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On the behaviour of the solutions of certain Schredinger equations for vanishing potentials Rome, Tovie Leon 1961

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ON THE BEHAVIOUR OF THE SOLUTIONS OF  CERTAIN SCHROEDINGER EQUATIONS FOR VANISHING POTENTIALS by T o v i e Leon Rome  B.A., U n i v e r s i t y o f B r i t i s h Columbia, 1959  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  i n the Department o f PHYSICS  We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d  THE  standard  UNIVERSITY OF BRITISH COLUMBIA September 1961  In presenting  t h i s t h e s i s 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 an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. f o r extensive  I f u r t h e r agree that permission  copying of t h i s t h e s i s f o r s c h o l a r l y purposes may  granted by the Head of my Department or by h i s  be  representatives.  I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission.  Department of  y£  /<?  S  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date  (i)  ABSTRACT In  s t u d y i n g the diamagnetism o f f r e e e l e c t r o n s i n a  u n i f o r m magnetic f i e l d i t was found t h a t r e d u c i n g the f i e l d to  zero i n the wavefunction  d i d n o t y i e l d the e x p e r i m e n t a l l y  i n d i c a t e d f r e e p a r t i c l e plane wave wavefunction.  However,  s o l v i n g the Schroedinger E q u a t i o n r e s u l t i n g from s e t t i n g the field  equal t o zero i n the o r i g i n a l e q u a t i o n d i d y i e l d a  p l a n e wave wavefunction.  T h i s paradox was n o t found to he  p e c u l i a r .to. the case o f a charged p a r t i c l e i n a u n i f o r m magnetic f i e l d but was found t o occur i n a number o f o t h e r systems. behaviour  I n o r d e r to g a i n an u n d e r s t a n d i n g s  dimensional  of this  unexpected  the f o l l o w i n g systems were analyzed: the onesquare w e l l p o t e n t i a l ; a charged,  spinless par-  t i c l e i n a Coulomb f i e l d and i n a uniform e l e c t r i c f i e l d ; one-dimensional  harmonic o s c i l l a t o r ;  p a r t i c l e i n a u n i f o r m magnetic f i e l d .  and a charged, From these  a  spinless  studies  •if t h e f o l l o w i n g were o b t a i n e d : c o n d i t i o n s f o r determining the r e s u l t o f r e d u c i n g the p o t e n t i a l i n a wavefunction;  the con-  d i t i o n under which the p o t e n t i a l o f a system may be switched off  w h i l e m a i n t a i n i n g the energy o f the system constant; the  r e l a t i o n s h i p between t h e r e s u l t ' o f p h y s i c a l l y s w i t c h i n g o f f a potential,  the r e s u l t o f r e d u c i n g i t i n the wavefunction,  and  t h e s o l u t i o n o f the Schroedinger E q u a t i o n o b t a i n e d by d e c r e a s ing  the p o t e n t i a l t o zero i n the o r i g i n a l wave equation; and  a g e n e r a l p r o p e r t y o f any w a v e f u n c t i o n w i t h r e s p e c t t o r e d u c i n g any parameter w i t h i n t h i s  wavefunction.  (ii)  ACKNOWLEDGEMENTS  I would l i k e to thank Dr. Robert B a r r i e , who suggested the problem i n t h i s t h e s i s , f o r h i s s u p e r v i s i o n and many i n t e r e s t i n g d i s c u s s i o n s . I a l s o w i s h t o g r a t e f u l l y acknowledge the f i n a n c i a l a s s i s t a n c e g i v e n me by t h e N a t i o n a l Research C o u n c i l i n t h e form o f " a b u r s a r y .  TABLE OF CONTENTS Page ABSTRACT  i  ACKNOWLEDGEMENTS  i  i  CHAPTER I  II  III  IV  V  INTRODUCTION  1  1.  General D i s c u s s i o n  1  2.  Boundary C o n d i t i o n s  3  3.  Singularities  5  4-.  Terminology  7  5.  Aim o f T h e s i s  9  ONE DIMENSIONAL SQUARE WELL POTENTIAL  10  1.  N e g a t i v e Energy S o l u t i o n s  10  2.  P o s i t i v e Energy S o l u t i o n s  15  COULOMB POTENTIAL  19  1.  N e g a t i v e Energy Case  19  2.  P o s i t i v e Energy S o l u t i o n s  24  UNIFORM ELECTRIC FIELD  28  1.  D e s c r i p t i o n o f System  28  2.  Wavefunction Behaviour and B l o c k Diagram  28  3.  Physical Analysis  36  THE HARMONIC OSCILLATOR  39  1.  Time Independent Treatment  39  2.  Time Dependent  41  Treatment  CHAPTER VI  Page UNIFORM MAGNETIC FIELD  51  1.  Quantum M e c h a n i c a l Treatment  51  2.  Classical  53  3.  Wavefunction  A n a l y s i s o f Switching O f f Essential Singularity for  58  Zero F i e l d VII  CONCLUSION  61  1.  D i s c u s s i o n o f Switching O f f Processes  61  2.  R e d u c t i o n o f the P o t e n t i a l i n Wave 64  Function 3.  Decreasing  the P o t e n t i a l i n the Wave  Equation 4.  70  D i s c u s s i o n o f the O r i g i n a l Problem and Paradox o f an E l e c t r o n i n a Uniform Magnetic  5.  A Comment on the Uniform E l e c t r i c  6.  73  Field Field  Case  75  Summary o f T h e s i s C o n c l u s i o n s  75  APPENDIX:  80  SYSTEMS CONTAINED WITHIN A PHYSICAL CONTAINER '  BIBLIOGRAPHY  84  CHAPTER I INTRODUCTION  1.  