and Dissertations

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 t h e DEPARTMENT of MATHEMATICS  We accept t h i s t h e s i s as conforming to t h e standard r e q u i r e d from c a n d i d a t e s f o r t h e degree o f MASTER OF ARTS.  Members o f the DEPARTMENT OF MATHEMATICS. THE  UNIVERSITY OF BRITISH COLUMBIA April,  1956.  ABSTRACT I n t r e a t i n g s o - c a l l e d "bounded quantum problems" one problems.  i s generally l e d to d i f f i c u l t  mechanical  eigenvalue  For example, the problem o f f i n d i n g t h e wave  e q u a t i o n f o r a hydrogen atom c o n f i n e d t o a sphere f i n i t e r a d i u s l e a d s one  to f i n d i n g s o l u t i o n s of  of Laguerre's  e q u a t i o n which v a n i s h f o r given f i n i t e v a l u e s o f the independent v a r i a b l e .  I n t h i s t h e s i s we d i s c u s s t h r e e  such problems i n some d e t a i l . Since 1937  a number o f papers on these problems  have appeared; we d e s c r i b e here the p r i n c i p a l methods of these papers, and a new  improve one  of them.  We t h e n  develop  method which i s s u f f i c i e n t l y g e n e r a l t o t r e a t  each  problem, and which can be e s t a b l i s h e d r i g o r o u s l y i n each case.  T h i s method p r o v i d e s us w i t h asymptotic  f o r the perturbed  eigenvalues.  expressions  ACKNOWLEDGEMENT  The author wishes t o thank Dr. T.E. H u l l f o r h i s i n v a l u a b l e h e l p i n t h e p r e p a r a t i o n of t h i s t h e s i s .  TABLE OF CONTENTS  I.  INTRODUCTION  I  II.  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 R o t a t o r  18  4.  The Hydrogen Atom  24  III.  FURTHER RESULTS  32  1.  General Procedure  32  2.  S p e c i a l Cases  43  i  The Harmonic O s c i l l a t o r  43  ii  The R i g i d Rotator  44  iii  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 i n t e r e s t e d  i n t h e s o l u t i o n s t o bounded quantum mechanical problems as t h e l i n e a r harmonic hydrogen atom.  such  o s c i l l a t o r , t h e r i g i d r o t a t o r and t h e  Even though such s i t u a t i o n s do not a r i s e i n  r e a l i t y , e n c l o s i n g a hydrogen atom i n a sphere o f f i n i t e r a d i u s might be expected t o g i v e , f o r example, an approximation to t h e e f f e c t of p r e s s u r e on s p e c t r a l 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 t h e quantum mechanical system does not a l t e r t h e d i f f e r e n t i a l e q u a t i o n to be s o l v e d , but a f f e c t s o n l y the boundary  conditions.  To i l l u s t r a t e , l e t us  c o n s i d e r a p a r t i c u l a r case, t h e l i n e a r harmonic  oscillator.  The d i f f e r e n t i a l e q u a t i o n to be s o l v e d i s  where we use the n o t a t i o n adopted by Auluck and K o t h a r i  (1).  The f i r s t few e i g e n f u n c t i o n s and e i g e n v a l u e s of t h e unbounded problem a r e  "a  8  * " X,  A=  I  U * = - 4 * \ x - | ) ( X + ») , X r X  u , = i * * x U " J3)(* + J3), x  2  3  -  2  -  I t i s c l e a r t h a t these "unperturbed"  s o l u t i o n s can  a l s o be used as p e r t u r b e d s o l u t i o n s i n some s p e c i a l F o r example,  circumstances.  U 2 g i v e n above, which i s the second e x c i t e d  s t a t e i n t h e unbounded problem, can a l s o be i n t e r p r e t e d as the ground s t a t e when t h e w a l l s of t h e e n c l o s u r e are a t x  0  = + 1 ;  X =» 2  i t f o l l o w s then, t h a t f o r t h i s p a r t i c u l a r  i s the lowest e i g e n v a l u e .  Similarly,  enclosure,  c a n be  thought  o f as t h e ground s t a t e when t h e w a l l s of t h e e n c l o s u r e  are at  x  0  = +  J 3 - Jo" , o r as the second e x c i t e d s t a t e  when they a r e a t \ = 4  x  Q  = +  J3 + J5  t h e n i n the former  case  i s the s m a l l e s t eigenvalue, and i n t h e l a t t e r i t i s  t h e second e i g e n v a l u e . We a r e l e d , t h e r e f o r e , to c o n s i d e r the f o l l o w i n g graph.  - 3  -  Other p o i n t s can be o b t a i n e d as we  d i d the f o u r  shown; s t i l l o t h e r s can.be o b t a i n e d by i n t e r p o l a t i o n methods, or by u s i n g t a b l e s g i v i n g zeros of o t h e r s o l u t i o n s o f  Hermite's  equation. While  some of the papers on t h e s u b j e c t are  concerned  w i t h the r e s u l t s of t h e s e numerical c a l c u l a t i o n s , most o f t h e papers are devoted t o f i n d i n g the asymptotic  behavior o f  X  near t h e ends o f the c u r v e s . Langer,  Cherry, E r d e l y i and o t h e r s have developed  a  t h e o r y c o n c e r n i n g the asymptotic b e h a v i o r , f o r l a r g e v a l u e s o f a parameter, of t h e s o l u t i o n s t o i n i t i a l v a l u e problems. Some of the papers which we w i l l d i s c u s s , and t h i s c o n s t i t u t e an attempt  t o i n i t i a t e an analogous  thesis,  theory f o r  boundary v a l u e 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 a p p l i e d to each of s e v e r a l c l a s s e s of problems.  