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On the quantum mechanical problem of a particle in two potential minima 1948

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Cop. I ON THE QUANTUM MECHANICAL PROBLEM OP A PARTICLE IN TWO POTENTIAL MINIMA by David Southard Carter A thesis submitted in partial fulfilment of the requirements for the degree of MASTER OF ARTS in the department of PHYSICS The University of British Columbia i April, 1948 ABSTRACT The problem of a particle in two adjacent one- dimensional rectangular, potential "boxes" i s an exactly solu- ble representative of a class of two-minima problems of con- siderable physical interest which have not been solved exactly.. It therefore affords a valuable opportunity for a c r i t i c a l examination of the extent of applicability of per- turbation theory methods to such problems. An exact im- p l i c i t solution of the problem is obtained, and i s 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 i s demonstrated by using the iden- t i c a l two-box problem as a basis for calculation of some con- stants of the ammonia molecule. AGKNOWLED GEMENT The author wishes to express his 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 in the Departments of Mathematics and Physics who have taken such kind interest i n the author's progress as a student s and have helped him in many ways. Finally, the author wishes to express his indebted- ness to the National Research Council for their award to him of a Studentship under which this research was conducted. TABLE OF CONTENTS Fag© I . INTRODUCTION 1 I I . DIRECT SOLUTION 1. Fo r m u l a t i o n o f the Problem . . . . . . . 3 2. S o l u t i o n o f the Problem i n I m p l i c i t Form 4 3. Problem A, The Single-Box A Case . . . . 6 4. Problem B, The Single-Box B Case . . . . 10 5. Problem C, I n Which the Two Boxes are Far Apart 10 6. Problem D, I n Which One Box i s Shallow . 17 I I I . PERTURBATION THEORY SOLUTION 1. Standard P e r t u r b a t i o n Theory Treatment o f Problem D . . 19 2. S p e c i a l P e r t u r b a t i o n Theory Treatment o f Problem C 21 IV. DISCUSSION 27 V. APPLICATION OF THE MODEL TO THE AMMONIA INVERSION SPECTRUM '30 APPENDIX A - N o r m a l i z a t i o n o f the E i g e n f u n c t i o n s of the Continuum . . . . 34 APPENDIX B - D e r i v a t i o n o f Equations (35a) and (35b) 38, APPENDIX C - C a l c u l a t i o n o f the C o e f f i c i e n t s E, and 6 A by Means o f Equations (43) and (45) 40 BIBLIOGRAPHY' 44 LIST OF PLATES Opposit PLATE I . The Two-Box Po t e n t i a l Function and an Example of an Energy Level System f o r Problem C 3 PLATE I I , A Rough Sketch of ^ as a Function of. £ . . . . 7 PLATE I I I The Eigenvalues of Problem D as a Function of X 27 PLATE IV * - as a Function of £ . . . . . 28 PLATE V Comparison of the Square Well Model and Manning's Model With the Ammonia Spectrum. . . . . 30 1. ON THE QUANTUM MECHANICAL PROBLEM OF A PARTICLE IN TWO POTENTIAL MINIMA I. INTRODUCTION The problem of a particle i n two potential ininima is of extensive interest i n theoretical physics since i t pro- vides a model for many physical systems. The simple one- dimensional case in which the minima are rectangular in shape serves as. a prototype by which we may understand many pheno- mena conneoted with metallic conduction3", van der Waals forces^, the sta b i l i t y of hydrogen-like ions 2, and the vibra- tion spectra of certain polyatomic molecules^. For this reason many authors, including those mentioned in the foot- notes, have discussed the model with a view to i t s physical significance. Manning and M.E. Bell , Rev.Mod.Phys. 12, 215 (1940). ^S, Dushman., "Elements of Quantum Mechanics", (Wiley), pp. 214-218, and references given there. Dushman1s approximation to the energy sp l i t t i n g i s incorrect (compare his_ equation ( 3 ) with our equation (26a) for c = o» and \- 1 ), since he f a i l s to consider the phase shift of the eigenfunction. ^See Part V below, where the present model i s applied to the ammonia-inversion spectrum. 2 . I t i s f e l t , however, t h a t none o f the p u b l i s h e d d i s c u s s i o n s have taken f u l l advantage o f 'the p o s s i b i l i t i e s o f the problem i n i l l u s t r a t i n g many mathematical methods which are c o n s t a n t l y used i n quantum mechanics. On the one hand, the problem i s one o f few n o n - t r i v i a l examples whioh may be s o l v e d by exact methods. On the o t h e r hand the s o l u - t i o n may be c a r r i e d out, w i t h c e r t a i n s i g n i f i c a n t l i m i t a t i o n s , by means o f p e r t u r b a t i o n theory. The complete knowledge ob- t a i n e d by the d i r e c t s o l u t i o n may then be employed t o i l l u s - t r a t e the nature of the p e r t u r b a t i o n t h e o r e t i c a l r e s u l t s ^ * The pedagogic u t i l i t y o f the d i s c u s s i o n i s enhanced by the n e c e s s i t y o f u s i n g wave-functions o f the continuum, and o f d e a l i n g w i t h a type o f p e r t u r b a t i o n theory t h a t i s not g e n e r a l l y d i s c u s s e d i n the l i t e r a t u r e . P.M. Morse and E..C.G. S t u c k e l b e r g , Helv.Phys.Acta, 4 , 337, (1931)i make.this k i n d o f i l l u s t r a t i o n u s i n g a model f o r ammonia. However, they c o n s i d e r a l e s s g e n e r a l case, and t h e i r model i s more complicated. PLATE I T h e Two-box Tbtential Function Voo, ^ and an example of an E n e r g y Level System -for Problem C REGION 1 EM, BOX A 2b REGION 2 REGION 3 BOX 6 2C — REGION A REGION 5 Legend: •The levels of problems A a n d B are d r a w n in the appropriate B O X •The Eoo.n °f problems A and B are