Cop. ON THE QUANTUM MECHANICAL PROBLEM OP A PARTICLE IN TWO POTENTIAL MINIMA by David Southard Carter A thesis submitted i n p a r t i a l f u l f i l m e n t of the requirements f o r the degree of MASTER OF ARTS i n the department of PHYSICS The University of B r i t i s h Columbia i A p r i l , 1948 I ABSTRACT The problem of a p a r t i c l e i n two adjacent onedimensional rectangular, potential "boxes" i s an exactly soluble representative of a class of two-minima problems of considerable physical interest which have not been solved exactly.. I t therefore affords a valuable opportunity f o r a c r i t i c a l examination of the extent of a p p l i c a b i l i t y of perturbation theory methods to such problems. An exact im- p l i c i t solution of the problem i s obtained, and i s reduced to e x p l i c i t approximate form i n two important special cases. These approximations are reproduced by perturbation theory methods, and t h e i r ranges of v a l i d i t y are demonstrated by comparison with the exact solution. The application of the model to a physical system i s demonstrated by using the ident i c a l two-box problem as a basis f o r c a l c u l a t i o n of some constants of the ammonia molecule. AGKNOWLED GEMENT The author wishes to express h i s deep appreciation of the many hours of patient and encouraging guidance given him by Professor G. M. Volkoff who suggested the problem and directed the research. The author also wishes to-thank a l l those members of the faculty i n the Departments of Mathematics and Physics who have taken such kind interest i n the author's progress as a student and have helped him i n many ways. s F i n a l l y , the author wishes to express his indebtedness to the National Research Council f o r t h e i r award to him of a Studentship under which t h i s research was conducted. TABLE OF CONTENTS Fag© I. INTRODUCTION 1 II. DIRECT SOLUTION 1. Formulation o f the Problem . . . . . . . 2. S o l u t i o n o f the Problem i n I m p l i c i t Form 3. P r o b l e m A, The S i n g l e - B o x A Case . . . . 4. P r o b l e m B, The S i n g l e - B o x B C a s e . . . . 5. P r o b l e m C, I n W h i c h t h e Two Boxes a r e Far Apart 6. P r o b l e m D, I n W h i c h One Box i s S h a l l o w . 3 4 6 10 10 17 I I I . PERTURBATION THEORY SOLUTION 1. Standard P e r t u r b a t i o n Theory Treatment of Problem D . . 2. S p e c i a l P e r t u r b a t i o n Theory Treatment of Problem C 19 IV. DISCUSSION 27 V. APPLICATION OF THE MODEL TO THE AMMONIA INVERSION SPECTRUM APPENDIX A - N o r m a l i z a t i o n o f t h e E i g e n f u n c t i o n s o f t h e Continuum . . . . 21 '30 34 APPENDIX B - D e r i v a t i o n o f E q u a t i o n s ( 3 5 a ) a n d (35b) 38, APPENDIX C - C a l c u l a t i o n o f t h e C o e f f i c i e n t s E, and 6 by Means o f E q u a t i o n s ( 4 3 ) and ( 4 5 ) 40 BIBLIOGRAPHY' 44 A LIST OF PLATES Opposit PLATE I . The Two-Box P o t e n t i a l F u n c t i o n and an Example o f an Energy L e v e l System f o r Problem C PLATE I I , A Rough Sketch o f ^ as a F u n c t i o n of. £ . . . . PLATE I I I 3 7 The Eigenvalues o f Problem D as a Function o f X 27 PLATE I V *- . . . . . 28 PLATE V Comparison o f the Square W e l l Model and Manning's Model With the Ammonia Spectrum. . . . . 30 as a F u n c t i o n o f £ 1. ON THE QUANTUM MECHANICAL PROBLEM OF A PARTICLE IN TWO POTENTIAL MINIMA I. INTRODUCTION The problem of a p a r t i c l e i n two potential ininima i s of extensive interest i n t h e o r e t i c a l physics since i t provides a model f o r many physical systems. The simple one- dimensional case i n which the minima are rectangular i n shape serves as. a prototype by which we may understand many phenomena conneoted with metallic conduction ", van der Waals 3 forces^, the s t a b i l i t y of hydrogen-like i o n s , and the v i b r a 2 t i o n spectra of c e r t a i n polyatomic molecules^. For t h i s reason many authors, including those mentioned i n the footnotes, have discussed the model with a view to i t s physical significance. Manning and M.E. B e l l , Rev.Mod.Phys. 12, 215 (1940). ^S, Dushman., "Elements of Quantum Mechanics", (Wiley), pp. 214-218, and references given there. Dushman s approximation to the energy s p l i t t i n g i s incorrect (compare his_ equation ( 3 ) with our equation (26a) f o r c = o» and \ - 1 ), since he f a i l s to consider the phase s h i f t o f the eigenfunction. 1 ^See Part V below, where the present model i s applied to the ammonia-inversion spectrum. 2. It is felt, however, t h a t none o f t h e d i s c u s s i o n s have t a k e n f u l l o f the c o n s t a n t l y u s e d i n quantum m e c h a n i c s . h a n d , t h e p r o b l e m i s one be a d v a n t a g e o f 'the p o s s i b i l i t i e s p r o b l e m i n i l l u s t r a t i n g many m a t h e m a t i c a l m e t h o d s which are may s o l v e d by t i o n may be exact carried o f few methods. out, with trate The by the direct the nature solution of the pedagogic u t i l i t y On the certain may The generally a type discussed other t h e n be solu- limitations, employed t o c o n t i n u u m , and that ob- illus- results^* d i s c u s s i o n i s e n h a n c e d by o f perturbation theory i n the one complete knowledge perturbation theoretical o f the the hand t h e significant n e c e s s i t y o f u s i n g wave-functions o f the dealing with On n o n - t r i v i a l examples whioh by means o f p e r t u r b a t i o n t h e o r y . tained published is the of not literature. P.M. M o r s e and E..C.G. S t u c k e l b e r g , Helv.Phys.Acta, 4, 337, (1931)i m a k e . t h i s k i n d o f i l l u s t r a t i o n u s i n g a m o d e l f o r ammonia. However, t h e y c o n s i d e r a l e s s g e n e r a l c a s e , and t h e i r m o d e l i s more c o m p l i c a t e d . ^ PLATE I T h e T w o - b o x Tbtential Function Voo, and a n example o f a n E n e r g y L e v e l System -for Problem C BOX A BOX 6 2b 2C — EM, REGION 1 Legend: REGION 2 REGION 3 REGION A REGION 5 •The levels of problems A a n d B are d r a w n i n the appropriate B O X •The Eoo.n ° f problems A a n d B a r e d r a w n in the a p p r o p r i a t e B o x - T h e l e v e l s of p r o b l e m C a c c o r d i n g -to e q u a t i o n s ( 2 3 ) a n d ( 2 4 ) 3. II. DIRECT SOLUTION 1. . ITOHMUXATION OF THE PROBLEM We