I t s a p p l i c a t i o n to each c l a s s can be  rigorously  j u s t i f i e d i n each case, and the problems mentioned above appear as s p e c i a l c a s e s . Before doing t h i s we w i l l  d e s c r i b e the more important  r e s u l t s obtained to date, and i n d i c a t e how  one of them can be  improved. II.  HISTORICAL SURVEY The  r e s u l t s we  are going t o d i s c u s s c o u l d be  e i t h e r a c c o r d i n g to the method used,  classified  or a c c o r d i n g t o t h e  problem f o r which t h e y were developed.  We  have a r b i t r a r i l y  - 4chosen the l a t t e r 1.  General Methods. We  it  classification.  b e g i n by d i s c u s s i n g a method due  i s the o n l y one which was developed  t o P r o e l i c h (Zt);  independently of  any  particular application.  We a l s o i n c l u d e i n t h i s s e c t i o n a  b r i e f d e s c r i p t i o n of two  numerical methods t y p i c a l of those  which c o u l d be a p p l i e d t o the problems b e i n g c o n s i d e r e d . I n 1938  P r o e l i c h (£) developed the method o u t l i n e d  below. Consider the two 1)  u" * I gu) + X)u * 0  2)  y " + (goo + X + A \ ) y s 0 ,  and the two or  equations  u'(0)  s e t s of boundary c o n d i t i o n s  u(0) = 0 , y(0) = 0  = 0 , y'(0) = 0 and u(b) = 0 , y(b - € ) - 0 .  M u l t i p l y i n g e q u a t i o n (1) by ( - y(x)) , e q u a t i o n (2) by u(x) , and adding, we get  j  x  |y'(x)uoo- u'umxjj. + A * uu)y(x) = 0.  I n t e g r a t i n g t h i s from for  £ X  x = 0  to x = b -e  and  solving  we o b t a i n  a. -y; ' -° • <)Y (b  (3)  J 0 Equation  UU)Y("fc)clt  (3) i s an exact e x p r e s s i o n f o r  , but, of course,  we do not know  y(t) .  I f , however,  i t i s r e a s o n a b l e t o expect t h a t  €  i s "small", then  y ( t ) 2z  u ( t ) ; then i f b  i s an o r d i n a r y p o i n t o f 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 c a s e t h e n  - u(b-e)u'(b-e)  -  —XI J u*(t)dt  •  e  I t f o l l o w s , then, t h a t t\ ^  € U'(b)) J  x  as e -  .  0  u (t)dt 4  o  (By  f (x) ^  lim x*x  ££*) S > (  0  g ( x ) as - 1  x  X Q - we s h a l l always mean  .)  X  The f i r s t  numerical method which we w i l l d i s c u s s was  developed p r i m a r i l y by Rauscher  (£) i n 1949•  The method  i n v o l v e s m a t r i c e s , and was o r i g i n a l l y designed f o r use w i t h a e r o n a u t i c a l v i b r a t i o n problems. m a t r i x elements  I n these s i t u a t i o n s , t h e  a r e determined e x p e r i m e n t a l l y .  The method  can, however, be adapted t o e i g e n v a l u e problems of t h e type under c o n s i d e r a t i o n . Convert t h e system  M  y " + 9<*>y = - A y , 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 t h e Green's F u n c t i o n a s s o c i a t e d w i t h  (6)  y  Choose Let  n  + tjtxiy s o , Ay(o) • 6y'u)* ytb) = o.  M  0 < x - L < X 2 < « . « < x  points  us now approximate t o  n  < b .  y ( x ) i n t h e f o l l o w i n g manner:  Y ( x ) = XI Si^^j , J = » , V w K « r «  (7)  y. y 5  U j  ).  i~\  The  g-;(x)  which  a r e t h e p o l y n o m i a l s of lowest p o s s i b l e degree  satisfy i)  the boundary c o n d i t i o n s of e q u a t i o n ( 4 ) ,  ii)  (8)  k  = 5  j k  .  b  YL {  Letting  (x )  X - - ^ , e q u a t i o n (5) becomes  I f we now put n  g j  r\  U V>3i(t><tt]/j = (  x • x^  uLv'yj .  j. « 1, 2, ..., n , (7) and (8) y i e l d t h e  matrix equation A Y = oo Y b  where and  A » (A<j) ,  A ^ = [ G(x t) o^UJdt , t|  Y~ ^ * V  I f we now s u b s t i t u t e  Yv. i n t o  (7) and t a k e  X i , «= - — ,K=1,2,.., n,  we get approximations  to the f i r s t  n  eigenvalues  and  corresponding e i g e n f u n c t i o n s of t h e o r i g i n a l e q u a t i o n ( 4 ) . I n g e n e r a l , a c o n s i d e r a b l e amount of work i s e n t a i l e d i n o b t a i n i n g t h e above G  (x, t )  i s of t h e f o r m  +  where  u-^  U.(X\Uxli)-Ux(X)U,(t) 2 W  and  U2  e q u a t i o n (6). (-)  when  approximations.  are l i n e a r l y independent  We take the  t < x < b .  must be determined  ( + ) when  0 < x < t  The c o e f f i c i e n t s  so t h a t  G(x, t )  solutions of and the  a^, a , p 2  t  t  s a t i s f i e s the boundary  c o n d i t i o n s o f t h e problem. G(XJL,  i  = 1,  2,  t)  ... n  w i l l have t o be t a b u l a t e d f o r  0  < t < h ,  and t h e m a t r i x elements J  ^(XL.t^ltjdt  0  evaluated numerically. I t i s o n l y a f t e r the above has been completed  that  one can b e g i n s o l v i n g t h e m a t r i x e q u a t i o n by any of s e v e r a l s t r a i g h t f o r w a r d methods. The f o l l o w i n g i s perhaps a more e f f i c i e n t  numerical  method of s o l v i n g the type o f e q u a t i o n w i t h which we dealing.  The method was  developed  i n 1950  by M i l n e  are (7).  - 8 We a r e g i v e n  U" • (31x1 + \\ u = 0 , u(o) = u l b ) * 0 , which we r e p l a c e by  o <„v-- 0 . + 9  Divide  the i n t e r v a l  (0,  length  h = ~-  L e t V ( i h , j h ) = V - , and l e t  (9)  (  .  b)  (n + 1)  into  subintervals of  i;j  V ( x , 0 c u U)cosJH t . K  K  K  The  initial  later  c o n d i t i o n s may be chosen a r b i t r a r i l y , and we w i l l  choose them i n a convenient f a s h i o n .  