drawn in the appropr ia te B o x - T h e levels of p rob lem 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 particle of mass jx subject to a potential V u ^ shown i n Plate I, and seek i t s bound energy levels together with the corresponding eigenfunctions. Vex) vanishes outside two potential "boxes" A and B ( 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 i n re- gion 2, V w = -\vj i n region 4). We shall f i r s t solve the general problem i n im- p l i c i t form and then consider the explicit solutions of the following special cases: Problem A, in 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 is present. Problem B, in which the width of box A is zero, and only box B. i s present. Problem C, in. which the distance 2,t» between the boxes i s large. Problem D, in which box B i s much shallower than box A ( X i s small compared to unity). 2. SOLUTION OP THE PROBLEM IN IMPLICIT FORM The eigenf unctions cflx) satisfy the equation which i s the Schrodinger equation multiplied by « - S-ir 2-^/^. (2) If ^ is the expression for 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 for eigenvalues by the boundary, conditions at i n f i n i t y (where (p must not be infinite) and at the boundaries of the boxes (where <p and i t s f i r s t derivative <f' must be continuous). The conditions at i n f i n i t y require that ^, and fts shall vanish, so that equations (3) may be rewritten with some changes in the constants as 5* cp, - ^ o ^ U - ^ e 1 J ( 6 a ) Cpz •= t\ Cos a . r _ t\ cose* Co.* ^ r (S(x-a^) 0 ( ^ ^ - a n (6b) (6c) 1 = i , ^ L e .1 (6d) <D - & cos*' ( x-a-zb- c -?,') •• (be) where the f i r s t three and the last 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 in which the two boxes occur in the problem. Applying" the condition that <p' shall be ©ontinuous at the boundaries of box A to the f i r s t three of equations (6) we obtain P/ot = W-Ua**) t ( 7b) which determine % and as functions of <x and (I, and consequently (in virtue of equations ( 4 ) ) as implicit functions of E . . Thus elimination of & between equations (7a)1 and (7b) yields for %^ : * r S ^ J = VS XT • Similarly we obtain from the last three of equations (6) (3/*' « *a.woc' ( c - } ( 7 c ) = _LL*JL w c c * C ) - f ( 7 d ) I f we now in s i s t that the two forms of <P3 i n equations (6c) and (6d) represent the same function we ob- tain the condition ^ - 1 * ) (9) where e = e'*P* ( 1 0 ) Equation (9) i s clearly a condition for eigenvalues.since i n accordance with equations (4) , ( 8 ) , and (10) i t i s an im- p l i c i t equation in E and the constants a. , b , c , ^ , and X of the potential energy function. If the values of E- satisfying equation (9) for a particular set of values of the constants are found, say by numerical methods, then ^ and may be found from equations (8), and ^ and may be found from equations (7). Thus the equations we have found' give us complete information about both the eigenvalues and the eigenfunctions i n implicit form. 3. PROBLEM A, THE SINGLE-BOX A CASE If c or ^ i s zero and box A alone i s present, i t i s clear that the eigenfunction in the whole region to P L A T E H A r o u g h s k e t c h o f tK as a Funct ion of E •for the c a s e i n w h i c h A< 32yja2\J/h <9 T h e Z e r o s o f XA a r e t h e S i n g l e - b o x L e v e l s Notice t h a t t h e s l o p e is N e g a t i v e a t e a c h Z e r o the right of box A may be represented by the function cp_ of equation (6c). The condition that <P shall not be i n f i n i t e at xs+co now implies that ^ - O , (11) whioh with equations ( 8 a ) and (4) i s the condition for eigen- values. ^ has been roughly sketched in Plate II as a function of E for the case in whioh V l i e s in the range Af< S2}Lo< zTJ/£: < 9 . (̂y(e') i s real only for E<0 and i s positive for a l l E , approaching •<*> as E-*--co . In the region of bound energy levels (-\J< g ) i t s zeros are separated by i n f i n i t e discontinuities at the points-1- E » , w = . * v _ . i ) 2 > 3 , ( 12) A s . E increases, the sign of ^ changes from - to * at each Eto,vv. a n < i from •*- to - at each zero. The zeros, when placed in increasing order, are alternately zeros of the f i r s t and second factors of ^ i n equation ( 8 a ) . The eigenfunctions ({^ corresponding to the bound levels of problem A may be obtained by setting ^ = O i n equation (71») and eliminating (_. by means of equation ( 7 a ) : •̂As "vl—>PO each zero approaches the E.OD,W which l i e s im- mediately above i t . Thus the may be though of as the levels of an i n f i n i t e l y deep box with base at - V . fcIn order to avoid extra notation we shall use the symbols (i to denote either the functions of E. defined i n equations (4) or the special values of the functions corres- ponding to eigenvalues of EL . 8 . where X i s an even or odd integer according as the f i r s t or second factor of ^ in equation (8a) vanishes. Substi- tution of equation (13) into equations (6) after setting ^ _ O in equation (6c) shows that the bound state eigen- functions are alternately multiples of the even and odd func- tions <fc* = c o s * * (14a) and 0>(x -vcO <M v ' * * x (14b) ) according as the f i r s t or second factor of %\ vanishes. In both cases i t may be shown that ,•00 which determines the normalizing factor. For later use we observe with the help of equation (8a) that for the even functions cos2-** -. °cV(** + (32} - oĉ /tAV > ( 1 5 d ) and for the odd functions The eigenfunctions of problem A for the continuum of free energy states ( E > O) must also be found since we w i l l require a complete set of eigenfunctions in applying perturbation theory. Setting 9 , we find that outside box A , (p no longer has the exponential form of expression (3) but that i t has the oscillatory form R w s V U i - ^ . Thus the condition that cp shall be bounded at i n f i n i t y is 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. It is clear that equation (1) and the con- tinuity conditions are satisfied for every, E > O by the following linearly independent pair 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 ( 1 6 ) . Further, we obtain a complete set of eigenfunctions by i n - cluding only the Cf)£ , since for given E , any three solu- tions of the second order differential equation (1) and the boundary conditions are linearly dependent. The factor. \//TF i n equations ( 1 7 ) has been chosen to make the normalized to "delta-functions" (see Appen- dix A). 