consider the problem of the one-dimensional motion of a p a r t i c l e of mass jx Vu^ subject to a p o t e n t i a l shown i n Plate I, and seek i t s bound energy l e v e l s together with the corresponding eigenfunctions. vanishes outside two p o t e n t i a l "boxes" A and B ( Vex) V(x^=o i n regions 1, 2> and J5) and has the constant values -1? and - inside boxes A and B respectively ( V(x) « - U gion 2, V w = -\vj i n re- i n region 4 ) . We s h a l l f i r s t solve the general problem i n imp l i c i t form and then consider the e x p l i c i t solutions of the following special cases: Problem A, i n which the width of box B i s zero ( C-O ) or the depth of box B i s zero ( \ - 0 ) and only box A i s present. Problem B, i n which the width of box A i s zero, and only box B. i s present. Problem C, in. which the distance 2,t» between the boxes i s large. Problem D, i n which box B i s much shallower than box A ( X i s small compared to u n i t y ) . 2. SOLUTION OP THE PROBLEM IN IMPLICIT FORM The eigenf unctions cflx) s a t i s f y the equation which i s the Schrodinger equation multiplied by « - S-ir -^/^. (2) 2 If ^ i s the expression f o r a bound state ( 6 < O ) eigen- function i n the region, the solutions of equation (1) are (3) where p .V ^ 7 > 0 V°- e<o ; ( 4 o ) so that The r \ i , ^ , and are constants to be determined together with a condition f o r eigenvalues by the boundary, conditions at i n f i n i t y (where (p must not be i n f i n i t e ) and at the boundaries of the boxes (where <p and i t s f i r s t derivative <f' must be continuous). that ^, and ft s The conditions at i n f i n i t y require s h a l l vanish, so that equations (3) rewritten with some changes i n the constants as may be 5* cp, - ^ o ^ U - ^ e Cp •= t\ Cos z _ r 1 J ( 6 a ) a. (6b) t\ cose* Co.* ^ r (S(x-a^) 0 (^^- n a (6c) 1 <D i = , L ^ .1 e (6d) - & cos*' ( x - a - z b - c -?,') •• (be) where the f i r s t three and the l a s t three expressions make up functions which are continuous at the boundaries of boxes A and B respectively. The two forms of cp_ have been chosen . to emphasize the symmetrical way i n which the two boxes occur i n the problem. Applying" the condition that <p' s h a l l be ©ontinuous at the boundaries of box A to the f i r s t three of equations (6) we obtain P which determine /ot = % and W-Ua**) as functions of <x and consequently ( i n v i r t u e of equations ( 4 ) ) of E . . 7 (I, and as i m p l i c i t functions Thus elimination of & between equations (7a) and (7b) y i e l d s f o r * ( b) t 1 %^ : rS^J = VS XT • (6) S i m i l a r l y we obtain from the l a s t three of equations (3/*' « *a.woc' ( c - ( ) } = _LL*JL w c c * C ) - 7 c f ( I f we now i n s i s t that the two forms of <P d ) in 3 equations (6c) 7 and (6d) represent the same function we ob- t a i n the condition ^ - 1* (9) ) where e = e'*P* ( 1 0 ) Equation (9) i s c l e a r l y a condition f o r eigenvalues.since i n accordance with equations ( 4 ) , ( 8 ) , and (10) i t i s an imp l i c i t equation i n E and the constants a. , b , c , ^ X of the potential energy function. , and I f the values of E- s a t i s f y i n g equation (9) f o r a p a r t i c u l a r set of values of the constants are found, say by numerical methods, then and may be found from equations (8), and ^ and be found from equations (7). ^ may Thus the equations we have found' give us complete information about both the eigenvalues and the eigenfunctions i n i m p l i c i t form. 3. PROBLEM A, THE SINGLE-BOX A CASE I f c or ^ i s zero and box A alone i s present, i t i s clear that the eigenfunction i n the whole region to PLAT E H A rough sketch •for t h e c a s e The Zeros Notice t h a t of the X A slope of t in which a r e the S i n g l e - b o x is N e g a t i v e a t K as a A< Function of Levels each E 32yja \J/h <9 2 Zero the right of box A may be represented by the function cp_ of equation (6c). at xs+co The condition that <P s h a l l not be i n f i n i t e now implies that ^ - O , (11) whioh with equations ( 8 a ) and ( 4 ) i s the condition f o r eigenvalues. ^ has been roughly sketched i n Plate II as a function of E f o r the case i n whioh V 9 l i e s i n the range . ^(y(e') i s r e a l only f o r i s positive for a l l E , approaching •<*> as Af< S2}Lo< TJ/£: z < In the region of bound energy l e v e l s (-\J< g E<0 and E-*--co . ) i t s zeros are separated by i n f i n i t e d i s c o n t i n u i t i e s at the points- 1 E»,w= As.E each . increases, the sign of Eto,vv. a n < i from ^ *v_.i ) 2 > 3 changes from •*- to - at each zero. , (12) - to * at The zeros, when placed i n increasing order, are a l t e r n a t e l y zeros of the f i r s t and second factors of The eigenfunctions ({^ l e v e l s of problem A may ^ i n equation ( 8 a ) . corresponding to the bound be obtained by s e t t i n g ^ =O equation (71») and eliminating (_. by means of equation ^•As in (7a): "vl—>PO each zero approaches the E.OD,W which l i e s immediately above i t . Thus the may be though of as the l e v e l s of an i n f i n i t e l y deep box with base at - V . In order to avoid extra notation we s h a l l use the symbols (i to denote either the functions of E. defined i n equations (4) or the special values of the functions corresponding to eigenvalues of EL . fc 8. where X i s an even or odd integer according as the f i r s t or second factor of ^ t u t i o n of equation ^ _ O (8a) i n equation (13) vanishes. into equations (6) Substi- after setting i n equation (6c) shows that the bound state eigen- functions are a l t e r n a t e l y multiples of the even and odd func- tions <fc* = c o s ** (14a) and 0>(x -vcO <M v '** (14b) x ) according as the f i r s t or second factor of In both cases i t may %\ vanishes. be shown that ,•00 which determines the normalizing factor. observe with the help of equation (8a) For l a t e r use we that f o r the even functions cos-** -. °cV(** + (3 } - oc^/tAV 2 2 > and f o r the odd functions The eigenfunctions of problem A f o r the continuum of free energy states ( E > O) must also be found since we w i l l require a complete set of eigenfunctions i n applying perturbation theory. Setting ( 1 5 d ) 9, we f i n d that outside box A , (p no longer has the exponential form of expression (3) but that i t has the o s c i l l a t o r y form RwsVUi-^ . cp Thus the condition that s h a l l be bounded at i n f i n i t y i s automatically f u l f i l l e d and we need impose only the four continuity conditions at the boundaries of the box. I t i s clear that equation (1) and the con- t i n u i t y conditions are s a t i s f i e d f o r every, E > O by the following l i n e a r l y independent p a i r of functions and of which the f i r s t i s even and the second i s odd: •J "XT (17) where $>* and are determined by the equations and where <x and -U are defined i n equations ( 4 ) and (16). Further, we obtain a complete set of eigenfunctions by i n cluding only the Cf)£ , since f o r given E , any three solu- tions of the second order d i f f e r e n t i a l equation (1) and the boundary conditions are l i n e a r l y dependent. The f a c t o r . \ / / T F i n equations to make the dix A). normalized (17) has been chosen to "delta-functions" (see Appen- 10. 