Mow make t h e f o l l o w i n g approximations:  Our  partial differential  do) where  (11)  v, iji+  equation  • V ., =  • Vc,^  t|i  g^ = g ( i h ) •  +  ^ -VCJ 9  The boundary c o n d i t i o n s a r e now  V«j = V  We now choose t h e i n i t i a l d i f f e r e n c e equation  now becomes  n+I>i  =0 .  c o n d i t i o n s t o be such t h a t i n t h e  approximation they become  - 9 -  (12)  »/, 0, 1*1  and  eS  V  l l =  .  =1,1.1 Using equations (10j^ matrix  V.  (11)  . i = 0,  and (12),  1,  The submatrix  i = 1,  ±  i s t r i a n g u l a r and n o n - s i n g u l a r . f o r the  n  j = 0,  n+1, 1 j,  2,  1,  ... n .  ... n , j = 0,  1,  ... ( n - 1 ) ,  We a r e a b l e , then, t o s o l v e  A , k - 0,  constants  we can now generate t h e  k  1,  2,  ... (n-1)  fromthe  set of equations  (13)  L  A  i ii v  ?  0  i--o  where  A  n  i s d e f i n e d t o be u n i t y .  The s o l u t i o n f o r the  A  i s e a s i l y achieved s i n c e t h e c o -  k  e f f i c i e n t matrix i s t r i a n g u l a r . We now r e - w r i t e equation (9)  (14)  as  Vij'X^UKicos/Jkjh . K: I  M u l t i p l y i n g both s i d e s o f (14) j = 0  (15)  to  j = n  by  Aj  and summing from  we o b t a i n , w i t h t h e a i d of e q u a t i o n  Lc»UHij^A co*jr k} = 0 , J  ilj  and i t can be shown t h a t (15) i m p l i e s t h a t w  J^AjCOsJT jK r 0 . j= 0 K  (13)  - 1 0 -  That i s ,  where  u^, k = 1,  characteristic  cosnyu  2, . . . n  are t h e  n  r o o t s of the  equation  + A„_,cos(n-i)/«  A,cos/« • A = o . 0  Having found t h e e i g e n v a l u e s , t h e e i g e n f u n c t i o n s a r e now easilydetermined i n t h e f o l l o w i n g manner. K  from  and  (We now omit t h e s u b s c r i p t  u.^ .) (10)  Into the d i f f e r e n c e e q u a t i o n  8  put  COS j / A  Since  equation  (10)  u  { 1 6 )  The  becomes  i + l  ,  U c o s ^ - h ^ U i - Ui., .  boundary c o n d i t i o n s imply t h a t U j _ , say  non-zero the other.  u  Q  = o , we choose any  U]_ « 1 , and s o l v e f o r the  The f a c t t h a t  ^ +± n  one a f t e r  should t u r n out t o be zero  serves a s a check on t h e c a l c u l a t i o n s * Equation  (16),  with  u. = u.^, k = 1,  2, • •• n , t o g e t h e r  t h e boundary c o n d i t i o n s p r o v i d e s t h e f i r s t  n  with  eigenfunctions.  - 11 2.  The Harmonic O s c i l l a t o r . In t h i s  s e c t i o n we w i l l summarize the r e s u l t s  of  two papers d e a l i n g with t h e l i n e a r harmonic o s c i l l a t o r .  Both  papers b e g i n by c o n s i d e r i n g the g r a p h i c a l method o u t l i n e d 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  essentially  by M i c h e l s , de Boer and B i j l ( 6 ) .  1937  in  that  invented  We w i l l develop  method h e r e , f o r the Harmonic o s c i l l a t o r ,  and a l s o  the  develop  a m o d i f i c a t i o n to the t e c h n i q u e , which c o n s i d e r a b l y extends its  utility. The  e q u a t i o n to be s o l v e d  U"4 { l i - i x * )  (17) where The  +  is  X}u =0  , E-(*+i)t»w  mass o f the o s c i l l a t o r i s  energy due t o a displacement  m,  {ct/  * , and  f u n c t i o n of an o s c i l l a t o r w i t h energy  and i s the  1  u(x-j X) E .  potential  i s the wave  The g e n e r a l  s o l u t i o n of e q u a t i o n (17) i s  (18)  We see t h a t as  x  odd  positive  + oo  i f t h e "natural" conditions that are imposed,  then we .must t a k e  integer  B = 0 , or  and  X  u(x, X  \)  0  equal to  equal t o an even  an  in  - 12 p o s i t i v e i n t e g e r o r zero, and c l a r i t y we w i l l c o n s i d e r  A = 0 .  F o r t h e sake of  o n l y t h e odd s o l u t i o n s t o t h e problem,  and l a t e r s t a t e t h e r e s u l t s f o r t h e even ones. I f we i n s i s t t h a t we expect t h a t  X  the approximation M - X = -  Q  w i l l be c l o s e to  any p o s i t i v e i n t e g e r .  case  u ( x , X) = 0 , where  L e t us put  M - X —  M - X  u  (  where  1  X = X  unless  T  = 2K - 1, K  T  A X , and make  +  1  X  i s large,  0  M • X  , i n which  f  A X , s u b s t i t u t i n g t h i s into e q u a t i o n (18)  (with B = 0 ) , we o b t a i n f o r  (19)  X  x  x  A  )  s  A  K = 1  * ^ { x - a | j ^ i x - } ,  and f o r K > 1  «»,>) *  A 4 - « * { » • f ' ^ - ' - ^ - v ) •(••V) „ » . .  - 4 M f f i * - ' - « ) & " *  Then, since.  2  1  )  Q  ZL  T  and  ,  u ( x , X) = u(xo, X* +  equations (19) and (20) f o r  (  E^ - ;^;>.'-' '"- x"4  1  vn  ,,  A X) = o  we c a n ' s o l v e  A X ; t h i s gives  ...  tn-Ql ir> v  v,  Q * W A X )  - 13 -  (22)  (A» V + *Xs  If  X - 3 +  A X , t h a t i s , when  , K >l).  k = 2 , e q u a t i o n (22)  reduces t o  (23)  U*3t**), Un*i)!  Expressions  (21) and ( 2 2 ) , and t h e s p e c i a l case  valid only f o r large  X Q ; u n f o r t u n a t e l y , t h i s i s j u s t when  t h e s e r i e s i n v o l v e d converge most s l o w l y . (22)  implies that  *X = 0  f o r m u l a i s not v a l i d f o r e a s i l y seen.  x  Q  reason that equation zeros o f  1  This i s  (23) i m p l i e s t h a t x  0  aX  i s greater than  T h i s , o f course, i s i n c o r r e c t . (22) i s not v a l i d f o r x  u ( x , X ) , i s t h a t the e x p r e s s i o n T  equation  u(x, X ) , the  near t h e s e z e r o s .  