GENERAL DISCUSSION One o f t h e problems which a r i s e s i n studying the  m a g n e t i c p r o p e r t i e s o f s o l i d s i s t h a t o f the o r b i t a l d i a magnetism o f f r e e e l e c t r o n s .  T h i s can be t r e a t e d  1  by s o l v i n g  t h e Schroedinger E q u a t i o n f o r an e l e c t r o n i n a u n i f o r m magnetic field.  S i n c e the case o f a weak f i e l d i s o f i n t e r e s t ,  t h e r e s u l t o f r e d u c i n g the f i e l d t o zero i n the s o l u t i o n o f t h i s S c h r o e d i n g e r E q u a t i o n was i n v e s t i g a t e d . The apparent  experimental r e s u l t o f s w i t c h i n g o f f the  magnetic f i e l d i s t h a t t h e e l e c t r o n d i r e c t l y and c o n t i n u o u s l y goes over t o a f r e e p a r t i c l e whose e i g e n f u n c t i o n i s a p l a n e wave.  The mathematical  treatment o f d e c r e a s i n g the f i e l d to  z e r o i s n o t so s t r a i g h t f o r w a r d . I f the f i e l d i s decreased i n t h e o r i g i n a l wave e q u a t i o n f o r the e l e c t r o n i n the f i e l d the r e s u l t i s an e q u a t i o n whose s o l u t i o n i s a p l a n e wave. if  t h e f i e l d i s reduced to zero i n the s o l u t i o n o f the o r i g i n a l  e q u a t i o n a p l a n e wave i s not o b t a i n e d . is  However,  Whereas the former  case  c o n s i s t e n t w i t h what i s expected the l a t t e r case i s i n c o n -  s i s t e n t w i t h what appears  to be experimental evidence.  So the  s i t u a t i o n i s t h a t the r e s u l t i s c o n s i s t e n t o r i n c o n s i s t e n t w i t h  2 t h e apparent  experimental o b s e r v a t i o n s depending a t which  stage o f the mathematics the f i e l d i s decreased  to zero.  p a r a d o x o f n o t o b t a i n i n g the e x p e r i m e n t a l l y i n d i c a t e d  This  free  p a r t i c l e plane wave s o l u t i o n by r e d u c i n g the f i e l d ,  o r poten-  tial,  equation  i n the e i g e n f u n c t i o n s o l u t i o n o f the o r i g i n a l  i s n o t p e c u l i a r to the case o f an e l e c t r o n i n a •uniform magnetic f i e l d .  I t a l s o o c c u r s i n o t h e r systems such as a one-  d i m e h s i o n a l harmonic o s c i l l a t o r , p o t e n t i a l and a charged, or  uniform e l e c t r i c  a p a r t i c l e i n a square w e l l  s p i n l e s s p a r t i c l e i n e i t h e r a Coulomb  field.  The p r e c e d i n g suggests the f o l l o w i n g q u e s t i o n s r e g a r d i n g a p a r t i c l e e x p e r i e n c i n g an e x t e r n a l f i e l d : (a)  Under what c o n d i t i o n s , i f any, i s the same r e s u l t o b t a i n e d by (i)  r e d u c i n g the f i e l d  to zero i n the s o l u t i o n to  the o r i g i n a l wave e q u a t i o n and by (ii)  s o l v i n g the e q u a t i o n o b t a i n e d from the o r i g i n a l one by l e t t i n g  (b)  the f i e l d go t o zero?  What i s the meaning o r s i g n i f i c a n c e o f those  situations  where the r e s u l t s a r e d i f f e r e n t depending on whether the f i e l d approaches zero i n the o r i g i n a l e q u a t i o n o r i n i t s solution? The  s i t u a t i o n d i s c u s s e d above may be i l l u s t r a t e d b y  t h e b l o c k diagram  3 corner 1  >  corner 3  solution i  field  to 0  ? | field  corner 2  >  to 0  corner U  solution where the c o r n e r s a r e o c c u p i e d as f o l l o w s :  corner one by the  o r i g i n a l wave e q u a t i o n f o r the p a r t i c l e i n the f i e l d ;  corner  two by the f r e e p a r t i c l e equation o b t a i n e d f r o n the e q u a t i o n in  corner one by d e c r e a s i n g the p o t e n t i a l to zero;  corner  t h r e e by the s o l u t i o n o f the e q u a t i o n i n corner one; and c o r n e r f o u r by the s o l u t i o n o f the e q u a t i o n i n corner two. If  the e n t r y i n corner f o u r may a l s o be o b t a i n e d by r e d u c i n g  t h e p o t e n t i a l i n the wavefunction occupying  corner three the  b l o c k diagram i s s a i d to be c l o s e d o r completed. I n terms o f t h i s b l o c k diagram the p r e c e d i n g q u e s t i o n s may be simply s t a t e d as i n the f o l l o w i n g . (a)  Under what c o n d i t i o n s can the b l o c k diagram be completed?  (b)  What i s the s i g n i f i c a n c e o f those s i t u a t i o n s i n which the b l o c k diagram cannot be completed?  2.  BOUNDARY. CONDITIONS To  completely  and u n i q u e l y d e s c r i b e a p h y s i c a l  system i n e i t h e r quantum o r c l a s s i c a l mechanics boundary cond i t i o n s must be i n t r o d u c e d i n a d d i t i o n to the d i f f e r e n t i a l equation.  S i n c e the Schroedinger  E q u a t i o n alone i s i n s u f f i -  c i e n t to f u l l y d e s c r i b e a p h y s i c a l s i t u a t i o n the block diagram,  4  as i t stands, d e a l s w i t h i n c o m p l e t e l y s p e c i f i e d systems. Unexpected r e s u l t s may t h e r e f o r e o c c u r .  