10. 4. PROBLEM B, THE SINGLE-BOX B CASE It i s clear from the symmetrical way i n which boxes A and B enter the problem that a l l the results derived i n seotion 3 for problem A may be transformed into corresponding results for problem B by making the following substitutions: <x »- c "VJ *v a, *~ tx' x >- x - zfe - c % %• \—- ^ and leaving a l l other quantities unchanged. We shall denote by C B̂ the bound-state eigenfunc- tions found in this 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 in which the distance £Y> between the two boxes i s large. For arbitrarily large b , e (see equation (10)) i s correspondingly small. Hence the value of i s . arbi t r a r i l y close to zero. It follows that every two-box level E. i s arbi t r a r i l y close to a level E 0 of problem A or B (a zero of or ). Indeed, for sufficiently large V> , every such E may be approximated by a solution of the equation where 11. ^ = > (20a) d6 2 V-KE. b> (20c) Two possible cases now present themselves. In the f i r s t case, which we shall c a l l "non-degenerate", E e i s a level of one single-box problem but not of the other. In the second case, which we shall call."degenerate", E o i s a level of both single-bos problems-1-. Since equation (9); i s symmetric in and ^tj,(we need only discuss those cases in which E „ = E i \ i s a level of problem A ( Vft (e^ - O J. 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 ^ in order that JUw*. E r • When b i s so large that In our special use of the word "degenerate" we refer to a case i n which there i s a simultaneous level of two different problems rather than one in which t.here i s more than one linearly independent eigenfunction belonging to a single level of the same problem. 1 2 « the approximation reduces to E - e A = < L Z / < \ . ( 2 3 ) 1 In the degenerate case we merely set ^ = ̂  = O in equation (19) and obtain E - V . W J W ( 2 4 ) The derivative H^ER) may be found by differen- tiating the expression (8a) using equations (4) and (5) and the fact that ^ ( E ^ = 0. The result i s Similarly for a level E a of problem B, - VS»\V(^(icV2oc ' * f 3 a . (25b) The last two equations show that the derivatives ^ and which appear in equations (23) and (24) are always negative. Hence the sign of ' E-E<* i n equation (23) i s always opposite to that of ^&(e<^ and the right hand side of equation (24) ^Condition (22) ensures that E . . i s much closer to E f l than to the nearest level E f t of problem B; i.e., there.is no approximate degeneracy. Eliminating e between equations (22) and (23), we have | E - E f t \ « W * / 4 ^ v so that even i f E t t is so close to E ^ that X * E „ ^ 4 UO * ^U*)= O , and we have 13. is always real. Stated in words, equation (24) implies that any level 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 far apart) be- comes s p l i t into two levels when the boxes are a large but f i n i t e distance apart. One of these levels l i e s above and the other l i e s below the original level, both by the same amount of order £ . On the other hand, equation (23) im- plies that each level which does not approach a level of problem B as t» —*. oo , differs from the corresponding problem A level -U only by an amount of order , and l i e s above or below in accordance with the following rule: The level i s "repelled" by the "closest" level of the other single-box problem, where we define the problem B* level which i s "closest" to• E * to be that level which i s not separated from E^ by an E«>|A of problem B (analagous to the Ea>,w of problem A given by equation (12)). If the original level l i e s "equally close" to two problem B levels (i.e., i f i t coincides with an E_,,n of problem B ), equation (23) shows that since ^oCEeo.^sCo , the problem C level E coincides with E*. at least to terms of order €_2 . Moreover, when we write the exact equation (9) in the form , i t i s clear that E, coin- cides exactly with , since " o^ l ^ " 8 O . The above results are illustrated by the sketched- in levels.in Plate I, where the boxes should be farther apart than they are in the sketch. The lowest level l i e s below 14.. Ec^o since i t is "repelled" by the higher level E f t o , and similarly the next lowest level l i e s below E a o since i t i s "repelled" by E«^ . The third level-coincides with E„ t since the latter coincides with Ea,, of problem B. The highest levels l i e above and below the coincident levels and • Making use of equations (8) and (2.5) we rewrite equations (23) and (24) in explicit form in order to compare these results with those of perturbation theory. Thus for degenerate levels we have: E - E, - + —-===L== ' ^/TTrrc^TTiTrjrr * { 2 6 a ) and for non-degenerate levels: E . £ = 2 f c z \«*CV ,(26b) We shall now discuss the eigenfunctions of problem C. Again we need only discuss those cases in which the problem C level E l i e s close to a problem A level E^ . In these cases the