F o r example, e q u a t i o n  J3 .  Even though  a t the zeros o f  i s p o s i t i v e o r n e g a t i v e , a c c o r d i n g as or l e s s t h a n  (23) a r e  0  The  near t h e s e  - 14  X  (24)  •  V)»n-l-V»"-»-V»jr»"*'  «»•  <*"+•).»  which o c c u r s i n e q u a t i o n (20) if  -  changes i n  \*  then,modify  the method of M i c h e l s , de Boer and  t h i s f a c t i n mind.  x  i s very sensitive t o small  i s near a zero of  I n (24),  (which i s the exact v a l u e o f of  A\  From (17)  l e t us r e p l a c e \)  , keeping o n l y the f i r s t we  u(x, \ ) » f  \*  L e t us,  B i j l , bearing by  \* +  and then expand i n powers two terms o f t h e  polynomial.  then obtain  where  "••  'x-i  1  and  Hence, p u t t i n g  x = x  Q  A\  and s o l v i n g f o r  & \  we  obtain  U = A'* AX = 2K-1 1-6X , K > » ) .  - 15 X  I n the s p e c i a l case t h a t  1  * 3 ( i . e . k=2)  we now  find  that X.lTf-1) '  U*3*AA).  A X = 0  I t i s e a s i l y seen from the above e x p r e s s i o n t h a t when  x  0  J3 , and t h a t  =  a c c o r d i n g as  x  Q  AX  i s p o s i t i v e or negative  i s l e s s than or g r e a t e r than  i s c l o s e enough to  J3  JJ  (if x  )•  The corresponding r e s u l t s f o r the even s o l u t i o n t o the problem are as f o l l o w s . If  X - X» +  AX  where  X » - 2k,  k - 0, 1,  then the method o f M i c h e l s , de Boer and B i j l g i v e s  and  A>c ii^fl F(x.)  where  and  {\,  V*AA,K>l),  '  2,  0  -  X  n.V  '\UK4l>!  16  -  L  y*  (2n)»  nsic*t  Our modification to this method gives  F(x.) • G ( X o )  '  J  where K  X^j,  GU)=  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 cases where (i)  x < < Jn+ ± Q  and x  En in the two limiting 0  >>  J n+1  .  X << Jn+T~ C  Under these circumstances the author develops two distinct approaches. a)  The first of these is as follows,  It can be shown that  where I **  T  and where  4)  )  4}  -  17  -  I t can a l s o be shown t h a t , f o r l a r g e M  = 2"* Tf"'* ri2m+1) K" "5 cos (JaK x - mn-  i\x ) x  K  m  Putting  m  i X + "5.  K =  and  m = +  t h a t t h e wave f u n c t i o n s vanish  b) x  Q  x  K  at  OWK)).  , we have, upon i n s i s t i n g x = + x  Q  ,  A second method of approximation i n t h e case t h a t < < by  Jn~+~I~ i s obtained x - x  Q  i n the f o l l o w i n g manner.  i n the differential  I n s i s t now, t h a t  equation;  we  Replace  obtain  u(0) * u ( 2 x ) = 0 , and l o o k f o r a s o l u t i o n Q  of t h e form  where  The  authors shew t h a t  and  that  A j ^ i s n e g l i g i b l e unless  k = n+1  ,  - 18 -  (ii)  X »/nTT 0  The authors show t h a t under t h e s e  conditions  V»*V* «(Mtttd^al ...) ,  t  t i l  ,^  and  l  , i^oi*iil ...)coia2l T  Using t h e above expressions we f i n d  3.  The R i g i d  #  that  Rotator  I n t h i s s e c t i o n we apply t h e method o f M i c h e l s , de Boer and B i j l t o the r i g i d r o t a t o r ; we a l s o t r e a t the  - 19 problem with our m o d i f i c a t i o n to the t e c h n i q u e .  Before  doing t h i s , we i n c l u d e f i r s t t h o s e graphs o f  X  v.s.  a r r i v e d at by Sommerfeld and Hartmann(ll) i n  1940.  The f u n c t i o n  0  s a t i s f i e s the equation  «" • ( & . • , } = & ) « • •  ( 2 5 )  where  m  i s a p o s i t i v e i n t e g e r , and t h e changes o f v a r i a b l e  x = cos 9 *@*  u ( x , X)  x  and  u(x, X) = s i n 9 • ^G  (9, X)  1  have been made.  i s the wave f u n c t i o n of the problem.  I t i s e a s i l y shown t h a t t h e g e n e r a l s o l u t i o n o f e q u a t i o n  (25)  i s g i v e n by  U(X,X>=  XHMzrt.f X)-" (M,- » xn)  A( I + f  + B(X  4. f  v  CMxr.-'XXMan.r X ) " ( < V X )  »M-»\\  (2r» »)l  'J  +  where  k-1).  = (m + k) (m +  I t c a n a l s o be shown t h a t i f we i n s i s t t h a t then we must put and  k  \ - %  i s even o r odd.  , where  A  or  B  u(+ 1,  X) = 0 ,  i s zero a c c o r d i n g  By u s i n g the method shown i n t h e  i n t r o d u c t i o n , Sommerfeld and Hartmann were a b l e to draw graphs s i m i l a r to t h o s e below, o f  X  v.s.  x  0  .  - 20  -  X  For  m  for  m = 1 , the only important d i f f e r e n c e being  4 —r  X  2  the  X  v.s. x  Q  graphs are s i m i l a r to the  one  that  I I  ,  = 0  if  m ^ 2 ; i t i s only f o r  m - 0  that  c) X —-=— -*._«> 6 *o ..  a  x.  s  -» i . ° ..  •  I t i s t h i s f a c t that  Sommerfeld  and Hartmann i n v e s t i g a t e d i n d e t a i l i n ( 1 1 ) . With t h e a i d of t h e asymptotic e x p r e s s i o n s which we s h a l l d e r i v e f o r A X , t h e i r r e s u l t s w i l l appear as s p e c i a l cases i n p a r t I I I . We w i l l now f i n d the M i c h e l s , de Boer, B i j l - t y p e approximations t o  AX;  l e t us d e r i v e them here f o r t h e even  s o l u t i o n s , and l a t e r s t a t e the r e s u l t s f o r the odd ones. The where  procedure i s as f o l l o w s .  