I f boundary c o n d i -  t i o n s a r e i n t r o d u c e d i n c o n j u n c t i o n w i t h the wave equations, c o r n e r s one and two o f the block diagram w i l l g i v e a complete d e s c r i p t i o n o f t h e i r r e s p e c t i v e p h y s i c a l s i t u a t i o n s and the problem w i l l be f o r m u l a t e d i n terms o f f u l l y systems.  specified  H e n c e f o r t h i n t h i s t h e s i s the b l o c k diagram w i l l be  c o n s i d e r e d o n l y i n terms o f f u l l y s p e c i f i e d systems, t h a t i s , where c o r n e r s one and two a r e o c c u p i e d by t h e boundary c o n d i t i o n s corresponding t o t h e i r r e s p e c t i v e systems i n a d d i t i o n t o the r e s p e c t i v e wave e q u a t i o n . S i n c e the t r a n s i t i o n from corner one to corner two i s made by d e c r e a s i n g t h e p o t e n t i a l to zero i n some manner, it  f o l l o w s t h a t the boundary c o n d i t i o n s i n corner two should  be o b t a i n e d by d e c r e a s i n g the p o t e n t i a l i n the boundary c o n d i t i o n s o f corner one.  W i t h t h e e x c e p t i o n o f the p o s i t i v e  t o t a l energy Coulomb and u n i f o r m e l e c t r i c f i e l d cases, the above procedure r e s u l t s i n the boundary c o n d i t i o n s being t h e same i n both c o r n e r s one and two.  These e x c e p t i o n s w i l l be  t r e a t e d i n s e c t i o n two o f chapter t h r e e and i n chapter f o u r . With r e g a r d to boundary c o n d i t i o n s two c a t e g o r i e s o f systems may be d i s t i n g u i s h e d . These types o f systems a r e t h o s e i n which (a) the o n l y p o t e n t i a l o r a f i e l d  the p a r t i c l e experiences i s  t h a t due to an e x t e r n a l source; o r  5 (b) i n a d d i t i o n to the p o t e n t i a l i n (a) the p a r t i c l e i s s u b j e c t to geometric c o n s t r a i n t s . The o n l y type o f geometric is  c o n s t r a i n t considered i n t h i s  thesis  t h a t o f a p a r t i c l e being contained i n a p h y s i c a l c o n t a i n e r .  I n the l a t t e r case, as h e r e i n c o n s i d e r e d , the p a r t i c l e i s always bound whereas i n the former case i t may o r may not be bound. ic  The o r i g i n a l system o f an e l e c t r o n i n a u n i f o r m  field  magnet-  can be made to i l l u s t r a t e e i t h e r type o f system.  1.  Corresponding  to (a) the system simply c o n s i s t s o f an o t h e r w i s e  unconstrained  e l e c t r o n moving i n a u n i f o r m magnetic f i e l d which  fills  a l l space.  I n t h i s case the e x t e r n a l source i s t h a t  w h i c h produces the magnetic f i e l d .  An example o f (b) i s an  e l e c t r o n c o n f i n e d w i t h i n a c r y s t a l and e x p e r i e n c i n g a c o n s t a n t magnetic f i e l d a t a l l p o i n t s w i t h i n the c r y s t a l . I n the main body o f t h i s t h e s i s o n l y the f i r s t of  system w i l l be a n a l y s e d .  system w i l l be d i s c u s s e d . to  type  I n the appendix the second type o f The o n l y t y p e . o f geometric c o n s t r a i n t  be t r e a t e d i n the appendix i s t h a t o f a p h y s i c a l c o n t a i n e r  which i s mathematically  d e s c r i b e d by an abrupt, i n f i n i t e w a l l  potential. 3.  SINGULARITIES S i n c e a d i f f e r e n t i a l e q u a t i o n i s c h a r a c t e r i z e d by the  number and type o f i t s s i n g u l a r i t i e s ,  the s i n g u l a r i t i e s o f the  d i f f e r e n t i a l equations i n c o r n e r s one and two o f the b l o c k diagram w i l l be s t u d i e d .  The s i t u a t i o n  the b l o c k diagram  6 r e p r e s e n t s i s t h a t o f comparing the r e s u l t o f a p p l y i n g a g i v e n procedure  to the s o l u t i o n  o f an i n i t i a l  equation with  t h e s o l u t i o n o f a d e r i v e d e q u a t i o n where the d e r i v e d e q u a t i o n is  o b t a i n e d by a p p l y i n g the same procedure  equation.  I n e f f e c t the s o l u t i o n s o f two d i f f e r e n t i a l  e q u a t i o n s , an i n i t i a l If  to the i n i t i a l  the i n i t i a l  and a d e r i v e d one, a r e being compared.  and d e r i v e d equations have d i f f e r e n t  s i n g u l a r i t i e s these equations a r e from d i f f e r e n t it  may n o t be r e a s o n a b l e t o a p r i o r i  solution  to go over to the s o l u t i o n  by a p p l i c a t i o n  types o f  c l a s s e s and  expect the i n i t i a l o f the d e r i v e d e q u a t i o n  o f the same procedure.  With the e x c e p t i o n o f the square w e l l case, i n a l l the cases h e r e i n c o n s i d e r e d the d e r i v e d e q u a t i o n d i f f e r s from the i n i t i a l  one w i t h r e g a r d to a s i n g u l a r i t y c l a s s i f i c a t i o n  as seen from the f o l l o w i n g t a b l e :  D i f f e r e n t i a l Eauation  Singularity at origin.  