4. PROBLEM B, THE SINGLE-BOX B CASE I t i s c l e a r from the symmetrical way i n which boxes A and B enter the problem that a l l the r e s u l t s derived i n seotion 3 f o r problem A may be transformed into corresponding r e s u l t s f o r problem B by making the following substitutions: <x »- c *v "VJ a, *~ tx' x >- x - zfe - c %• % ^ \—- and leaving a l l other quantities We s h a l l denote by unchanged. C^ B the bound-state eigenfunc- tions found i n t h i s way from equations (14). 5. PROBLEM C, IN WHICH THE TWO BOXES ARE FAR APART We return now to the two-box problem and consider the oase i n which the distance For a r b i t r a r i l y large b , e large. correspondingly small. E. V> , every such E or of is. I t follows that every two-box i s a r b i t r a r i l y close to a l e v e l or B (a zero of equation (see equation (10)) i s Hence the value a r b i t r a r i l y close to zero. level £Y> between the two boxes i s ). E 0 of problem A Indeed, f o r s u f f i c i e n t l y large may be approximated by a solution of the 11. where ^ = > (20a) d6 2 V-KE. b> (20c) Two possible cases now present themselves. f i r s t case, which we s h a l l c a l l "non-degenerate", E In the e is a l e v e l of one single-box problem but not of the other. the second case, which we s h a l l call."degenerate", l e v e l of both single-bos problems- -. 1 In E o is a Since equation (9); i s symmetric i n and ^tj, we need only discuss those cases i n which i s a l e v e l of problem A E „ = E i \ ( ( V ( e ^ - O J. ft In the "non-degenerate case equation (19) becomes where the functions and derivatives are evaluated at E The sign i n the denominator must agree with the sign of i n order that JUw*. E r • When b . ^ i s so large that In our special use of the word "degenerate" we r e f e r to a case i n which there i s a simultaneous l e v e l of two d i f f e r e n t problems rather than one i n which t.here i s more than one l i n e a r l y independent eigenfunction belonging to a single l e v e l of the same problem. 1 2 « the approximation reduces to E-e <L /<\ = A Z . (23) In the degenerate case we merely set ^ =^ 1 = O i n equation (19) and obtain E - V . W J W ( 2 4 ) H^ER) may be found by d i f f e r e n - The derivative t i a t i n g the expression (8a) using equations (4) and (5) and the f a c t that ^ ( E ^ = 0. Similarly for a level E The r e s u l t i s of problem B, a - VS»\V(^(icV2oc'*f3 . (25b) a The l a s t two equations show that the derivatives ^ and which appear i n equations (23) and (24) are always negative. i n equation (23) i s always opposite Hence the sign of ' E-E<* to that of ^&(e<^ and the r i g h t hand side of equation (24) ^Condition (22) ensures that E . . i s much closer to E than to the nearest l e v e l E of problem B; i . e . , there.is no approximate degeneracy. Eliminating e between equations (22) and (23), we have f l f t |E -E so that even i f E X and we have t t f t \« W*/4^ v i s so close to E ^ that * E „ ^ 4 UO * ^U*)= O , 13. i s always r e a l . Stated i n words, equation (24) implies that any l e v e l which i s common to both single-box problems (or to the two-box problem when the boxes are i n f i n i t e l y f a r apart) becomes s p l i t into two l e v e l s when the boxes are a large but f i n i t e distance apart. One of these l e v e l s l i e s above and the other l i e s below the o r i g i n a l l e v e l , both by the same amount of order £ . On the other hand, equation (23) im- p l i e s that each l e v e l which does not approach a l e v e l of problem B as t» —*. oo , d i f f e r s from the corresponding problem A level -U only by an amount of order or below , and l i e s above i n accordance with the following r u l e : The l e v e l i s "repelled" by the "closest" l e v e l of the other single-box problem, where we define the problem B* l e v e l which i s "closest" to• E * not separated from to the Ea>,w E^ to be that l e v e l which i s E«> by an of problem B (analagous |A of problem A given by equation I f the o r i g i n a l l e v e l (12)). l i e s "equally c l o s e " to two problem B l e v e l s ( i . e . , i f i t coincides with an E_,,n of problem B ) , equation (23) shows that since the problem C l e v e l E of order €_ . 2 coincides with ^oCEeo.^sCo , E*. a t least to terms Moreover, when we write the exact equation (9) i n the form cides exactly with , i t i s c l e a r that , since E, coin- "o^l^" O . 8 The above r e s u l t s are i l l u s t r a t e d by the sketchedi n l e v e l s . i n Plate I , where the boxes should be farther apart than they are i n the sketch. The lowest l e v e l l i e s below 14.. Ec^o since i t i s "repelled" by the higher l e v e l s i m i l a r l y the next lowest l e v e l l i e s below "repelled" by E«^ . E a o E , and f t o since i t i s The t h i r d level-coincides with since the l a t t e r coincides with E ,, a E„ of problem B. t The highest levels l i e above and below the coincident l e v e l s and • Making use of equations (8) and (2.5) we rewrite equations (23) and (24) i n e x p l i c i t form i n order to compare these results with those of perturbation theory. Thus f o r degenerate l e v e l s we have: E - E, - + —-===L== ' ^/TTrrc^TTiTrjrr and f o r non-degenerate l e v e l s : E . £ = 2fc z * {26a) \«*CV ,(26b) We s h a l l now discuss the eigenfunctions of problem C. Again we need only discuss those cases i n which the problem C l e v e l E l i e s close to a