X' •  , and  Put  k * 1, 2, ..., (B - 0 ) .  we make the approxai mat i o n s  ^2L_i - X —  L = R , i n which case  - X = -  is,  i f X» = (m+l)m  and  i f k  > 1  (27)  where  B U ) - 11  and  we  X = X* +  we f i n d  that  obtain  UU.Xl^:  As b e f o r e ,  ^21,-1 " ^  a X . I f  All-x'J^BOrt-A*CU))  a X ,  T  u n  less  B = 1 , that  - 22 u ( x , \) - 0 , s o l v i n g (26)  since  0  and (27)  for  A X  we  obtain (28)  !  * i " + J (M . -yKMt,.rV)"-cM -v) *« 1)t  B  t  for  k = 1 , and f o r k  > 1  we  obtain  sia  (29)  c c Xo)  Equations (28)  and (29)  we showed i n s e c t i o n 1, an e x p r e s s i o n u(x,  x  (in) I  2  \ )  for  A X  are v a l i d f o r . x  near u n i t y .  Q  t h e method can be m o d i f i e d  this modification  (30)  t o provide  which i s v a l i d near those zeros of  which l i e between zero and u n i t y .  T  The r e s u l t of  gives  A X *  )  K>l,  where  If  m = 0  and  As  k = 1 , that i s ,  reduces t o  &\±—L f  \» = 0 ,  (28)  -23 Let us now compare (29) k • 2 , ( i . e . X' = 3 x 2  (3D  = 6).  and (30)  when  E q u a t i o n (29)  m • 0  and  becomes  -61 *a > V ((2i>-i>un-t)-fcH(»-iX»-'H-fr)-(?»»-fe) i«) x  whereas (3®) becomes  (32)  ^  , - J X : '  I t i s e a s i l y seen t h a t equal as while  (31)  x  Q  equations (31)  -> 1 , but t h a t  (32)  and (32)  are asymptotically  i s valid f o r x  near  Q  ~ -  i s not.  The c o r r e s p o n d i n g r e s u l t s f o r t h e odd s o l u t i o n s t o the problem are as f o l l o w s . K = 1,  2,  If  X = X  T  +  then i f k = 1  AX*=  *a X? + 7 <  3' &  X'H M>n.j V) • • ( M» - V) ^ »»+,  u^Toi—*'  A X , X  T  =  ,  - 24 For  -  1 , the a p p r o x i m a t i o n due to the method o f  k >  Michels,  de Boer and B i j l i s  e(x.) F(x.) whereas our m o d i f i c a t i o n t o the method g i v e s  E, F  and  G  are d e f i n e d  4  below:  la •»•»)!  *  t>«0  4.  The Hydrogen Atom. In 1937  Michels,  de Boer and B i j l  method which, i n s e c t i o n s t o the harmonic o s c i l l a t o r  (2)  and  (3)  (6) developed the  we have seen a p p l i e d  and r i g i d r o t a t o r r e s p e c t i v e l y .  I n t h i s s e c t i o n we o u t l i n e the r e l e v a n t p o r t i o n of t h e i r paper (they  a l s o d i s c u s s a p h y s i c a l a p p l i c a t i o n of t h e s i t u a t i o n ,  - 25 and  -  do some numerical c a l c u l a t i o n s ) and g i v e t h e r e s u l t s  of our m o d i f i c a t i o n to t h e i r method. a paper w r i t t e n i n 1946 one w r i t t e n i n 1953  We w i l l then  discuss  by de Groot and t e n Seldam(j>) and  by D i n g l e  (2.).  We s t a r t w i t h t h e e q u a t i o n  u'» • (I - £i!*L> + X ) u * o where  u(x)  i s t h e wave f u n c t i o n d e n s i t y ,  I t i s e a s i l y shown t h a t the s o l u t i o n which i s f i n i t e a t the origin i s  where  X = -  Since  u(x, X)  - — g » M- >  0 •  w i l l d i v e r g e as  x-*•  0 0  , unless the s e r i e s  the c o n d i t i o n that l i m u(x, X) » 0 r e s t r i c t s ^ x-*» to t h e s e t - ~ 2 > u - p + k , k * l , 2, ... . I f t h e  terminates, X  boundary c o n d i t i o n s a r e  a  u(0, X) = u ( x , X) = 0 , t h e method  o f M i c h e l s , de Boer and B i j l  0  gives  - 26 which i s the r e s u l t  s t a t e d i n ( 6 ) . I n g e n e r a l we  A/*  U = 2.  For  K  > 1  we  —  obtain  .y^***')  ni»pt»*i)i  obtain  where  and  For  p = 0  A> ^  and  K = 2 , that i s  ^  which i s v a l i d f o r l a r g e  ,  x  •  u- - 2 , t h i s reduces t o f  ( > .  )  - 27 Our m o d i f i c a t i o n of t h e method g i v e s  where  j  K->  For  p = 0  and  n  ^ ^ ^ ^  K = 2  A/* ta=  7  "  the above r e d u c e s t o  ,  1  (X =  ),  rt»3  which i s v a l i d f o r l a r g e  x  0  and f o r  x  0  near  2 .  The second paper on t h e hydrogen atom w h i c h we summarize i s by de Groot and t e n Seldam (j>). The a u t h o r s d i s c u s s t h e method of M i c h e l s , 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 n o t a t i o n .  They a l s o f i n d M i c h e l s , de Boer and B i j l - t y p e to H  T  & \  when  p = 0  and  jj, » 2 , and when f  = 2 (see the p r e v i o u s p a r t o f t h i s The  approximations p = 1  and  section).  authors note t h a t t h e s e r i e s - i n v o l v e d i n t h e  M i c h e l s , de Boer and B i j l - t y p e approximations converge most s l o w l y when they a r e most a c c u r a t e ; i t i s f o r " t h i s r e a s o n t h a t t h e y f i n d an i n t e g r a l r e p r e s e n t a t i o n f o r these  series.  -  The  28  -  authors show t h a t sums o f t h e form 00  Ox) '- Z can be expressed i n terms o f t h e f u n c t i o n  x  A s t r a i g h t f o r w a r d r e c u r r e n c e r e l a t i o n i s t h e n developed connecting  Fi  m  +  (x) and  1  F i ( x ) , where m  Fi,(x) = E i l x ) - E i 10  E i(x)  i s g i v e n by  -(  and  x  has been p r e v i o u s l y The  tabulated.  authors t h e n c o n s i d e r t h e f o l l o w i n g  a)  E  < 0 ,  r  0  large  b)  E  < 0 ,  r  Q  small  c)  E ±  d)  E  e)  E -> + oo  0  > 0 ,  The  E  cases.  