Singularity at i n f i n i t y type  order  Free particle  none  irregular  fourth  Square w e l l  none.  irregular  fourth  regular  irregular  fourth  none  irregular  fifth  Harmonic o s c i l l a t o r  none  irregular  sixth  U n i f o r m magnetic  none  irregular  sixth  Coulomb Uniform  potential  field electric  field  field  7 S i n c e the wave e q u a t i o n f o r both the f r e e p a r t i c l e and the square w e l l p o t e n t i a l have the same s i n g u l a r i t y p a t t e r n the square w e l l p o t e n t i a l w i l l  be t r e a t e d f i r s t .  The Coulomb  e q u a t i o n i s t r e a t e d next s i n c e i n a d d i t i o n to i t s r e g u l a r s i n g u l a r i t y a t the o r i g i n i t has the same type o f s i n g u l a r i t y at  infinity  as does the f r e e p a r t i c l e e q u a t i o n .  n e g a t i v e and p o s i t i v e t o t a l energy  The cases o f  are treated separately f o r  both the square w e l l and the Coulomb p o t e n t i a l s .  The u n i f o r m  e l e c t r i c f i e l d , whose e q u a t i o n has a s i n g u l a r i t y a t i n f i n i t y one o r d e r g r e a t e r than has t h e f r e e p a r t i c l e equation, i s t r e a t e d next.  Chapters  s i x and seven are devoted  to the  harmonic o s c i l l a t o r and u n i f o r m magnetic f i e l d cases whose e q u a t i o n s have s i n g u l a r i t i e s a t i n f i n i t y then the f r e e p a r t i c l e 4.  two o r d e r s l a r g e r  equation.  TERMINOLOGY B e f o r e completing  t h i s i n t r o d u c t i o n the terminology  a s s o c i a t e d w i t h the p o t e n t i a l going to zero w i l l  be s p e c i f i e d .  The word "reduce"' (and i t s d e r i v a t i o n s ) r e f e r s o n l y to the p o t e n t i a l going to zero i n the wavefunction.  That i s ,  " r e d u c t i o n " i s a s s o c i a t e d w i t h the step from corner three to  corner f o u r i n the b l o c k diagram.  p u r e l y mathematical  T h i s term r e f e r s to a  procedure w i t h no dependence on, o r r e l a -  t i o n t o , any parameter o r v a r i a b l e ; f o r example, no c o n n e c t i o n w i t h time. be  "reduced"  An example o f r e d u c t i o n i s l i m f ( x ) ; x i s s a i d to to b.  8 The  step from  corner one to corner two, t h a t i s , the  p o t e n t i a l going to zero i n the wave equation, w i l l n o t a t p r e s e n t have any d e f i n i t e term a s c r i b e d to i t .  Non-committal  terms such as " t h e p o t e n t i a l i s decreased" o r the " p o t e n t i a l goes to zero" w i l l be used. The  e x p r e s s i o n " s w i t c h o f f " and i t s d e r i v a t i v e s  r e f e r s o n l y to the p h y s i c a l p r o c e s s o f the p o t e n t i a l d i m i n i s h e d to z e r o .  being  The p h y s i c a l p r o c e s s o f " s w i t c h i n g o f f "  a p o t e n t i a l i s a time dependent p r o c e s s i n which the p o t e n t i a l is  a f u n c t i o n o f the time.  F o r example, the s w i t c h o f f may be  e x p o n e n t i a l w i t h a time constant, a step f u n c t i o n w i t h r e s p e c t to  time o r l i n e a r over a time i n t e r v a l .  To i n c o r p o r a t e the  time dependence o f the s w i t c h o f f i n the wave e q u a t i o n r e q u i r e s a tiiae dependent H a m i l t o n i a n . in  However the p o i n t o f i n t e r e s t  t h i s t h e s i s i s t o d e s c r i b e the r e s u l t o f s w i t c h i n g o f f  r a t h e r than to d e s c r i b e the behaviour o f t h e system w h i l e the p o t e n t i a l i s being switched o f f .  Hence the p r e c i s e time  dependence o f t h e s w i t c h o f f i s n o t o f i n t e r e s t and a l l the H a m i l t o n i a n s w i l l be independent  o f time r e g a r d l e s s o f whether  t h e time dependent o r independent chapter f i v e a d i s t i n c t i o n w i l l types o f switch o f f .  wave e q u a t i o n i s used.  In  be drawn between two d i f f e r e n t  5.  AIM OF THESIS  In t h i s thesis the various aforementioned  systems  are analyzed with the following intentions: (a) to obtain general c r i t e r i a f o r determining the wavefunction obtained by reducing the p o t e n t i a l to zero i n the i n i t i a l  wavefunction;  (b) to determine under what conditions the block diagram i s completed  and the meaning of such a completion;  (c) to determine the meaning of those situations i n which the block diagram i s not closed; and (d) to determine the relationship between reducing the p o t e n t i a l i n the wavefunction, decreasing the p o t e n t i a l i n the wave equation and the method of switching o f f the p o t e n t i a l .  CHAPTER I I ONE DIMENSIONAL SQUARE WELL POTENTIAL  1.  