problem A l e v e l E ^ these cases the values of & . In determined by equations (7a) and (7b) d i f f e r l i t t l e from those given by expression (13), and the f i r s t three of equations (6) d i f f e r l i t t l e equations (14). Thus i n the neighborhood from of the box to which the unperturbed l e v e l belongs, the two-box eigenfunct i o n d i f f e r s l i t t l e from the corresponding single-box eigenfunction. To determine the nature of the perturbed eigenfunct i o n near box B we f i n d the r a t i o \ fc/Rl of the amplitudes of the o s c i l l a t o r y parts of the eigenfunctions inside the' two 15. boxes. F i r s t we equate expressions ( 6 0 ) and ( 6 d ) at and eliminate fe by means of equation (9); then we "x=<\ simpli- fy by means of equations ( 5 ) , (7), and (8$ to give:. In the non-degenerate case we may t 0 g i v s/ e ~ ^ 1 use equation (9) /V so that which shows that the r a t i o of amplitudes of the eigenfunction i n the regions of boxes B. and A i s of order 6 and henoe the chance of f i n d i n g the p a r t i c l e near box B i s of order e * compared to the chance, of finding i t near box A. In our solution of problem 0 by perturbation theory, we s h a l l take the problem A eigenf unction C|> to be the unA perturbed eigenfunction corresponding to the perturbed eigenfunction dO belonging to the problem C l e v e l E clear from equations (14) and (28) that both of order £ hear box B.. . <^ It is and Thus the difference <j> A are between the perturbed and unperturbed wave-function i s of the same order of size as the unperturbed wave-function i t s e l f ; f a c t whioh w i l l be very troublesome a i n our application of perturbation theory. In the degenerate case we obtain with the help of equation (24): » U-E^ACE*) « t € Ai/^T . ( ) 29a 16. and similarly: Further, since expressions (8) are almost zero, we obtain with the help of equations [5): 1 V,nZ*o. I « 2*0/tVU , \ S»YV£*'C\ «2ot'p/rtAU (30) (30) into equation (27) Substituting equations (29) and and then s u b s t i t u t i n g (23) into the r e s u l t , we obtain f i n a l l y : which d i r e c t l y shows that i n the degenerate case the r a t i o of amplitudes of the problem C eigenfunction i n the regions of the two boxes, i s of order unity, i r r e s p e c t i v e l y of how f a r apart the boxes are or of how small the function becomes i n the region between the boxes. I t i s clear from previous arguments that i f and <p R are the problem A and problem B eigenfunctions (given by equations (14) and t h e i r analogues f o r problem B) which belong to , the eigenf unctions tp belonging to the two problem C l e v e l s which are close to E<* be approximated by constant multiples of and Thus equation (31) the appropriate boxes.. must both <P a near t e l l s us that apart from an a r b i t r a r y m u l t i p l i c a t i v e constant, each <p must approximate the normalized single-box eigenfunctions near the corresponding boxes: C{ « and (p /^a+ \/Q> A CP « *HPH/VC + I/O near box A, near box B. (32a) (32b) 17. But since <^ and <$ B are of order e near t h e i r opposite boxes, i t i s clear from these equations that. <p may be approximated everywhere by the sum or difference of the normalized and : F i n a l l y the r a t i o Pn/P of the p r o b a b i l i t y of B f i n d i n g the p a r t i c l e near box A to that of f i n d i n g i t near box B may be approximated by means of equations (15a), (52a), and"(32b). We f i n d : i+eo (33) " 6. \ f i *-«_JZ : i =I PROBLEM D, IN WHICH ONE BOX IS SHALLOW In problem D, where we suppose that (box B i s shallow! and b X i s small i s a r b i t r a r y , we obtain an e x p l i c i t approximate expression f o r the eigenvalues by considering the f a c t that equation (9) defines E . of X . as a many-valued I t i s clear that f o r every l e v e l function E n of problem A, there i s a branch of t h i s function which approaches as \—> o » E H But each of these branches may be expanded i n a series of the form which implies that of accuracy i n X may be determined to any order , provided X i s s u f f i c i e n t l y small. The range of application of these series w i l l be discussed l a t e r 18. (see P a r t IV). The pansion coefficients A. and * i n the above z have b e e n d e t e r m i n e d f r o m e q u a t i o n differentiation direct of result theory. ( s e e A p p e n d i x B) of They e e q u a t i o n '(34) i n order with (9) by ex- partial t o compare the that of perturbation are: «=l|fU = £*<* (e'^ -0/2K(.*(>M z C (33a) and where E, denotes e x p r e s s i o n f e r e n t i a t i o n with respect to evaluated at E= E A . ( 3 5 a ) , the E , and primes denote a l l the functions difare 19. III. 1. PERTURBATION THEORY SOLUTION STANDARD PERTURBATION-THEORY TREATMENT OF PROBLEM D" In solving problem D we s h a l l make use of the r e - sults of the usual kind of perturbation theory which i s found i n most books on quantum mechanics^. In order to apply the standard theory, the perturbation of the Hamiltonian operator must be expressible as a power series i n some parameter such that the perturbation vanishes when the parameter- i s zero. Let V«^(*} a n V (x) d B be the p o t e n t i a l functions of problems A and B r e s p e c t i v e l y : V * ^ * 1 V U^ = U a v \V (x] B (36a) (36b) where V,(x) s ) I O (36c) Although most texts do not consider the case i n which the unperturbed problem has a continuous spectrum, t h e i r discussions may e a s i l y be generalized with the help of equations (81) and (82) of Appendix A, to give the r e s u l t s stated i n equations (43)-(45). Application of the theory to our problem i s s i m p l i f i e d by the f a c t that the bound states are not degenerate, that a l l the eigenfunctions considered are r e a l , and that the perturbation operator i s simply proportional to the expansion parameter (see equation (36b)). 20, •k * 6 H^U^ » ' , and Ulxi be the Hamiltonian opera- tors of problems A, B, and D respectively: • v » * V„ • l 3 7 a ) , (37b) so that c l e a r l y (38) In view of equations (38) that X and (36b) and the fact i s small f o r problem D, i t i s clear that we may take problem A as the unperturbed problem, " V operator, and X as the perturbing Q as the expansion parameter. Accordingly, - we suppose that i f box B i s s u f f i c i e n t l y shallow ( A i s suf- . f i c i e n t l y small), there i s a normalized bound-state eigenfunction cp of problem G corresponding to each normalized bound-state eigenfunction .