not l a r g e  l a s t p o r t i o n o f the paper i s devoted t o the  p h y s i c a l importance of t h e problem. i n t r o d u c t i o n t o (10)  I n t h i s c o n n e c t i o n the  may a l s o be of i n t e r e s t .  - 29  -  I t w i l l be seen i n p a r t I I I t h a t the i n t e g r a l s Fi (x)  o b t a i n e d by de Groot and t e n Seldam are v e r y s i m i l a r  m  t o those which we w i l l o b t a i n i n a more n a t u r a l and d i r e c t fashion. ' The t h i r d paper, w r i t t e n by D i n g l e (£) i n c o n t a i n s p o s s i b l y the most complete, from a p o i n t of view, treatment of t h e hydrogen  1953  mathematical  atom problem.  The author s t a r t s w i t h t h e r a d i a l e q u a t i o n  (33)  £  (TR)  + 1 ^  +  -  iiiL£  *1£±:MM  0.  and makes the f o l l o w i n g changes of v a r i a b l e  These reduce  (33)  t o the form  5ll ( t « + l - i t t t .  (34)  ?  The to a)  s 0  .  %  author t h e n p r o v i d e s the f o l l o w i n g forms o f t h e  solution  (34): a convergent for  b)  Cl£t2))^*)  s e r i e s expansion which i s convenient o n l y  s m a l l ©,  an e x p r e s s i o n found by t h e W e n t z e l - K r a m e r s - B r i l l o u i n method, which i s asymptotic to the s o l u t i o n , and which is valid for  ^  near a zero o f  ^  - 4$  n + 4p(p+U),  - 30 c)  an asymptotic expansion i n powers of valid f o r large  $  f  which i s  •  Each o f t h e s e s o l u t i o n s i s t h e n used t o p r o v i d e an approximation for  An  i n terms o f  ? •  The convergent t h a t approximation t o  s e r i e s i s used t o p r o v i d e e s s e n t i a l l y An  found by M i c h e l s , de Boer and  The W e n t z e l - K r a m e r s - B r i l l o u i n s o l u t i o n i s  cosn(n-p)(-iA( ^ ^)  t sinncn- )(A( ,^  e  ( 3 5 )  S  P  6 ( s l  ?  )  where  AW  feai  -  | T p l p + » ) - ¥ n y +• j J " * x  and  { n - p * l » * ^ } *• x  Then p u t t i n g  A H *  $ S s  0  and equating (35) to zero we  £t*ATTU-P>  F o r the lowest s t a t e  ~  Ln. •  Tir*  (p = 0 ,  n  get  88 0  l)  t h i s reduces t o  Bijl.  -'31  An ^  The  asymptotic  by the author  1111211  •  #  - Z c o s K W t j . - i ) )  ( f R)  expansion of  c o snCn-P)*  put  z e r o , we  n ^ i t  ?  i s given  as  * sM | - ifill^^li....] ?  ,  now  ,  f o r large  Jj(apTann-P-i)isinin«i-p»wi** s-  I f we  A  (1*(J;-T-)±  < * p -  ±  -  $ = $  0  , n =  n  and  J l•  ' " • P ^ + ' l  .  »...}  equate the above t o  obtain  f  l  « m « - P ) i  -  K-p-im«.«rM  I n t h e case where  p = 0 , n  I  Q  = 1  ^ Q T P ) ( n , - f - » ) / f , . . .  ' -  #  | +> ( n, - ?) (n,+ p+«)/j « #  , t h i s reduces t o  tii.-*  t=0  The  author a l s o d i s c u s s e s two  an e x p r e s s i o n f o r and zero  An  when  S  0  other c a s e s .  i s such t h a t  He f i n d s E  he uses t h i s to f i n d t h a t t h e smallest: e i g e n v a l u e (when  p = 0)  i f t h e r a d i u s of t h e  enclosure  is  0 is  ,  r .o  1.8354  .  f o r m u l a f o r use when  $  He a l s o w r i t e s down P r o e l i c h * s ( / t ) i s near a zero o f the u n r e s t r i c t e d  0  solution. III.  Further  Results.  I n s e c t i o n 1 of t h i s p a r t we w i l l d e v e l o p an approximate e x p r e s s i o n Froelich.  for  & X  which i s s i m i l a r t o t h a t o f  We w i l l t h e n prove t h a t , u n l i k e F r o e l i c h ' s , our  approximation t o  A X  i s v a l i d near s i n g u l a r as w e l l as.  ordinary.; p o i n t s o f t h e d i f f e r e n t i a l  equation.  In s e c t i o n 2 we w i l l g i v e t h e asymptotic for  aX  expressions  f o r 'the harmonic o s c i l l a t o r , t h e r i g i d r o t a t o r , and  the hydrogen atom problems. 1.  G e n e r a l Procedure The  e q u a t i o n s a t i s f i e d by t h e s o l u t i o n t o t h e  u n r e s t r i c t e d problem i s of t h e form  u" + (91x1  ( 3 6 )  where one c o n d i t i o n on  u  f  A)U-0  i s t h a t . u ( 0 , X) = 0  the other boundary c o n d i t i o n i s u ( x ' , X) = 0 , x where  x  f  or 1  >  u ( 0 , X) = 0 T  0 ,  may or may not be a s i n g u l a r i t y o f t h e equation;  the o r i g i n (as i n t h e hydrogen atom problem) may be a s i n g u l a r point.  A l l points  equation.  Let  problem, where  x, 0 < x < x  y ( x , X + t X)  T  are ordinary points of the  be t h e s o l u t i o n t o t h e r e s t r i c t e d  y(xo, X + & X) = 0 , t h e c o n d i t i o n on  A X) a t the o r i g i n being t h e same as t h a t on  y(x, X + u(x, X) •  y ( x , X + A X)  Then y  M  s a t i s f i e s the equation  + ( q u ) + A+AA)y=o  ,  which may be r e w r i t t e n as  y " + lgu> U ) y = -*Xy .  (37)  By v a r i a t i o n o f parameters i t i s easy t o convert  equation  (37) i n t o t h e i n t e g r a l e q u a t i o n x (3$)  y(x) = uix)~ ~  j (vix)ult>-uux>vi*)J y ii\ dt o  where  u ( x ) i s the s o l u t i o n t o t h e u n r e s t r i c t e d problem,  and where  v ( x ) i s a s o l u t i o n o f e q u a t i o n (36)  I n (38) l e t us put x = x for  AX  ; since  y(x ) = 0 Q  we c a n s o l v e  to get  . \ (39)  0  such t h a t  W U (Xo)  A A =  V(x,)J "uit)yit)dK  " uix.i  0  J\wy(-t>c|t  .  