NEGATIVE ENERGY SOLUTIONS A p a r t i c l e having negative t o t a l energy I n a square v e i l  p o t e n t i a l , whose p o t e n t i a l i s zero a t i n f i n i t y , corresponds to the p h y s i c a l  s i t u a t i o n o f a p a r t i c l e whose k i n e t i c energy i s  l e s s than the absolute value o f i t s negative p o t e n t i a l energy i n the r e g i o n where the p o t e n t i a l i s non-zero.  I n the r e g i o n  where the p o t e n t i a l i s non-zero the c r i t e r i a f o r the above system are: (2.1) (2.2)  E -V  0*(E|  >/0  ^VO  and  0* T* Va  where E denotes the t o t a l energy, V the p o t e n t i a l energy and T the k i n e t i c energy.  Both c o n d i t i o n s f o l l o w from the conserva-  t i o n o f energy and the f a c t that the k i n e t i c energy i s p o s i t i v e w h i l e both the t o t a l and p o t e n t i a l energies are negative.  The  p a r t i c l e ' s negative t o t a l energy, caused by the p o t e n t i a l energy dominating over the k i n e t i c energy, i m p l i e s that the p a r t i c l e i s confined w i t h i n the r e g i o n o f the p o t e n t i a l .  In  c l a s s i c a l mechanics there i s no p o s s i b i l i t y o f the p a r t i c l e leaving  the r e g i o n of the w e l l .  I n quantum mechanics the  p r o b a b i l i t y of the p a r t i c l e being outside the r e g i o n o f the  11 p o t e n t i a l i s s m a l l and decreases e x p o n e n t i a l l y as the d i s t a n c e f r o m the r e g i o n o f p o t e n t i a l i n c r e a s e s . The follows:  V  where  square w e l l p o t e n t i a l , V, may be d e s c r i b e d as  v-f- - Vo 0  i s positive.  (2.3)  £?  (2.4)  & *  To m a i n t a i n  and To  E  *  ^  V  P A^«noix B  The wave e q u a t i o n  - J * - °  V  ' -°  >  M  >  i s then  *  ' "'"I  are imposed. +  B sin  ccx  conditions  The s o l u t i o n o f ( 2 . 3 )  where  c{ - +1  2m j£+v ) ' 0  i s r e a l by (2.1) . a v o i d an unbounded s o l u t i o n and to p e r m i t  the boundary c o n d i t i o n s t h a t t o i n f i n i t y a r e imposed.  C e' *  <f> = U and  •  a  c o n t i n u i t y a t x = ± a the boundary  <p'(-«.) =  and is  ^  (  -<K < x <: a  p  e  <^  normalization  tends to zero as |*/  The s o l u t i o n o f ( 2 . 4 )  tends  i s then  xxx.  x<-a  i s r e a l and p o s i t i v e .  By matching the s o l u t i o n s a t  two c o n d i t i o n s and t h e i r corresponding  x=»±a. 2  s o l u t i o n s are o b t a i n e d .  T h e s e c o n t i n u i t y c o n d i t i o n s on the w a v e f u n c t i o n determine the allowed  v a l u e s f o r the t o t a l energy E .  12  (a) The  s o l u t i o n corresponding to the  condition  «. t a n oca. • 0  B cos «<x  (2.5)  By  - a<  x<a.  matching a t e i t h e r boundary the r e l a t i o n between the O B e ^  coefficients i s  obtained  from the n o r m a l i z a t i o n  (b)  s o l u t i o n a s s o c i a t e d w i t h the  The  ci c o t oca = -  [ C e" **  (2,5')  X  p  cos  condition  .ca.  f  + oo  1^1  dx*/.  condition  is  x>a. -a <X<a  x< - a f __ * A v> ~  A / f/ * — l n " v * cosot.a + a^'oc"-^j sin<*a. cos a: a 3  cos oca  ^  are obtained  and  as before  1  by matching a t a boundary and  from  normalization. As order and  oc  V  i s reduced to zero,  0  ft  goes to i«.  In  t h a t the wavefunction be w e l l behaved the energy, must be determined from the a p p r o p r i a t e  condition.  The  s o l u t i o n when  first a  p e r m i s s i b l e values  c o n d i t i o n i s ct tan oca- (3 .  becomes -ip i s ot=(3= E = o .  p  continuity An  obvious  The  other  o f E are determined by s u b s t i t u t i n g -i(3  13 for  o(  tan  (-ip  The  solution of tills  Since  i n t h e above c o n d i t i o n and s o l v i n g  a)  |3  TMs  V  zero  c<  the  a  i s reduced  0  and  potential  noted  zero  this  ^  to zero.  go t o z e r o .  does.  yields  going  i s an  unacceptable  f o r E i s zero.  V  0  result  i s reduced  to  Thus A, B, and C go t o z e r o  Hence f o r e i t h e r  as  c o n d i t i o n t h e wave-  as t h e p o t e n t i a l  to zero  result  equation w i l l  does.  consistent with  o f decreasing  I t should that E  the ensuing decreased  also being  potential  (2.2).  i n t h e wave  t o d e t e r m i n e what  goes t o zero  wave e q u a t i o n i s s o l v e d .  such  criteria  the p o t e n t i a l  now be s t u d i e d i n o r d e r  h a p p e n s when t h e p o t e n t i a l  i n t h e wave  that f o r a l l intermediate values  the c r i t e r i a  that the f i n a l  to  zero i s a (free) p a r t i c l e w i t h n e i t h e r k i n e t i c  r e s u l t o f decreasing  energy, t h a t i s , t h e t o t a l potential  zero.  o f the  (2.2) o f the system a r e s a t i s f i e d i t  apparent  tial  equation  I f the p o t e n t i a l  is  the  |S = - c o ,  i s  the i d e n t i c a l  Hence as  tanh|5a»-A  i s i t s only acceptable value i s i n accord with the  The  is  a  that the c o n t i n u i t y conditions implying  potential  and  positive  positive  acot«.a -^  f u n c t i o n goes t o zero be  i s e q u i v a l e n t to  Hence t h e o n l y a c c e p t a b l e v a l u e  condition  when  forfinite,  i s d e f i n e d as being  solution. The  I .  