<Pi» = < U j a * i/p (39) E^ « E of problem A belonging to the eigenvalue that (p and the eigenvalue E pressible as power series i n X <P* <P» * *<P. to which i t belongs are exof the form * A*<fc. v E = E v XE, a ' E . v 0 , such R - - - - - ~ , . (40) (41) We f i n d the c o e f f i c i e n t s of these series from standard perturbation theory, expressed i n terms of "matrix 2 1 . elements" of the form Thus, E. * e -r , VoV f vcA • , Vo^ n (45) where the summations are over a l l the bound states of the perturbed problem, whose eigenvalues dix C. tion. The values of E, and The. c a l c u l a t i o n of E, In c a l c u l a t i n g equation (45) E E^' un- E ^ • d i f f e r from E ^ are calculated i n Appeni s a straightforward integra- , the range of integration i n t i s f i r s t extended from -«> contour integration i s employed. to •»» , and then Investigation of the poles of the integrand shows that there are terms from the i n t e g r a l which exactly cancel out the terms of the summation, and the balance of the i n t e g r a l involves only the constants and (i of the l e v e l . <* Thus i t i s shown that the r e s u l t s of perturbation theory agree exactly with the d i r e c t l y (35)* tained r e s u l t s of equations 2. ob- SPECIAL PERTURBATION THEORY TREATMENT OP PROBLEM C When the distance finite, VU) reduces to 2.^ V^M between the boxes i s i n . for a l l f i n i t e , and 22. problem 0 reduces ly c h o s e n the box B, when the t o p r o b l e m A. origin of the x. . A c c o r d i n g l y we eigenvalue E a p p r o a c h e s an ' Further, i f E„ o f p r o b l e m A, we say belonging to E where i s the E to- c - numerical E is a the B of problem A normalized or level, eigenfunotion form problem A ^VUjb) eigenfunotion belonging i s a f u n c t i o n whose maximum b—»•«> (i.e., . a s b — * co ), proaches zero uniformly i n \ is a of every } "non-degenerate" s i n g l e - b o x v a l u e approaches z e r o as hand, i f E , initial- to problem •)»—»<» eigenvalue normalized , and p assume t h a t a s assume t h a t t h e has had coordinate at the centre two-box p r o b l e m w o u l d h a v e r e d u c e d VJ=OO B. I f , h o w e v e r , we "degenerate" On ^ ap- the single-box l e v e l , other we assume that (S}M\ where A and .= F\ R<J«U) * i s a linear We may C approximately E combination problem B eigenfunctions belonging p r o a c h e s z e r o u n i f o r m l y as value + now use i f we E„ =• E ^ E . perturbation theory assume t h a t f o r e v e r y (defined i n equation - t 0 problem , and ap- to solve problem single-box (say o f problem A ) , there E of the (47) b — » oo . o f p r o b l e m C w h i c h i s e x p r e s s i b l e as parameter to , + e t , e x E i s an x • eigenvalue a power s e r i e s (20)) o f the - - - eigen- i n the form # (48) 23.. If £.<«, i s a "non-degenerate" l e v e l we may substi-. tute equations (46) and (48) into the Schrodinger equation for problem C to obtain: i n which we may regard "V* as a perturbing operator; ^ as the perturbation of the wavefunction, and as the perturbation of the energy l e v e l . In t h i s problem, however, we cannot employ standard perturbation theory, f o r although the perturbing V^("v,fe.^ operator vanishes when the expansion parameter 6 = 0 i s not expressible as a power series i n € function . , it Moreover, the need not be expressible as a power, series i-n 6. . We need only assume that the function * i s bounded as fe—*• « */« (50) O . Rewriting the i d e n t i t y (49) with the help o f we f i n d that Multiplying both sides of t h i s i d e n t i t y by over a l l x , and d i v i d i n g by e , integrating , we obtain the i d e n t i t y i n fe : . + 00 U. •fcE**---M<M<Mr)Axt Since and \ •« t'A ^ V . ^ ^ ^ A x . (52a) are o f order fe (or less) near box B (see equations (14) and ( 5 0 ) ) , and E, i s independent of £ , we 24. f i n d on taking l i m i t s as t i o n E, that the f i r s t order correc- fc—*0 vanishes: E l s • 0 In order to determine equation (52a) Now *V may (52b) E , we set E, - O 2 and divide again by €. . in We "find that as be expressed i n terms of the unperturbed eigen- functions i n the form (see Appendix A): «v = Z ^ % v fe % +% <?») A Thus we f i n d on substitution of equations (50) equation (53), . (5*J and (54) into that i n a notation s i m i l a r to that of equation (42) above: ° where a vanishing term of the form fc."' ^^^'"Vp ^ 1 h a s been neglected. In order to obtain an e x p l i c i t expression f o r E we would have t o . f i n d e x p l i c i t expressions f o r the attempt to f i n d the asymptotic values of the t u t i n g equation (54) multiplying by <^ into equation (51)» that . of Appendix A. , We by s u b s t i - s e t t i n g -E,*0 , integrating over a l l X fc. , and using equation (81) cy^ z t , d i v i d i n g by We f i n d then, 25. Thus the «y£ approach the solutions of a p a i r of simultaneous i n t e g r a l equations, which we have been unable to solve exactly. We may check the v a l i d i t y of our r e s u l t by f i n d i n g the f i r s t two c o e f f i c i e n t s of the expansions of the powers of \ c^j* i n : V * + *1h * tf<y& • f 1581 Substituting equations ( 5 8 ) and ( 3 6 b ) into the r e l a t i o n s h i p ( 3 7 ) , and equating the f i r s t two powers of. * on either side, we obtain i n the notation of equation ( 4 2 ) : <tf. te<r xY*li E • o ,^ * " e ( w a ) . (59b) Equations ( 3 6 b ) , ( 5 5 ) , ( 5 8 ) and ( 5 9 ) now t e l l us that to the second order i n X : Comparing t h i s r e s u l t with equations given i n Appendix C, we f i n d our r e s u l t to be i n agreement with that of the l a s t section. In the special but important case that E-o i s - a degenerate single-box l e v e l , we are rewarded with greater success. Substituting equations ( 4 7 ) and ( 4 8 ) into the Schrddinger equation f o r problem C, we f i n d : Substituting H^vV^ grating over a l l x for H , multiplying by dO ft , inte- j d i v i d i n g by fe , and taking l i m i t s as 26.. O. we obtain: 62a) - OO S i m i l a r l y , using we H . *V„ 8 for H > and multiplying by ((>, obtain: . \ CO too -00 '-CD But from equations (14), (62b) (15b), (15c) and t h e i r analogues f o r problem B,. we f i n d that <M*V*A* * \ <M»v«4x - •-2** pe/«\jyx .