0  E q u a t i o n (39) i s exact, b u t , o f course, we don't know y ( t ) . However, i f x  0  i s c l o s e enough t o x  1  , u(t)  should be a  y ( t ) (0 < t < x ) .  good approximation t o  Q  d e f i n e the f o l l o w i n g approximation t o  (40)  -  x  =  Vix ) 0  J  A X :  w  uMOtitt -  •  u(Xo)  "  j**v<t)uU)alt  *  .  We would now l i k e t o show two t h i n g s ; t h a t as  x  0  -*• x  A X  a X.  , and t h a t , i n t h e asymptotic approximation, we x  1  may n e g l e c t t h e term denominator  L e t us, t h e n ,  u(x ) • \ Q  of ( 4 0 ) . I t w i l l  i)  s u f f i c e to show t h a t  o » Xp  /•X.  VlXoij ^  i i i .)  i n the  \ u<*)y<*)dt * \ V i t i J t 0  ii)  v ( t ) u ( t ) dt  uM-udt - U(X«) I v ( * ) w ( t ) t J t  V(x ) J «  u K)dt v  0  v i x > j u«t) yct)4t 0  o  Vw l X ) i  - - . | t  J V(i)<At X  - uix.)( v K ) y ( * ) d t  .  p  To s i m p l i f y the following- d i s c u s s i o n , l e t us now c o n s i d e r o n l y the harmonic o s c i l l a t o r , and i n d i c a t e any changes necessary f o r t h e treatment Then i n e q u a t i o n ( 3 6 )  and  u ( x , X)  i s o f the form  of t h e o t h e r  later problems.  35 -  7  where  P«(x) '-  n 21  a  t  3^ ;  we may assume t h a t  a  n  > 0 .  k=0  L e t us f i r s t prove i ) , i i ) and i i i )  i n t h e case t h a t  x  T  - + oo  y ( x , X ) i s o f the form  We proceed as f o l l o w s .  K>n*i  where  GlnUl r £ b X* K  and  b  k  ^  0  when  k  >, n + 1 i f X Q i s l a r g e enough ( i t can  be seen f r o m e q u a t i o n ( 1 8 ) A  X  <  t h a t i f X Q i s l a r g e enough,  2 ) .  Now, t o show ( i ) u s i n g t h e Lebescjue bounded convergence theorem we need  and t h i s i s t r u e f o r each f i x e d  x , s i n c e any s o l u t i o n o f  the o r i g i n a l d i f f e r e n t i a l e q u a t i o n i s , f o r each f i x e d an a n a l y t i c f u n c t i o n of  x  .  0  To show t h a t  \u(x) y ( x ) \ i s  bounded by an i n t e g r a b l e f u n c t i o n , we i n t r o d u c e y*(x).  -  o ,  x , n .  x ,  the f u n c t i o n  - 36 .Now,  -  i t i s e a s i l y shown t h a t  for a l l positive l a r g e enough.  |y*(x)|  x , independently of  x  0  i s bounded  , i f x  0  is  We proceed to do t h i s as f o l l o w s . oo  and  therefore  £ <•**•" $ I I M X. . K  Now,  if  0 < x < x  Ib x  0  , then  I_b«x  K  K  fc e  (42)  Equations  (41)  and (42)  t o g e t h e r imply t h a t  |y*u)| « 2iAU"  i x X  £  ib »x* K  Kro  for a l l positive not depend on  x  x ; t h a t is?, f o r some f i n i t e 0  if x  0  |Y*U)| 4 M  i s l a r g e enough,  }  x>, o .  M  which does  - 37  -  We have shown, then, t h a t where  )u(x)|  |u(x) y * ( x ) | < M | u ( x ) |  i s i n t e g r a b l e from zero t o i n f i n i t y .  Lebescjue's  qonvergence theorem s t a t e s that j uu)y*(x)dx  ^  juMxjdx  O  as x.-»°°;  p  c l e a r l y , then, f  Xe  To prove ( i i ) we need o n l y prove t h a t  V(x *M u ( x )d x - > t « ° <•)) w (x)  cis x--»  l l  while  x. x'^oo I  w u , )  )  v^x)uix)dx I <  The f i r s t of t h e s e statements i s e a s i l y proved. Let  a  be some number l a r g e r t h a n t h e l a r g e s t zero o f u ( x ) .  For  a < XQ  we can w r i t e  v(x ) Q  V ( X . ) = U(X.> \ J  i n the form  U»IX)  •  Since U <(x) X> =  it  A * i^PnCX)  i s easy to show t h a t •» xS" V(x ) ^ (  o s x,-»  -  Therefore,  38  -  since ) uMx)dx ?  VU.l ) U ( X ) d x v  0  -> *  0  X -S>oo.  OS  0  E  0  To prove the second statement we note t h a t  sinee  i v K  VU) ^  ^ 2 —  os *-»«> <*  9  \ vtx)ucxjdx -  \vujii(x>elx v  + \v<x)u(x)dx  (vtx)mx)dx  t  X„  \  ^i x  )  dx  as *.->*>.  Now,  5^*,  u<x.) (  and  d x -» o  as  since .«<  j  I if*  X.-»«o  j vix)U(x)dx 9  j  <  0 0  MIX.) I V ( x J M ( X ) d x '  =0  .  C l e a r l y , t h e n , we have proved ( i i ) . The p r o o f of ( i i i ) i s s i m i l a r t o t h a t o f ( i i ) . have a l r e a d y shown by ( i ) t h a t  We  - 39 -  j uix)yu>o(x and  i n proving  *  ( u*(X)olx  a s x -> oo 0  t  ( i i ) we showed t h a t f * x  V ( X . ) \ U ( X ) o l x "* t  •*  l  *• ~*  0 0  y  therefore  V i x ) I u i x ) y ( x ) c l x -* * oo  «s  e  A l s o i n the course o f p r o v i n g  where  M  does not depend on  x  * . - » «*> .  ( i ) we showed t h a t  i f x  Q  i s l a r g e enough.  Q  Therefore ly(x)| $ M'luix)!  (  «< x s x N  0  J  and so  W(x ) 0  j vix)yix)olx|  $ | u i X o ) 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  It i s clear that  uu.) j v u ) y u)oix -> o o s x. -> <> * ,  |  -  40  -  and M'uu.) ] | V(x)u(x)|olx  and  -*o  qs x  t h i s was a l l t h a t remained to be done t o complete the  proof of ( i i i ) .  J  We have shown, then, t h a t  and  also that  I f we now put vix)*  it  u(x)  i s easy t o show t h a t  (* J UMt)  ,  '  W = 1 ; i f we a l s o choose  A  such  that  1 ) U (X)fllX = I j l  our  approximate e x p r e s s i o n  for  A \  reduces t o t h e simple  form  )  where, o f course, u(x)  .  a  uMx)  i s l a r g e r t h a n t h e l a r g e s t zero of  - 41 I t i s c l e a r now, t h a t s t e p s ( i ) , ( i i ) and ( i i i ) are a l s o e a s i l y proved when  x  1  (where  u ( x ) = 0) f  ordinary point of the d i f f e r e n t i a l equation.  