a  the obtained  goes t o zero  S i n c e E goes t o zero  the potential nor poten-  energy, E, i s z e r o .  a l l the energy l e v e l s as V does  Hence as  collapse to  u  dl^  (2.6)  = 0  is  the e q u a t i o n d e s c r i b i n g the r e s u l t o f d e c r e a s i n g p o t e n t i a l  in  the above manner.  apparent  +  dx*  That i s , i t i s (2.6) [ r a t h e r than the  z*E T =o , K  ( E * o) '  l  ]  '  v  w h i c h d e s c r i b e s the system when the p o t e n t i a l i s decreased 3 as  above.  By a w e l l known theorem  r e g a r d i n g the s o l u t i o n o f  L a p l a c e ' s E q u a t i o n w i t h boundary c o n d i t i o n s , the s o l u t i o n o f y  (2.6) w i t h the boundary c o n d i t i o n s t h a t Ixl is  tends to i n f i n i t y i s T=0.  I f , however, the p o t e n t i a l  decreased w i t h o u t e x p l i c i t l y r e q u i r i n g  o f - t h e system be s a t i s f i e d  goes to zero as  t h a t the c r i t e r i a  f o r a l l i n t e r m e d i a t e v a l u e s o f the  p o t e n t i a l the r e s u l t i n g e q u a t i o n i s  (2.7)  4i£ i-EY=o. +  dx  1  T h i s i s the e q u a t i o n o b t a i n e d i f the p o t e n t i a l i s mathematically (2.7)  s e t equal to zero i n ( 2 . 3 ) . is  f Ae" + Be* B  < r x  where  w i t h the r e a l p a r t non-negative. x = *• 00 B i s zero.  i m p l i e s A i s zero and  IT i s i n g e n e r a l complex  The c o n d i t i o n t h a t M-^o  Y=o  x=- 00  implies  r e g a r d l e s s whether the c r i t e r i a  a r e e x p l i c i t l y i n t r o d u c e d o r not.  obtaining  at  4^ =»o a t  Hence the o n l y s o l u t i o n c o n s i s t e n t w i t h the  boundary c o n d i t i o n s i s (2.2)  The g e n e r a l s o l u t i o n o f  V-O  The reason f o r  both times i s t h a t the boundary c o n d i t i o n s  15 a r e the same. (2.2)  That i s , even though i n the l a t t e r  treatment  was n o t e x p l i c i t l y a p p l i e d to E i t was i m p l i c i t l y  a p p l i e d s i n c e the boundary c o n d i t i o n s f o r a bound system were  maintained. The  b l o c k diagram can now be c o n s i d e r e d .  The b l o c k  diagram f o r the n e g a t i v e energy s o l u t i o n s f o r the square w e l l p o t e n t i a l i s o c c u p i e d as f o l l o w s : corner one by equat i o n s (2.3) wavefunction  and (2.4)  and the boundary c o n d i t i o n s t h a t the  be zero a t  %- ± 00  ; corner two by the same  boundary c o n d i t i o n s and by e q u a t i o n (2.6)  o r (2.7)  depending  o n what i s s t i p u l a t e d r e g a r d i n g d e c r e a s i n g the p o t e n t i a l ; c o r n e r t h r e e by wavefunction by  Y=0  .  (2.5)  o r (2.5 ) ; f  and corner f o u r  As has been demonstrated the r e s u l t o f r e d u c i n g  t h e p o t e n t i a l i n the wavefunction  (2.5) o r (2.5^) i s V » 0 .  Hence the r e s u l t o f r e d u c i n g the p o t e n t i a l i n corner t h r e e is  t h e same as s o l v i n g corner two and the b l o c k diagram i s  closed. 2.  POSITIVE ENERGY SOLUTIONS The p h y s i c a l s i t u a t i o n which the mathematics o f  t h i s s e c t i o n d e s c r i b e s i s t h a t o f a p a r t i c l e f e e l i n g the e f f e c t o f a square w e l l p o t e n t i a l whose v a l u e i s such t h a t t h e p a r t i c l e ' s t o t a l energy i s p o s i t i v e .  I f the p o t e n t i a l i s  assumed n e g a t i v e the k i n e t i c energy i s then g r e a t e r than the a b s o l u t e v a l u e o f the p o t e n t i a l energy. s c a t t e r i n g by a square w e l l p o t e n t i a l .  T h i s i s the case o f  16  The procedure f o r treating t h i s case i s s i m i l a r to the negative energy case. potential i s different.  However the r e s u l t of reducing the The p o t e n t i a l i s as i n section one  since i t i s assumed to be negative.  For the region  |x/>a  the equation i s again ( 2 . 4 ) but with the boundary conditions <P  that  behaves as a s i n u s o i d a l l y o s c i l l a t i n g function at  X=±co.  Hence  where  f  C sin  is f>- ^ j  2  " ^ '  + D cos (3X For 1*1 < a-  •  the equation i s again (2.3) with the s o l u t i o n  V=  A SinotX + BeOS <*X  where  OL = +  (2m(£ + Vo)  r  The functions <p and V and t h e i r f i r s t derivatives are again matched at  X= ± a  to produce two sets of solutions each  corresponding to a d i f f e r e n t r e l a t i o n between (a)  c£ and f$  <L t a n oi CL = |3 tan ficc  Associated with the condition  i s the s o l u t i o n  5  COS  D cos  OCX  /x/<a  0x  /x/ > a  To maintain continuity of the wavefunction at  V  0  i s reduced to zero,  oC  approaches  and, i n order  that the wavefunction be well-behaved, D approaches B. Bc0S|3X zero.  the  B COS cLQ. = 0 COS ($& .  