( 3) , -00 6 •'-On Accordingly, with equation (15a) and i t s analogue, the condit i o n f o r simultaneous solutions of equations.(62) takes the form (62) which y i e l d s two possible solutions f o r E, : (65) This r e s u l t agrees exactly with equation (26a). F i n a l l y we obtain from equations (62), (65), (63), and the r a t i o 1 *» which agrees with equation also derive equation (33) VT7(^ (31). . I t i s clear that we (67) could from our perturbation theory re- s u l t s , using the same arguments as before. PLATE HI ' T Eigenvalues of Problem C as a function of \ h e 27 17. DISCUSSION The d i r e c t l y obtained r e s u l t s f o r the eigenvalues of the two-box problem are i l l u s t r a t e d i n Plates I I I and IV. Plate I I I i s a sketch of the eigenvalues £ as a many- valued function of X f o r the case i n which problem A has three l e v e l s ft , E , narrower than box A. , and e Q l , and box B i s s l i g h t l y The horizontal dashed l i n e s represent the problem A eigenvalues, while the sloping dashed l i n e s r e present the problem B eigenvalues. S i m i l a r l y the dot-dashed l i n e s represent the i n f i n i t i e s &<»,*v of and ^ heavy l i n e s represent the two-box l e v e l s themselves. . The In accordance with equation ( 9 ) , i t i s clear that i f V» i s not i n f i n i t e ( Z. i s not zero), i s i n f i n i t e , and vice versa. ^ i s zero i f and only i f ^ Accordingly the heavy curves . must cross the discontinuous curves at the encircled points, which mark the intersections of the single-box. l e v e l s with the A , a n d can cross the discontinuous curves only at these points. The approximate r e s u l t s obtained by both d i r e c t and perturbation theory methods f o r problem C, i n which V» i s large, are i l l u s t r a t e d by the fact that i n t h i s case the heavy curves closely follow the dashed curves. In accor- dance with equations (23) and (24), and equations (52b) and X = « a as a f u n c t i o n of € f o r fhe PLATE case in which a V K U =3-5, showing the approximations to the first order in €. 40 40 3-5 3-5 30 30 — ——' ______ — 25 25 20 20 I-5 — — — — — I-5 l-O I0 OS Approxirru ttion to fin t order OS Computed values O 0-2 0-6 0 4 ~ 6 0-8 l-O 28. ( 6 ^ ) , the heavy curves l i e farther (at distances of order £ ) from the nearest dashed curves near the points of intersect i o n of two dashed curves (the points where "degenerate" single-box: l e v e l s occur), than at other points (where the distances from the dashed curves are of order € ). 2 The approximate r e s u l t s obtained by the two methods for problem D, i n which X i s small, are i l l u s t r a t e d by the behaviour of the three curves whioh follow before t h e i r f i r s t turning points. when V> , , and I t i s c l e a r that i s l a r g e the curves turn so sharply that the series ; (34) and (41) cannot be expected to be v a l i d beyond the f i r s t turning point- of each curve, and c e r t a i n l y the approximations given by the f i r s t two terms of the series w i l l not be v a l i d beyond these points. Further, since the c o e f f i c i e n t s of the series depend upon b , and the curves must pass through the encircled points regardless of the value of t» , i t i s clear that the series can never be v a l i d beyond the f i r s t encircled point of each curve, and that the approximations to the second order i n \ can never be trusted beyond the f i r s t turning points. Plate IV shows the behaviour of the l e v e l s of a two-box problem i n which both boxes are i d e n t i c a l ( A- I C* <X ), as functions of £ , and thereby i l l u s t r a t e s the r e s u l t s of equations ( 2 6 a ) and ( 6 5 ) . In order to show numerical r e s u l t s , the dimensionless constant Va 1 W given the value 3*5 , and the dimensionless quantity X* <xci = V a l ^ i s used instead of EL. . is and 29. The of * . continuous curves represent the exact values In accordance w i t h equations e a c h c o n t i n u o u s c u r v e has 1 (4)» (9), a n d an e q u a t i o n o f t h e iz.zs I (20c), form •* * (68a) ». where y _ Q,CK •= V and y 0 .is the value of (where €. * o The y 1 2-2-- * a t one , 2 of the s i n g l e - b o x the levels ). dashed curves represent the approximations the exact curves i n accordance w i t h equations and (68b) (26a) to (65), and approximation 2* > • (69) obtained from equation (4a). The the lower level, to levels 3-3 value of i s surprisingly 0<^ • The , and b/a. value of h e n c e we o f the approximation f o r large. y<> and the is valid for this f i n d from equation = - lowest fcro from curve i s (20c), that the O\4-0il (70) C l * i s , the approximation i s v a l i d infinity For f o r the p o i n t of departure i s V o . that of a p p l i c a b i l i t y f o r i n s t a n c e , the approximation i « % range f o r values, of b between PLATE C o m p a r i s o n o f -the S q u a r e Well M o d e l a n d M a n n i n g ' s M o d e l w i t h t h e A m m o n i a Spectrum 30. V. APPLICATION OF THE MODEL TO THE AMMONIA INVERSION SPECTRUM In order to i l l u s t r a t e . t h e value of the two-box problem as a prototype model f o r a physical system, we shall use i t to calculate some of the constants of the ammonia molecule. Many authors 1 have pointed out that the motion of the NH3 molecule which contributes to the inversion spectrum i s that i n which the nitrogen atom moves back and f o r t h through the t r i a n g l e formed by the three hydrogen atoms. There i s an equilibrium p o s i t i o n for-the nitrogen atom on either side of the t r i a n g l e , and a p o t e n t i a l barrier with a maximum i n the plane of the t r i a n g l e which the nitrogen atom must traverse. I t has been shown that although the molecule i s 2 three dimensional, the method of normal coordinates may be used, and hence the l e v e l s of the inversion spectrum closely approximate the l e v e l s of a one-dimensional blem. two-minimum pro- Many authors have used t h i s fact to estimate some constants of the ammonia molecule. Manning^, f o r instance, assumed a potential function similar to that shown on the right-hand side of Plate V, and by assuming a reduced mass "Infrared and Raman Spectra", (Van Nostrand), pp. 221 to 224, and references given there. N* Rosen and P.M. Morse, Phys.Rev.42, 210, (1932). ^M.F. Manning, Jour.of Chem.Phys. 3, 136, (1935). 