i s an  In this  case  we f i n d t h a t A \  ^  as  x, -» x  1  and t h a t as  if  u  i s nomalized «  x„-»x*  so t h a t  \ u ( x ) o l x = \, l  0  I n t h i s case, and  x  1  ;  a  u(a) ^ 0 , and no zero.', o f  u  may be g r e a t e r than o r l e s s than  separates x  1  a  .  The p r o o f s o f the p r e v i o u s r e s u l t s may e a s i l y be adapted t o the o t h e r problems i n w h i c h we a r e i n t e r e s t e d . The p r o o f s a r e v i r t u a l l y unchanged i f we a l l o w u(x)  to be o f the form  U(X>-  A-Jt^PrtU)  ?•'  >  K>o,rv\>o*  J  our r e s u l t s , then, a r e v a l i d f o r t h e bounded hydrogen atom. The r i g i d r o t a t o r problem i s d i f f e r e n t The  e q u a t i o n t o be s o l v e d i s of t h e form  i n two ways.  42 -  where  h ( x ) f 1 , but i s continuous i n t h e open i n t e r v a l  under c o n s i d e r a t i o n ; and t h e range o f t h e independent v a r i a b l e i s from zero t o t h e s i n g u l a r i t y a t For the r i g i d  x = 1 (i.e. ©  rotator  The s o l u t i o n s o f the u n r e s t r i c t e d problems a r e o f the f o r m  where  P ( )  I s t h e n t h Legendre p o l y n o m i a l .  x  n  A X. to be  I t i s e a s i l y shown t h a t i f we d e f i n e  *x  -  wm*.>  ,  V(x„) Jv»tx)uMx)dx - Utx.)j MX) V(x)M(x)otx o  then  6> X  o  AX  as  x  0  -> x  not be g r e a t e r t h a n u n i t y . )  f  .  (Both  uMx)  where, i n t h i s case x  0  We can a l s o f i n d  Feu  J  x  1  j h l x ) u M x ) d x = I.  and that  x*  must  43 2.  -  S p e c i a l Cases I n t h i s s e c t i o n we  a p p l y the r e s u l t s developed  above to f i n d the asymptotic f o r m u l a s 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 t h e hydrogen atom problems.  The d i f f e r e n t i a l  equations f o r the t h r e e  problems are found i n S c h i f f (£, Ch. 4);  i n the  derivation  o f t h e asymptotic formulas, t h e necessary summations and d e f i n i t e i n t e g r a l s are t o be found i n S c h i f f (£, Ch. and i n Whittaker and Watson (12, i.  The  Ch.  15).  Harmonic O s c i l l a t o r The  differential  2m  ol-r*  e q u a t i o n f o r the wave f u n c t i o n i s  x  CT*U  = E U  and the f o l l o w i n g changes of v a r i a b l e  c o n v e r t t h e e q u a t i o n i n t o the form  *( ( t - i » M • * ) « « < » . The  e i g e n f u n c t i o n s o f the unbounded problems are  5i: h where  4)  H (t) n  i s the n t h Hermite p o l y n o m i a l .  - 44 u(+ x , X ) = 0  I f we i n s i s t t h a t  Q  , i t i s easily-  shown t h a t  X » n +  where ii.  A X .  The R i g i d Rotator The wave f u n c t i o n  -L d ( sine £ & \ sine ote V Xe*  I f we put  x • cos 9  s a t i s f i e s the equation  ( \ . j £ \^ V* si**-* '  + T  and  -  u(x) - s i n 9 •  0 u  (9)  *  this  becomes  s*ls* • ( JL  * )u - o. x  +  The s o l u t i o n t o t h e u n r e s t r i c t e d problem i s  uu)  where  = U-x*)*Pf(x) = U-x ) ^' 1  Pn( ) x  2  j  ^  U  )  i s t h e nth Legendre p o l y n o m i a l , and where  \ - n ( n +1)  >  Upon i n s i s t i n g t h a t find that, f o r m • 0  n ^ m>, u(+ X Q ,  o  ^  X ) • 0 , we c a n e a s i l y  - 45 -  where  X • n(n+l) +  ft  5  Sdfi 2  a X •  f  M  where  r -  where  X = n(n+l) +  0  r  n  " ** 2 m  W  1 t  For  m >, 1  we f i n d  that  U n - l K ) !  whichever i s an i n t e g e r , and  A X .  We note here t h a t <} A ~  -I  o<  r»» s o,  which i s i n agreement w i t h t h e graphs g i v e n i n s e c t i o n 3 , part I I . iii.  The Hydrogen Atom The d i f f e r e n t i a l  the wave f u n c t i o n  equation f o r t h e r a d i a l part of  R is  and t h e changes o f v a r i a b l e  -  convert  t h i s i n t o the  2pi The  +  46  -  equation  U - £i*i!> + X)IA = o.  s o l u t i o n s t o the unbounded problemr  where  X - —ir n*  and  L  i s an a s s o c i a t e d Laguerre  Insisting that \  JL  0  a w  X ) = 0 , we  u(xo, X  2  are  polynomial,  f i n d that  *  =  as  x„ -> <*>  n*"* in-p-i)Kntpj! 3  where and  X = - — p = 0  9  +  AX  .  For the ground s t a t e where  t h i s reduces to  x\ = 1  BIBLIOGRAPHY (1) .  F.C. Auluck and D.S. K o t h a r i , P r o c . Camb. P h i l . S o c , 41  (1945),  175.  (2) .  S. Chandrasekhar, A s t r o p h y s . J . , SI  (1943),  263.  (2.).  R.B. Dingle, P r o c . Camb. P h i l . S o c , A £ ( 1 9 5 3 ) ,  (AJ.  H. F r o e l i c h , Phys. Rev., j & ( 1 9 3 8 ) , 945-  (£).  S.R. de Groot and C.A. t e n Seldam, P h y s i c a , 12  103  (1946),  669.  (6).  A. M i c h e l s , J . De Boer and A. B i j l , P h y s i c a , 1± ( 1 9 3 7 ) , .981.  (1950),  245-  (J).  W.E. M i l n e , J . Research U.S. Bur. Stand.,  (8).  M. Rauscher, J . Aero. Sc., 16 ( 1 9 4 9 ) , 3 4 5 -  (£).  L . I . S c h i f f , Quantum Mechanics (McGraw-Hill, 1 9 4 9 ) , C h . 4 .  (1O0.  C.A. t e n Seldam and S.R. de Groot, P h y s i c a , IB ( 1 9 5 2 ) , 8"91.  (11) . A. Sommerfeld (1940),  and H. Hartmann, Ann. £ . Phys. (5) 37  333.  (12) . E.T. Whittaker and G.N. Watson, Modern A n a l y s i s 1915),  Ch. 15.  (Cambridge,  

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