{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Campus":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Degree":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","DegreeGrantor":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","Genre":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Program":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Science, Faculty of","@language":"en"},{"@value":"Physics and Astronomy, Department of","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Campus":[{"@value":"UBCV","@language":"en"}],"Creator":[{"@value":"Carter, David Southard","@language":"en"}],"DateAvailable":[{"@value":"2012-03-21T18:42:21Z","@language":"en"}],"DateIssued":[{"@value":"1948","@language":"en"}],"Degree":[{"@value":"Master of Arts - MA","@language":"en"}],"DegreeGrantor":[{"@value":"University of British Columbia","@language":"en"}],"Description":[{"@value":"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\r\nsolution 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.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/41659?expand=metadata","@language":"en"}],"FullText":[{"@value":"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\u00a9 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. \u00a3 . . . . 7 PLATE I I I The Eigenvalues of Problem D as a Function of X 27 PLATE IV * - as a Function of \u00a3 . . . . . 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\u00bb 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 \u2014 REGION A REGION 5 Legend: \u2022The levels of problems A a n d B are d r a w n in the appropriate B O X \u2022The Eoo.n \u00b0f 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) \u00ab-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\u00bb 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 \u00ab - 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 \u00b0 - 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
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 \u00bb , w = . * v _ . i ) 2 > 3 , ( 12) A s . E increases, the sign of ^ changes from - to * at each Eto,vv. a n < i from \u2022*- 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\u00bb) and eliminating (_. by means of equation ( 7 a ) : \u2022^As \"vl\u2014>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 o \u00bb 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 \u00ab = l | f U = \u00a3*<* 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\u00ab^(*} 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, \u2022k6* H ^ U ^ \u00bb ' , and Ulxi be the Hamiltonian opera-tors of problems A, B, and D respectively: \u2022 v \u00bb \u2022 l 3 7 a ) * V\u201e , (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 .