1 G. Herzberg, 2 31. /x r ^ . ( P O * to his gtrij and model t o t h o s e found fitting the lowest three l e v e l s f r o m t h e ammonia s p e c t r u m , " e q u i l i b r i u m h e i g h t o f the NR3 pyramid" mined the s e p a r a t i o n o f the minima), the h e i g h t o f the between t h e m i n i m a , and lated he. d e t e r - (half the minima. some o f t h e h i g h e r l e v e l s and found the "potential the asymptotic v a l u e o f the f u n c t i o n at l a r g e d i s t a n c e s from He of hump" potential then them t o be calcu- i n good agreement w i t h t h o s e o f ammonia. In tial o u r c a l c u l a t i o n s we first f u n c t i o n o f t h e type d i s c u s s e d above, i n w h i c h boxes a r e i d e n t i c a l . T h e n , by 0. , b , andV . Our f o r t h e two the v a l u e s of lowest l e v e l s (26a) t o be v a l i d , we the s p l i t t i n g o f one o f t h e s e l e v e l s when t h e distance apart, then € eC i f *o and X, s i n g l e - b o x l e v e l s , we X,* notice that z and problem the i f A£ is boxes a r e a ' z^c - or But X * oca 3 - use Then, assuming approximation 2 b to o f the s i n g l e - b o x 1*1 a o.zWtT . various values of mass the constants, n u m e r i c a l method i s f i r s t g r a p h i c a l methods t o d e t e r m i n e Y s ^A- determine poten- both u s i n g t h e same r e d u c e d and m a k i n g t h e same f i t a s M a n n i n g , we for assume a two-box (71) Y\ &E ( 1+yV are the values of X (72) f o r the f i r s t two have - * o X = a . l * ( E , -EO , (73) 32. and hence 6 * A (74) £ Thus, by using the r a t i o s lowest two ammonia l e v e l s , we and then f i n d the two £ f o r each l e v e l , corresponding values of b _ H _ By t r y i n g d i f f e r e n t fVs f e (-75) and i n t e r p o l a t i n g , we f i n d one which the two r a t i o s b/a. 1*1 and the assumed calculate found from the a s AE/(E,-E«,) are equal. , we calculate for F i n a l l y , using t h i s a , b , and U . We f i n d that our value f o r the "equilibrium height of. the. NH^ pyramid" ( a * b ) agrees very well with Manning*s, and that our value f o r the height of the "potential hump" agrees f a i r l y w e l l ; but we f i n d no higher bound7levels. Therefore we next assume a potential function, as shown on the left-hand side of Plate V. A f t e r f i n d i n g the necessary equations f o r t h i s problem, and making numerical calculations s i m i l a r to those of the l a s t paragraph, we go on to calculate higher l e v e l s . The r e s u l t s of our calculations are shown i n the following table and are i l l u s t r a t e d i n Plate V. 33. Levels (cm:') o- 1 12+ 2" + 961 1610 1870 2360 2840 3 3" + 0 0.66 932.4 968.1 1597.5 1910 2380 2861 0 0.83 936 961 0 0.83 0+ NH3- Square Well Manning 1640 2170 2650 3290 Shapes of Potential Functions Manning Square Wells - Widths of Boxes (2 a.) Separation .of Boxes 0.28 A _ - 0.44 <.2.fe) Equilibrium Height of Pyramid 0.37 Height of Potential Hump 2071 0.36 A A 0.41 cm" 1 45100 c n r 1 A 0.38 A O.36 A 1 Square Wells With I n f i n i t e Sides 1640 cm" 1640 cm" 1 1 1650 00 cm"" 1 34. APPENDIX A . It NORMALIZATION OF THE EIGENFUNCTIONS OF THE CONTINUUM i s w e l l known t h a t we may choose n o r m a l i z e d p r o - blem A e i g e n f u n c t i o n s (17)) and <fy£ ( e q u a t i o n s ( 1 4 ) , (15) and such t h a t a l a r g e c l a s s of. f u n c t i o n s be e x - pressed i n the form — 00 ^ where t h e *C cy *\ and o + ^ i ^ ^, O j ^ t are functions of p (76) and j( given by: r+oo ) -00 <V£ - (TuKjjmAx Conversely, i f a r b i t r a r y functions . ^ and l 7 7 b ) ^ a r e chosen to define a function by means o f e q u a t i o n (7&), equations hold. i n t o equation and necessarily (77) (77b), then S u b s t i t u t i n g equation we o b t a i n f o r a r b i t r a r y o^^ , (76) , ^ '-00 But s i n c e V ^^jfc v a n i s h e s a t ' x = ± co , we know f r o m t h e o r t h o g o n a l i t y theorem f o r e i g e n f u n c t i o n s that "'"For example, E . C . Kemble, "Fundamental P r i n c i p l e s o f Quantum M e c h a n i c s " , C h . 7 1 , where t h e i n t e g r a t i o n i s c a r r i e d o u t over E . r a t h e r t h a n -V • 35. (79) '—•ft and h e n c e e q u a t i o n (73) may be written •Vv • -co Further, since functions (80) C\* pK^ ' uoav*- o ' ) since the implies a r e r e s p e c t i v e l y even and odd , i t follows '-CO Therefore, Jo and of * (80) that for arbitrary <j,(V), (8D ' 0 ^ are a r b i t r a r y functions, equation that \ < W \ ^ M ^ ^ * . <fcW which i n turn (82) , & '-co implies that P^- i n the symbolism o f d e l t a - f u n c t i o n s , A> , t u - w (83) M •00 Knowing malized tyfc that , we may equation (82) s t a r t w i t h unnormalized f u n c t i o n s , tyj = <Lo<*k (* + and derive the n o r m a l i z i n g f a c t o r s as 0<A<>k . Let A. > O 6 Then, i f C £ on s u b s t i t u t i n g i s the normalizing \{k) for <fyU<) a n d (84) follows: , and a number L\ be d e f i n e d yv say x>a for Choose a p a r t i c u l a r that holds f o r s u i t a b l y nor- such by t h e e q u a t i o n s tV-M*.A factor for , we C J (fjj CJ)j{ i n for find 36. equation (82) and dropping the , U+ "VCO \ Cv^vA c t signs, A « I V<¥k' < ^ * i Thus s i n c e -(86) . A (87) i s an even f u n c t i o n o f x , e q u a t i o n (86) implies that .00 2. \ X U,A> A x = I (88) '0 f o r a l l A>o » and consequently, 2L « 2 \ r<U » i A-*0 But we . (89) )o s h a l l show t h a t and hence t h a t To prove e q u a t i o n (90) r e a l number ^ we n o t i c e t h a t f o r every , 4^0 since I approaches L we choose $ (92) Q zero and the p a t h o f i n t e g r a t i o n i s o f Hence e q u a t i o n (89) f i n i t e length. Now - implies that • *' i n such a way (93) that (94a) t94b) 37. Substituting equation ( 8 4 ) into equation (8.6), changing the " variable of integration to , and expanding the t r i - ^=*-<s gonometric functions with the help of equations ( 9 4 ) , we f i n d that „ 5, S V * (93) where c vUA***) PIV) , V O^V) c c v c*sVl*+S) _ Integrating by parts i n equation (95) [ " s f t \ v , ^ x ( 9 6 a ) ( 9 6 b ) and using the formulae ^o<Y><a, Ax * - I J L ^ ^ (97a) and I * ^ 0<b<<x * * V ; we obtain ( But i n accordance with equations J94) the f i r s t term approaches ~^^fz term vanishes since qjOk*&) order L\ . and (9?)» 