values of & determined by equations (7a) and (7b) di 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 from equations (14). Thus in the neighborhood of the box to which the unperturbed level belongs, the two-box eigenfunc- tion differs l i t t l e from the corresponding single-box eigen- function. To determine the nature of the perturbed eigenfunc- tion near box B we find the ratio \ fc/Rl of the amplitudes of the oscillatory parts of the eigenfunctions inside the' two 1 5 . boxes. First we equate expressions ( 6 0 ) and ( 6 d ) at "x=<\ and eliminate fe by means of equation (9); then we simpli- fy by means of equations ( 5 ) , (7), and (8$ to give:. In the non-degenerate case we may use equation (9) t 0 g i v e s/ ~ ^ 1 /V so that which shows that the ratio of amplitudes of the eigenfunction in the regions of boxes B. and A i s of order 6 and henoe the chance of finding the particle 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 shall take the problem A eigenf unction C|>A to be the un- perturbed eigenfunction corresponding to the perturbed eigen- function dO belonging to the problem C level E . It i s clear from equations (14) and (28) that both <̂  and <j>A are of order £ hear box B.. Thus the difference 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 ; a fact whioh w i l l be very troublesome i n our application of perturbation theory. In the degenerate case we obtain with the help of equation (24): » U - E ^ A C E * ) « 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) Substituting equations (29) and (30) into equation (27) and then substituting (23) into the result, we obtain f i n a l l y : which directly shows that in the degenerate case the ratio of amplitudes of the problem C eigenfunction i n the regions of the two boxes, i s of order unity, irrespectively of how far apart the boxes are or of how small the function becomes in the region between the boxes. It i s clear from previous arguments that i f <pR and are the problem A and problem B eigenfunctions (given by equations (14) and their analogues for problem B) which belong to , the eigenf unctions tp belonging to the two problem C levels which are close to E<* must both be approximated by constant multiples of and <Pa near the appropriate boxes.. Thus equation (31) t e l l s us that apart from an arbitrary multiplicative constant, each <p must approximate the normalized single-box eigenfunctions near the corresponding boxes: C{ « (pA/^a+ \/Q> near box A, (32a) and CP « *HPH/VC + I/O near box B. (32b) 17. But since <^ and <$B are of order e near their opposite boxes, i t i s clear from these equations that. <p may be ap- proximated everywhere by the sum or difference of the normalized and : Finally the ratio Pn/PB of the probability of finding the particle near box A to that of finding i t near box B may be approximated by means of equations (15a), (52a), and"(32b). We find: i+eo ( 3 3 ) " \ f i *-«_JZ : i = I 6. PROBLEM D, IN WHICH ONE BOX IS SHALLOW In problem D, where we suppose that X i s small (box B is shallow! and b i s arbitrary, we obtain an explicit approximate expression for the eigenvalues by considering the fact that equation (9) defines E. as a many-valued function of X . It i s clear that for every level E n of problem A, there is a branch of this function which approaches E H as \—> o » But each of these branches may be expanded in a series of the form which implies that may be determined to any order of accuracy in X , provided X is sufficiently small. The range of application of these series w i l l be discussed later 18. (see P a r t I V ) . The c o e f f i c i e n t s o f A. and * z i n the above ex- pansion have been determined from equation (9) by p a r t i a l d i f f e r e n t i a t i o n (see Appendix B) i n order to compare the d i r e c t r e s u l t o f equation '(34) w i t h t h a t o f p e r t u r b a t i o n theory. They a r e : e « = l | f U = £*<* z(e'^ C-0/2K(.*(>M (33a) and where E , denotes e x p r e s s i o n ( 3 5 a ) , the primes denote d i f - f e r e n t i a t i o n w i t h r e s p e c t to E , and a l l the f u n c t i o n s are e v a l u a t e d a t E= E A . 19. III. PERTURBATION THEORY SOLUTION 1. STANDARD PERTURBATION-THEORY TREATMENT OF PROBLEM D" In solving problem D we shall make use of the re- sults of the usual kind of perturbation theory which is found in 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 in some parameter such that the perturbation vanishes when the parameter- i s zero. Let V«^(*} a n d V B (x) be the potential functions of problems A and B respectively: V * U ^ * 1 v (36a) V a U ^ = \ V B ( x ] (36b) where V , ( x ) s ) (36c) I O Although most texts do not consider the case in which the un- perturbed problem has a continuous spectrum, their discus- sions may easily be generalized with the help of equations (81) and (82) of Appendix A, to give the results stated i n equations (43)-(45). Application of the theory to our problem is simplified by the fact that the bound states are not degenerate, that a l l the eigenfunctions considered are real, and that the perturbation operator i s simply proportional to the expan- sion parameter (see equation (36b)). 20, •k6* H ^ U ^ » ' , and Ulxi be the Hamiltonian opera- tors of problems A, B, and D respectively: • v » • l 3 7 a ) * V„ , (37b) so that clearly (38) In view of equations (38) and (36b) and the fact that X is small for problem D, i t i s clear that we may take problem A as the unperturbed problem, "V Q as the perturbing operator, and X as the expansion parameter. Accordingly, - we suppose that i f box B is sufficiently shallow ( A i s suf- . fi c i e n t l y small), there i s a normalized bound-state eigen- function cp of problem G corresponding to each normalized bound-state eigenfunction .<Pi» = < U j a * i/p (39) of problem A belonging to the eigenvalue E^ « E R , such that (p and the eigenvalue E to which i t belongs are ex- pressible as power series in X of the form <P* <P» * *<P. * A*<fc. v - - - , (40) E = E 0 v XE, a ' E . v - - ~ . (41) We find the coefficients of these series from stan- dard perturbation theory, expressed in terms of "matrix 2 1 . elements" of the form Thus, E. * , e - r V o V , f vcA • Vo^ n (45) where the summations are over a l l the bound states of the un- perturbed problem, whose eigenvalues E^' differ from E ^ • The values of E, and E ^ are calculated in Appen- dix C. The. calculation of E, i s a straightforward integra- tion. In calculating E t , the range of integration i n equation (45) is f i r s t extended from -«> to •»» , and then contour integration is employed. Investigation of the poles of the integrand shows that there are terms from the integral which exactly cancel out the terms of the summation, and the balance of the integral involves only the constants <* and (i of the level . Thus i t i s shown that the results of perturbation theory agree exactly with the directly ob- tained results of equations (35)* 2 . SPECIAL PERTURBATION THEORY TREATMENT OP PROBLEM C When the distance 2.̂  between the boxes is i n - f i n i t e , VU) reduces to V ^ M . for a l l f i n i t e , and 2 2 . problem 0 reduces to problem A. I f , however, we had i n i t i a l - l y chosen the o r i g i n of the x. c o o r d i n a t e a t the c e n t r e o f box B, the two-box problem would have reduced to problem B when VJ=OO . A c c o r d i n g l y we assume t h a t as •)»—»<» } every eigenvalue E approaches an eigenvalue o f problem A o r B. ' F u r t h e r , i f E„ i s a "non-degenerate" s i n g l e - b o x l e v e l , say o f problem A, we assume t h a t the normalized e i g e n f u n o t i o n b e l o n g i n g to E has the form where i s the normalized problem A e i g e n f u n o t i o n b e l o n g i n g to- E c - E p , and ^VUjb) i s a f u n c t i o n whose maximum numer i c a l v a l u e approaches zero as b—»•«> ( i . e . , ^ ap- proaches zero u n i f o r m l y i n \ . as b —* co ), On the o t h e r hand, i f E , i s a "degenerate" s i n g l e - b o x l e v e l , we assume t h a t (S}M\ .= R<J«U) * + , (47) where F\ i s a l i n e a r combination of the problem A and problem B e i g e n f u n c t i o n s b e l o n g i n g to E . , and ap- proaches zero u n i f o r m l y as b — » oo . We may now use p e r t u r b a t i o n theory t o s o l v e problem C approximately i f we assume t h a t f o r every s i n g l e - b o x e i g e n - v a l u e E„ =• E^ (say o f problem A ) , there i s an e i g e n v a l u e E o f problem C which i s e x p r e s s i b l e as a power s e r i e s i n the parameter ( d e f i n e d i n equation (20)) o f the form E - t 0 + e t , e x E x • - - - # (48) 23.. If £.<«, is a "non-degenerate" level we may substi-. tute equations (46) and (48) into the Schrodinger equation for problem C to obtain: in which we may regard "V* as a perturbing operator; ^ as the perturbation of the wavefunction, and as the perturbation of the energy level. In this problem, however, we cannot employ standard perturbation theory, for although the perturbing operator V^("v,fe.^ vanishes when the expansion parameter 6 = 0 , i t is not expressible as a power series in € . Moreover, the function need not be expressible as a power, series i-n 6. . We need only assume that the function * « * / « (50) i s bounded as fe—*• O . Rewriting the identity (49) with the help of we find that Multiplying both sides of this identity by , integrating over a l l x , and dividing by e , we obtain the identity i n fe : . + 00 •« U. •fcE**---M<M<Mr)Axt t'A ̂ V . ^ ^ ^ A x . (52a) Since and \ are of order fe (or less) near box B (see equations (14) and ( 5 0 ) ) , and E, is independent of £ , we 24. find on taking limits as fc—*0 that the f i r s t order correc- tion E, vanishes: E l s 0 • (52b) In order to determine E 2 , we set E, - O i n equation (52a) and divide again by €. . We "find that as Now *V may be expressed in terms of the unperturbed eigen- functions in the form (see Appendix A): «v = Z % ^ v fe % +% <?») A . (5*J Thus we find on substitution of equations (50) and (54) into equation (53), that in a notation similar 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 explicit expression for E z , we would have to.find explicit expressions for the . We attempt to find the asymptotic values of the cy^ by substi- tuting equation (54) into equation (51)» setting -E,*0 t multiplying by <^ , integrating over a l l X , dividing by fc. , and using equation (81) of Appendix A. We find then, that 25. Thus the «y£ approach the solutions of a pair of simultaneous integral equations, which we have been unable to solve exactly. We may check the validity of our result by finding the f i r s t two coefficients of the expansions of the c^j* i n powers of \ : V * + *1h * tf<y&f • 1581 Substituting equations (58) and (36b) into the relationship ( 3 7 ) , and equating the f i r s t two powers of. * on either side, we obtain in the notation of equation ( 4 2 ) : <tf. • o , ^ ( w a ) te<rExY*li * e " . (59b) Equations ( 3 6 b ) , ( 5 5 ) , (58) and (59) now t e l l us that to the second order in X : Comparing this result with equations given i n Appendix C, we find our result to be in agreement with that of the last section. In the special but important case that E-o is-a degenerate single-box level, we are rewarded with greater success. Substituting equations (47) and (48) into the Schrddinger equation for problem C, we find: Substituting H ^ v V ^ for H , multiplying by dOft , inte- grating over a l l x j dividing by fe , and taking limits as 26.. O. we obtain: - OO 62a) Similarly, using H8. *V„ for H > and multiplying by ((>, we obtain: . \ CO t o o (62b) - 0 0 ' - C D But from equations (14), (15b), (15c) and their analogues for problem B,. we find that <M*V*A* * \ <M»v«4x - •-2** ,pe/«\jyx .