8 ) we f i n d that as L\ —*0 , and the second and fyU^-a) must be of F i n a l l y , v/e f i n d that we may interchange the order of integration i n the l a s t term, since exists. 9 ^ ^• J ' ^ < x x f Hence the l a s t term vanishes also, and I- fe 1 5 ^ C * / z . (99) /D/ 38. DERIVATION OF EQUATIONS (35a) APPENDIX B. (35b) AMD V From e q u a t i o n ( 9 ) , we have -^vn/^w^), (4), which w i t h equations (8), (9), a n d (20) (100) becomes " e ' - i H ^ - ^ ^ V C ^ ^ ^ - ' ^ W ) . (io!! Hence f o r N O S and E = E * , E, - (£V***TJj\«^ Setting (4c)) E*6 A i n equation and <X.'* 1(V (8b) we o b t a i n (N^V-o Hence w i t h e q u a t i o n s ( = ^ ^ v ( » - e ^ ) (20a) (4b) ( c . f . equations and (25a) To o b t a i n E For \= O we E w 2 ( 1 0 © differentiate ^ equation 3) (35a) we o b t a i n e q u a t i o n E, - fc*** ( e * ^ - \ ) / 2 Y \ { \ « V * ) ) and . # 102 ( 1 0 4 ) (101): "*0 • («X C^UV«W).c»5) + find * - , 1 1 0 7 1 39. and where a l l the expressions are to be evaluated at N*0 . £ = e» and Substituting into equation (105) we f i n d with the help of equation (102) that <*%., *..{-«. F i n a l l y , we f i n d equation (8b), set to obtain equation from equation (4b) and OI'-A^ (35b). , and use equations from (35a) and (109) 40.' APPENDIX C. CALCULATION OP THE COEFFICIENTS E, AND Ei BY MEANS OF EQUATIONS (43) AND (45) We f i n d E, E, from equations ( 1 4 ) , (15), and ( 4 2 ) : , -[VocVwCa^/^L-^-)^ = On e v a l u a t i n g the i n t e g r a l , we f i n d t h a t equation (110) e x a c t l y w i t h equation To f i n d and E z (110) agrees (35a). , we f i r s t (42) t o c a l c u l a t e the m a t r i x use equations. ( 1 4 ) , (15), elements from which we o b t a i n . Y<£, r (S(^ [^ ^ ^]/KM^^)^K(^l+^) / S i m i l a r l y we use equations Z ( C ( 1 4 ) , (15), (17), (112) and (42) t o f i n d the m a t r i x elements W r i t i n g the c o s i n e f u n c t i o n i n e x p o n e n t i a l form, and u s i n g the equation (114) I n t h i s equation ot' r e p r e s e n t s the v a l u e o f oUej f o r 41. we obtain where N = V/ATT Y (116) (117) T,U0 = »t-*0 T (118) T \ U ) - - Ci - e 4(iC )/(JUw^CU-;(>) 2 (119) I* , Vik) and From equations (118) even function of h Xjl-Jc) -'\/(^)(JU-.-^\ (120) . ( 1 2 1 ) and (121), and the f a c t that T, i s an , the i n t e g r a l i n equation (45) takes the form X -_ ti W a r , , , T, 4 T ; V T ; ) ^ (122) We now evaluate the i n t e g r a l i n the l a s t equation by means of contour integration. runs along the r e a l axis from Choosing the contour which - ft. to + ^. (where ft > O ) and then around the semi-circle i n the upper h a l f plane from + ft. to -^ , we f i n d that the i n t e g r a l around the semi-circle approaches zero as X - R —• a> . ZW*l4 (Sum of residues of the i n t e - • grand at i t s poles i n the upper half-plane.) Hence (123) 4-2. T\ and T-s have poles at residue of 3>T, while that of only. /L-^n> The : is is = (125) 21*;. IV* , we f i n d that * 'P Adding ft, and and multiplying by the contribution of T, and T to 3 X i s exactly equal to (35k). the f i r s t term of equation In order to evaluate the residues of T** • and we f i n d from equations (18) ^kt****) T* A i that Jk^) ^..p C . ( 6) 12 and hence that e Equation (127) +Q. t e l l s us that ponding to each l e v e l to E P ( = A*.=-;^ ). (127) -2*^/Lkr*\^**<-ol*<*) *T\~ has a pole corres- ( i . e . , f o r -V = XQf ) as w e l l as The.residue at k.- + ^' i s . , (128) V^H*""^"^^^^ Evaluating t h i s residue with the help of equations (4) and (25a), >e with equation we f i n d that i n accordance W/(»-o') e e (112), (129) 43. Thus every term of equation (45) whioh arises from a discrete level i s cancelled by an equal and opposite term which arises from the continuum l e v e l s . F i n a l l y we evaluate the residue of <W -i[W(tiVl "V • T V at (130) where Vk) ( k tip)' Vil^/zl&^'PKi'l* (W) Using the equation we obtain with the help of equations (4): (133) But on substituting X = A(b the help of equation (35a) the second term i n equation i n equation (133) that (35b). 2ir;NR (Jk^) A we f i n d with i s exactly 44. BIBLIOGRAPHY 1. S. Dushman, "Elements of Quantum Mechanics", Wiley, pp. 214-218, and r e ferences given there. 2. G.. Herzberg, "Infrared and Raman Spectra", Van Nostrand, pp. 221-224, and references given there. 3. E. C. Kemble, "Fundamental P r i n c i p l e s of Quantum Mechanics", McGraw-Hill, Ch.VI. 4. M. F. Manning, Jour.of Chem.Phys., 5. M.. F. Manning and M. E. B e l l , Rev.Mod.Phys, P. M. Morse and E. C. G. Stuckelberg, Helv;Phys.Acta, N. Rosen and P. M. Morse, Phys.Rev., 6. 7. 3, 136 (1935). 12, 215 (1940). 4, 337 (1931). 42, 210 (1932). j
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On the quantum mechanical problem of a particle in two potential minima Carter, David Southard 1948
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Title | On the quantum mechanical problem of a particle in two potential minima |
Creator |
Carter, David Southard |
Publisher | University of British Columbia |
Date Issued | 1948 |
Description | The problem of a particle in two adjacent one-dimensional rectangular, potential "boxes" is an exactly soluble representative of a class of two-minima problems of considerable physical interest which have not been solved exactly. It therefore affords a valuable opportunity for a critical examination of the extent of applicability of perturbation theory methods to such problems. An exact implicit solution of the problem is obtained, and is reduced to explicit approximate form in two important special cases. These approximations are reproduced by perturbation theory methods, and their ranges of validity are demonstrated by comparison with the exact solution. The application of the model to a physical system is demonstrated by using the identical two-box problem as a basis for calculation of some constants of the ammonia molecule. |
Subject |
Quantum theory |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-03-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085420 |
URI | http://hdl.handle.net/2429/41659 |
Degree |
Master of Arts - MA |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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