(63) -00 •'-On Accordingly, with equation (15a) and i t s analogue, the condi- tion for simultaneous solutions of equations.(62) takes the form which yields two possible solutions for E, : (62) This result agrees exactly with equation (26a). Finally we obtain from equations (62), (63), and (65), the ratio (65) 1 *» V T 7 ( ^ . (67) which agrees with equation (31). I t i s clear that we could also derive equation (33) from our perturbation theory re- sults, using the same arguments as before. P L A T E H I ' T h e Eigenvalues of Problem C as a function of \ 27 17. DISCUSSION The directly obtained results for the eigenvalues of the two-box problem are illustrated in Plates III and IV. Plate III i s a sketch of the eigenvalues £ as a many- valued function of X for the case i n which problem A has three levels , Eft, , and e Q l , and box B i s slightly narrower than box A. The horizontal dashed lines represent the problem A eigenvalues, while the sloping dashed lines re- present the problem B eigenvalues. Similarly the dot-dashed lines represent the i n f i n i t i e s &<»,*v of and ^ . The heavy lines represent the two-box levels themselves. In accordance with equation ( 9 ) , i t i s clear that i f V» i s not in f i n i t e ( Z. i s not zero), ^ i s zero i f and only i f ^ is i n f i n i t e , and vice versa. Accordingly the heavy curves . must cross the discontinuous curves at the encircled points, which mark the intersections of the single-box. levels with the A , a n d can cross the discontinuous curves only at these points. The approximate results obtained by both direct and perturbation theory methods for problem C, in which V» i s large, are illustrated by the fact that i n this case the heavy curves closely follow the dashed curves. In accor- dance with equations (23) and (24), and equations (52b) and P L A T E X = « a as a func t ion of € for fhe case in which a V K U =3-5, showing the approximations to the f irst order in €. 40 3-5 30 25 20 I-5 I0 OS — — — ' _ _ _ _ _ _ — — — — — — Approxirru Computed ttion to fin values t order O 0-2 0 4 0-6 0-8 ~ 6 4 0 3-5 30 25 20 I-5 l-O OS l-O 28. ( 6 ^ ) , the heavy curves l i e farther (at distances of order £ ) from the nearest dashed curves near the points of intersec- tion of two dashed curves (the points where "degenerate" single-box: levels occur), than at other points (where the distances from the dashed curves are of order € 2 ). The approximate results obtained by the two methods for problem D, in which X i s small, are illustrated by the behaviour of the three curves whioh follow , , and before their f i r s t turning points. It i s clear that when V> i s large ; the curves turn so sharply that the series (34) and (41) cannot be expected to be valid beyond the f i r s t turning point- of each curve, and certainly the approximations given by the f i r s t two terms of the series w i l l not be valid beyond these points. Further, since the coefficients 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 valid beyond the f i r s t encircled point of each curve, and that the approximations to the second order in \ can never be trusted beyond the f i r s t turning points. Plate IV shows the behaviour of the levels of a two-box problem in which both boxes are identical ( A- I and C* <X ), as functions of £ , and thereby illustrates the results of equations (26a) and ( 6 5 ) . In order to show numerical results, the dimensionless constant Va 1 W i s given the value 3*5 , and the dimensionless quantity X * <xci = V a l ^ is used instead of EL. . 2 9 . The continuous curves r e p r e s e n t the exact v a l u e s o f * . I n accordance w i t h equations (4)» (9), and (20c), each continuous curve has an equation o f the form 1 i z . z s I • * * ». (68a) where y _ Q,CK •= V 1 2 - 2 - - * 2 , (68b) and y0 . i s the v a l u e o f y a t one o f the s i n g l e - b o x l e v e l s (where €. * o ). The dashed curves r e p r e s e n t the approximations to the exact curves i n accordance w i t h equations (26a) and (65), and the approximation 2 * > • (69) obtained from e q u a t i o n (4a). The range of a p p l i c a b i l i t y o f the approximation f o r the lower l e v e l s i s s u r p r i s i n g l y l a r g e . For the lowest l e v e l , f o r i n s t a n c e , the approximation i s v a l i d from fcro to i « 0 < ^ • The value o f y<> f o r t h i s curve i s % 3 - 3 , and hence we f i n d from e q u a t i o n (20c), t h a t the value o f b/a. f o r the p o i n t o f departure i s V o . = - C l * (70) t h a t i s , the approximation i s v a l i d f o r values, of b between i n f i n i t y and O\4-0il P L A T E C o m p a r i s o n o f -the S q u a r e Well Model and M a n n i n g ' s Mode l w i t h the Ammonia Spectrum 30. V. APPLICATION OF THE MODEL TO THE AMMONIA INVERSION SPECTRUM In order to illustrate.the value of the two-box problem as a prototype model for 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 spec- trum i s that in which the nitrogen atom moves back and forth through the triangle formed by the three hydrogen atoms. There i s an equilibrium position for-the nitrogen atom on either side of the triangle, and a potential barrier with a maximum in the plane of the triangle which the nitrogen atom must traverse. It has been shown2 that although the molecule i s three dimensional, the method of normal coordinates may be used, and hence the levels of the inversion spectrum closely approximate the levels of a one-dimensional two-minimum pro- blem. Many authors have used this fact to estimate some constants of the ammonia molecule. Manning^, for instance, assumed a potential function similar to that shown on the right-hand side of Plate V, and by assuming a reduced mass 1G. Herzberg, "Infrared and Raman Spectra", (Van Nostrand), pp. 221 to 224, and references given there. 2N* Rosen and P.M. Morse, Phys.Rev.42, 210, (1932). M̂.F. Manning, Jour.of Chem.Phys. 3, 136, (1935). 31 . /x r ^ . ( P O * to gtrij and f i t t i n g the lowest three l e v e l s o f h i s model to those found from the ammonia spectrum, he. d e t e r - mined the " e q u i l i b r i u m h e i g h t o f the N R 3 pyramid" ( h a l f 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 " p o t e n t i a l hump" between the minima, and the asymptotic v a l u e o f the p o t e n t i a l f u n c t i o n a t l a r g e d i s t a n c e s from the minima. He then c a l c u - l a t e d some of the higher l e v e l s and found them t o be i n good agreement w i t h those of ammonia. I n our c a l c u l a t i o n s we f i r s t assume a two-box poten- t i a l f u n c t i o n o f the type d i s c u s s e d above, i n which both boxes are i d e n t i c a l . Then, by u s i n g the same reduced mass and making the same f i t as Manning, we determine the c o n s t a n t s , 0. , b , a n d V . Our n u m e r i c a l method i s f i r s t to use g r a p h i c a l methods t o determine the v a l u e s of X * oca and Y s ^A- f o r the two lowest l e v e l s o f the s i n g l e - b o x problem f o r v a r i o u s v a l u e s o f 1*1 a o.zWtT . Then, assuming the approximation (26a) to be v a l i d , we n o t i c e t h a t i f A£ i s the s p l i t t i n g o f one o f these l e v e l s when the boxes are a d i s t a n c e 2 b a p a r t , then € - z^c3- ' (71) o r eCz Y\ &E ( 1+yV ( 7 2 ) But i f * o and X , are the v a l u e s of X f o r the f i r s t two s i n g l e - b o x l e v e l s , we have X , * - * o X = a . l * ( E , - E O , (73) 3 2 . and hence 6 * A £ (74) Thus, by using the ratios AE/(E,-E«,) a s found from the lowest two ammonia levels, we calculate £ for each level, and then find the two corresponding values of b _ _ H f e (-75) By trying different fVs and interpolating, we find one for which the two ratios b/a. are equal. Finally, using this 1*1 and the assumed , we calculate a, b , and U . We find that our value for the "equilibrium height of. the. NĤ  pyramid" ( a* b ) agrees very well with Manning*s, and that our value for the height of the "potential hump" agrees f a i r l y well; but we find no higher bound7levels. Therefore we next assume a potential function, as shown on the left-hand side of Plate V. After finding the necessary equations for this problem, and making numerical calculations similar to those of the last paragraph, we go on to calculate higher levels. The results of our calculations are shown in the following table and are illustrated in Plate V. 33. Levels (cm:') Manning Square Well NH3- 0+ 0 0 0 o- 0.83 0.83 0.66 1 + 936 932.4 1- 961 961 968.1 2+ 1610 1640 1597.5 2" 1870 2170 1910 3 + 2360 2650 2380 3" 2840 3290 2861 Shapes of Potential Functions - Manning Square Wells Square Wells With Infinite Sides Widths of Boxes (2 a.) _ 0.28 A 0.36 A Separation .of Boxes <.2.fe) - 0.44 A 0.41 A Equilibrium Height of Pyramid 0.37 1 O.36 A 0.38 A Height of Potential Hump 2071 cm"1 1640 cm"1 1650 cm""1 45100 cnr 1 1640 cm"1 0 0 34. APPENDIX A . NORMALIZATION OF THE EIGENFUNCTIONS OF THE CONTINUUM I t 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 unc t ions and <fy£ (equat ions (14 ) , (15) and (17)) 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 ex- pressed i n the form 00 — ^ * C * \ + ̂ i ^ ^ , ( 7 6 ) where the c y o and O j ^ t are f u n c t i o n s o f p and j( g i v e n b y : r+oo ) - 0 0 <V£ - (TuKjjmAx . l 7 7 b ) C o n v e r s e l y , i f a r b i t r a r y f u n c t i o n s ^ and ^ are chosen t o d e f i n e a f u n c t i o n by means o f e q u a t i o n (7&), then equat ions (77) n e c e s s a r i l y h o l d . S u b s t i t u t i n g e q u a t i o n (76) i n t o e q u a t i o n (77b), we o b t a i n f o r a r b i t r a r y o^^ , , and ^ ' - 0 0 V But s i n c e ^ ^ j f c vanishes a t ' x = ± co , we know from the 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 t h a t "'"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 " , Ch. 7 1 , where the i n t e g r a t i o n i s c a r r i e d out over E. r a t h e r than -V • '—•ft 35. (79) and hence eq u a t i o n (73) may be w r i t t e n •Vv • (80) -co Jo F u r t h e r , s i n c e and a r e r e s p e c t i v e l y even and odd f u n c t i o n s o f * , i t f o l l o w s t h a t f o r a r b i t r a r y <j,(V), C\* pK'̂ uoav*- o ' (8D ' ' - C O ) 0 T h e r e f o r e , s i n c e the ^ are a r b i t r a r y f u n c t i o n s , e q u a t i o n (80) i m p l i e s t h a t \ < W \ ^ M ^ ^ * . <fcW , (82) ' - co & which i n t u r n i m p l i e s t h a t i n the symbolism o f d e l t a - f u n c t i o n s , P̂ - A> , t u - w M (83) •00 Knowing t h a t equation (82) holds f o r s u i t a b l y nor- m a l i z e d tyfc , we may s t a r t w i t h unnormalized f u n c t i o n s , say tyj = <Lo<*k (* + f o r x > a (84) and d e r i v e the n o r m a l i z i n g f a c t o r s as f o l l o w s : Choose a p a r t i c u l a r A. > O , and a number L\ such t h a t 0<A<>k . Let be d e f i n e d by the equations y 6v t V - M * . A Then, i f C£ i s the n o r m a l i z i n g f a c t o r f o r , we f i n d on s u b s t i t u t i n g \{k) f o r <fyU<) and C J (fjj f o r CJ)j{ i n 36. equation (82) and dropping the t signs, "VCO , U+ A \ C v ^ v A cV<¥k' < ^ * i « I . -(86) Thus since A (87) i s an even function of x , equation (86) implies that .00 2. \ X U,A> A x = I ' 0 (88) f o r a l l A>o » and consequently, 2 L « 2 \ r<U » i . (89) A-*0 )o But we s h a l l show that and hence that To prove equation (90) we notice that f o r every r e a l number ^ , 4̂ 0 - Q (92) since I approaches zero and the path of integration i s of f i n i t e length. Hence equation (89) implies that L *' • (93) Now we choose $ i n such a way that (94a) t94b) 37. Substituting equation ( 8 4 ) into equation (8 .6) , changing the " variable of integration to ^ = * - < s , and expanding the t r i - gonometric functions with the help of equations ( 9 4 ) , we find that „ 5, S V * (93) where PIV) c V v U A * * * ) , ( 9 6 a ) O^V) c c v c*sVl*+S) _ ( 9 6 b ) Integrating by parts in equation (95) and using the formulae [ s " f t \ v , ^ x Ax * - I J L ^ ^ ^o<Y><a, (97a) and * I * ^ 0 < b < < x ; V*/D/ we obtain ( 9 8 ) But in accordance with equations J94) and (9?)» we find that the f i r s t term approaches ~^^fz as L\ —*0 , and the second term vanishes since qjOk*&) and fyU^-a) must be of order L\ . Finally, v/e find that we may interchange the order of integration i n the last term, since ^ ̂ • <J f' x ̂ x exists. Hence the last term vanishes also, and I- fe 1 5 ^ C * / z . (99) 38. APPENDIX B. DERIVATION OF EQUATIONS (35a) AMD (35b) V From eq u a t i o n ( 9 ) , we have - ^ v n / ^ w ^ ) , ( 1 0 0 ) which w i t h equations (4), (8), (9), and (20) becomes " e ' - i H ^ - ^ ^ V C ^ ^ ^ - ' ^ W ) . (io!! Hence f o r N S O and E = E * , E, - (£V***TJj\«^ ( 1 0 2) S e t t i n g E * 6 A and <X.'* 1(V ( c . f . equations (4b) and (4c)) i n equation (8b) we o b t a i n ( N^V - o = ^ ^ v ( » - e ^ ) # . ( 1 0 3 ) Hence w i t h equations (20a) and (25a) we o b t a i n e q u a t i o n (35a) E, - fc*** ( e * ^ - \ ) / 2 Y \ { \ « V * ) ^ ( 1 0 4 ) To o b t a i n E 2 w © d i f f e r e n t i a t e e q uation (101): E "*0 • («X+C^UV«W).c»5) For \= O we f i n d * - , 1 1 0 7 1 39. and where a l l the expressions are to be evaluated at £ = e» and N * 0 . Substituting into equation (105) we find with the help of equation (102) that <*%., * . . { - « . Finally, we find from equation (4b) and from equation (8b), set OI'-A^ , and use equations (35a) and (109) to obtain equation (35b). 40.' APPENDIX C. CALCULATION OP THE COEFFICIENTS E, AND Ei BY MEANS OF EQUATIONS (43) AND (45) We f i n d E, from equations (14), (15), and (42): E , = , - [ V o c V w C a ^ / ^ L - ^ - ) ^ (110) On evaluating the i n t e g r a l , we f i n d that equation (110) agrees exactly with equation (35a). To f i n d E z , we f i r s t use equations. (14), (15), and (42) to calculate the matrix elements from which we obtain . Y<£, r ( S ( ^ / [ ^ Z ( ^ C ^ ] / K M ^ ^ ) ^ K ( ^ l + ^ ) (112) S i m i l a r l y we use equations (14), (15), (17), and (42) to f i n d the matrix elements Writing the cosine function i n exponential form, and using the equation (114) I n t h i s equation ot' represents the value of oUej f o r 41. we obtain where N = V/ATT Y T,U0 = T»t-*0 (116) (117) (118) (119) and T \ U ) - - Ci - e 4 ( i C ) / ( JUw^CU-; (> ) 2 I* - ' \ / ( ^ ) ( JU- . - ^ \ (120) Vik) , Xjl-Jc) . ( 1 2 1 ) From equations (118) and (121), and the fact that T , i s an even function of h , the integral in equation (45) takes the form X -_ ti W a r , , T , 4 T ; V T ; ) ^ , (122) We now evaluate the integral in the last equation by means of contour integration. Choosing the contour which runs along the real axis from - ft. to + ̂ . (where ft > O ) and then around the semi-circle in the upper half- plane from + ft. to - ̂  , we find that the integral around the semi-circle approaches zero as R —• a> . Hence X - ZW*l4 (Sum of residues of the inte-• grand at i t s poles i n the (123) upper half-plane.) 4-2. T\ and T-s have poles at / L - ^ n > only. The : residue of 3>T, i s while that of i s * = ' P (125) Adding ft, and and multiplying by 21*;. IV* , we find that the contribution of T, and T 3 to X i s exactly equal to the f i r s t term of equation (35k). In order to evaluate the residues of T** • and TA* i we find from equations (18) that ^kt****) CJk^) ^ . . p . ( 1 26) and hence that e +Q. = -2*^/Lkr*\^**<-ol*<*) (127) Equation (127) t e l l s us that * T \ ~ has a pole corres- ponding to each level (i.e., for -V = XQf ) as well as to E P ( A*.=-;̂  ). The.residue at k.- + ̂ ' i s V^H*""^"^^^^ . , (128) Evaluating this residue with the help of equations (4) and >e with equation (112), W/(e»-eo') (129) (25a), we find that i n accordance 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 levels. Finally we evaluate the residue of " V • T V at <W -i[W(tiVl where V k ) ( k tip)' Vil^/zl&^'PKi'l* ( W ) (130) Using the equation we obtain with the help of equations (4): But on substituting X = A(b in equation (133) we find with the help of equation (35a) that 2ir;NRA(Jk^) i s exactly the second term in equation (35b). (133) 44. BIBLIOGRAPHY 1. S. Dushman, 2. G.. Herzberg, 3. E. C. Kemble, 4. M. F. Manning, 5. M.. F. Manning and M. E. B e l l , 6. P. M. Morse and E. C. G. Stuckelberg, 7. N. Rosen and P. M. Morse, "Elements of Quantum Mechanics", Wiley, pp. 214-218, and re- ferences given there. "Infrared and Raman Spectra", Van Nostrand, pp. 221-224, and references given there. "Fundamental Principles of Quantum Mechanics", McGraw-Hill, Ch.VI. Jour.of Chem.Phys., 3, 136 (1935). Rev.Mod.Phys, 12, 215 (1940). Helv;Phys.Acta, 4, 337 (1931). Phys.Rev., 42, 210 (1932). j


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