r e l a t i o n between B and D i s As  |xt=-a  Hence  i s the s o l u t i o n everywhere when the p o t e n t i a l i s  Since t h i s s o l u t i o n i s applicable i n a l l space the  +*>  /  and B i s a r b i t r a r y .  -00  lV\  z  dx = I  i s not applicable  B i s usually chosen to be unity as t h i s  normalizes the function to u n i t f l u x .  (b)  Corresponding is  ai cotaca. = (3 co t pa.  to the c o n d i t i o n  the s o l u t i o n  (2.8')  Asinocx  |x|<a  si n /3x  /x/ > a  C  .  s i n p%  By  the same argument as i n the p r e c e d i n g the s o l u t i o n  is  found to apply a t a l l p o i n t s when the p o t e n t i a l i s reduced  to  zero. As the p o t e n t i a l i s reduced  and  the c o n t i n u i t y c o n d i t i o n s a t  to zero x =* a  a l l o w e d v a l u e s o f E become i d e n t i t i e s . is  ct goes to /S  which determine the Hence t h e end r e s u l t  a f r e e p a r t i c l e w i t h an a r b i t r a r y t o t a l energy E and a  trigonometric  wavefunction.  The r e s u l t o f the p o t e n t i a l going wave e q u a t i o n w i l l now be i n v e s t i g a t e d . energy  to zero i n the U n l i k e the n e g a t i v e  case, E i s n o t bounded by the p o t e n t i a l .  Hence  d e c r e a s i n g the p o t e n t i a l to zero does n o t i n f l u e n c e E and the r e s u l t i n g wave e q u a t i o n i s ( 2 . 7 ) . conditions that  Y  When the boundary  oscillates sinusoidally at  x = - co a r e  imposed the s o l u t i o n n o r m a l i z e d to u n i t f l u x i s (2.9)  COS / 2 m E ' x  (2.9')  s i n jzm£'  o r a l i n e a r combination  X  o f the two.  The p h y s i c a l s i g n i f i c a n c e o f decreasing, the potential  i s t h a t the p a r t i c l e i n q u e s t i o n i s acted upon by a  diminishing force.  When the p o t e n t i a l reaches zero there  18 1 s no f o r c e a c t i n g on t h e p a r t i c l e and i t i s t h e n a f r e e one. T h i s i s c o n s i s t e n t w i t h t h e above r e s u l t s . The b l o c k diagram  f o r t h e p o s i t i v e energy  square  veil  case i s o c c u p i e d as f o l l o w s : c o r n e r one by e q u a t i o n s (2.3) and (2.4) and t h e boundary c o n d i t i o n s t h a t o s c i l l a t e s a t X=ico  <p  sinusoidally  ; c o r n e r two by e q u a t i o n (2.7) w i t h t h e  above boundary c o n d i t i o n s on  4^  j c o r n e r t h r e e by wave-  f u n c t i o n (2.8) o r (2.8'); and c o r n e r f o u r by w a v e f u n c t i o n o r (2.9').  As has been demonstrated  (2.9)  (SIS) reduces t o (2.9), o r  (2.8') t o (2.9')* when t h e p o t e n t i a l i s reduced i n t h e wavef u n c t i o n (2.8) o r (2.8') .  Hence t h e b l o c k diagram i s c l o s e d .  CHAPTER I I I COULOMB POTENTIAL  1.  NEGATIVE ENERGY CASE A p a r t i c l e w i t h n e g a t i v e t o t a l energy i n a Coulomb  field  corresponds to an a t t r a c t i v e  p o t e n t i a l b i n d i n g the  p a r t i c l e t o the source o f the p o t e n t i a l as i n the example of  the hydrogen  atom.  I n t h i s Coulomb case the p o t e n t i a l i s - A A i s a non-negative  where  constant and r i s the d i s t a n c e from the  s o u r c e o f the p o t e n t i a l to the p a r t i c l e .  Corresponding to  t h e c r i t e r i a (2.2) i n chapter two the t o t a l energy i n t h i s c a s e i s bounded as f o l l o w s :  -mA  1  (3.1) In  < E < 0.  f a c t the exact a l l o w e d energy v a l u e s f o r the n e g a t i v e  energy case are (3.2) The  £ = -mA  1  n  o n , z, ,... 3  allowed e n e r g i e s being n e g a t i v e corresponds to the  p a r t i c l e being bound w i t h i n a f i n i t e r e g i o n o f apace.  Hence  the w a v e f u n c t i o n s a t i s f i e s the n o r m a l i z a t i o n c o n d i t i o n  all  space  This normalization condition i n  20 Y goes to zero as any s p a t i a l  t u r n implies that  coordinate  approaches i n f i n i t y . The wave e q u a t i o n f o r t h i s system i s Vf  (3.3) and  l  i s expressed  (E*A)4  - l a  /  =0  i n s p h e r i c a l coordinates.  By i n t r o d u c i n g  t h e quantum numbers I and m the e q u a t i o n i s separated i n the u s u a l manner i n t o angular and r a d i a l equations.  I  f o r a given where  R (r) j s  and m i s the product  ^ (^f)  The s o l u t i o n  t  ltn  i s a s p h e r i c a l harmonic.  m  The r a d i a l  e q u a t i o n f o r the Goulomb f i e l d i s i _  The  <J  MJfU  Lsss\( E+A  -hhMAn))=  o.  P ImlEl s  independent V a r i a b l e r i s r e p l a c e d by  I n terms o f p  the s o l u t i o n i s ^  Kt (fO  C  =  n i  e'  Pk  L i s an a s s o c i a t e d Laguerre  L j ' * ' ( p) polynomial  where and  C  ftt  n o r m a l i z i n g c o e f f i c i e n t to be determined from £ * R*< (*') f  The  normalized  1  <lr = /  and the r e l a t i o n  t o t a l wavefunction i s  isa  21 p = Af* A r  where  i s used  as i t i s e q u i v a l e n t to the  o r i g i n a l d e f i n i t i o n when (3.2) i s used. If  the p o t e n t i a l i s reduced  t o zero then the wavefunction Furthermore zero.  (3.4) a l s o goes to z e r o .  by (3.2) a l l the energy  As the energy  to 'zero through A going  e i g e n v a l u e s c o l l a p s e to  goes to zero the l i n e a r and angular  momentum, and t h e r e f o r e t  go to zero.  t  Hence the wave-  f u n c t i o n goes to zero a: i s r\ The  treatment o f d e c r e a s i n g the p o t e n t i a l i n the  wave e q u a t i o n i s s i m i l a r to t h a t used i n the n e g a t i v e square w e l l case. criteria  I f the p o t e n t i a l i s decreased  energy  such t h a t the  (3.1) and (3.2) a r e s a t i s f i e d , E goes to zero as the  p o t e n t i a l does and the wave e q u a t i o n d e s c r i b i n g the r e s u l t o f d e c r e a s i n g the p o t e n t i a l to zero i n t h i s manner i s  V t - 0  (3.5) in  l  analogy  to ( 2 . 6 ) .  The s o l u t i o n i s  f = 0 by the same  theorem^ s i n c e the boundary c o n d i t i o n i s zero as  r  goes to i n f i n i t y .  goes to  I f the p o t e n t i a l i s decreased  w i t h o u t e x p l i c i t l y s t i p u l a t i n g t h a t E goes to zero the wave e q u a t i o n becomes (3.6) in  analogy  maintained  V to ( 2 . 7 ) .  l  f + £mE  V *o  S i n c e the same boundary c o n d i t i o n i s  the s o l u t i o n i s  s  O  by an argument e s s e n t i a l l y  tlie same as the one f o l l o w i n g e q u a t i o n (2.7) i n chapter two.  » -0  Again to  i s the s o l u t i o n when the p o t e n t i a l i s decreased  zero i n the wave e q u a t i o n r e g a r d l e s s o f the e x p l i c i t  c o n d i t i o n s on E. (that  y  T h i s i s because the boundary c o n d i t i o n  goes t o zero as r goes to i n f i n i t y such t h a t \y\  dp =•1  ) i m p l i e s the p a r t i c l e  i s bound, the  energy i s n e g a t i v e and ( 3 . 1 ) i s s a t i s f i e d . The  b l o c k diagram f o r the n e g a t i v e energy case o f  t h e Coulomb p o t e n t i a l i s populated  as f o l l o w s : corner one by  e q u a t i o n (3.3) and the boundary c o n d i t i o n t h a t z e r o as;] Y  f  goes to i n f i n i t y ; corner two by e i t h e r  goes to equation  (3.5) o r (3.6)(depending on the c o n d i t i o n s e x p l i c i t l y imposed on d e c r e a s i n g the p o t e n t i a l ) and the above boundary c o n d i t i o n ; corner t h r e e by wavefunction f o u r by  4^ ~ 0.  p o t e n t i a l i n (3.4) i s The  (3.4-) j a^d- corner  S i n c e the r e s u l t o f reducing the ^=0  the b l o c k diagram i s c l o s e d .  c l a s s i c a l switching o f f o f t h i s p o t e n t i a l i n a  manner such t h a t the c r i t e r i a o f the system a r e s a t i s f i e d f o r i n t e r m e d i a t e v a l u e s o f the p o t e n t i a l w i l l now be c o n s i d e r e d . The  c l a s s i c a l c r i t e r i a f o r the n e g a t i v e t o t a l energy Qoulomb  case a r e the f o r c e e q u a t i o n  (3.7) and  the i n e q u a l i t y  23 The  l a t t e r f o l l o w s from the c o n s e r v a t i o n o f energy  bound system w i t h zero p o t e n t i a l a t i n f i n i t y . based  on (3.7)  i t w i l l now  for a  By an a n a l y s i s  be shown t h a t the v e l o c i t y goes to  z e r o as the p o t e n t i a l i s switched o f f .  As the p o t e n t i a l i s  s w i t c h e d o f f , A goes to zero and hence  vf 2  S i n c e A being decreased i m p l i e s the f o r c e ,  goes to z e r o . -A  ,  d e c r e a s e d i n magnitude, the r e s u l t i n view o f the t a n g e n t i a l v e l o c i t y , v, i s t h a t r w i l l Hence f o r  v r l  Is finite  tend to i n c r e a s e .  to go to zero v must go to zero w i t h the  r e s u l t t h a t as the p o t e n t i a l i s switched o f f the k i n e t i c energy  goes to z e r o .  Hence the r e s u l t o f s w i t c h i n g o f f i n  t h e above manner i s a p a r t i c l e without k i n e t i c o r p o t e n t i a l energy,  t h a t i s w i t h zero t o t a l  energy.  B e f o r e -concluding the c l a s s i c a l s w i t c h i n g o f f o f t h i s Coulomb p o t e n t i a l some r e s u l t s , which w i l l on i n t h i s t h e s i s , w i l l  be p r e s e n t e d .  be u s e f u l  further  Of the v a r i o u s ways o f  d i s c r e t e l y s w i t c h i n g o f f the p o t e n t i a l c o n s i s t e n t w i t h the criteria  (3.7)  and  (3.3)  the l e a s t f a v o r a b l e one i s i f a g i v e n  decrease occurs instantaneously.  I t i s assumed t h a t the  p o t e n t i a l parameter i n s t a n t a n e o u s l y goes from A to Aj. where A i < A.  A t the i n s t a n t o f decrease  and  (3.8)  must be s a t i s f i e d by both A and A i .  and  (3.8)  at t h i s i n s t a n t  ,(3.9)  (3.7)  yields  i *w"V - A < A, .  i s s a t i s f i e d by A Combining  (3.7)  24 This  inequality  i m p l i e s that the p o t e n t i a l  switched  o f f to:• z e r o i n a f i n i t e number  criteria  o f t h e s y s t e m a r e t o be s a t i s f i e d  values was  o f the p o t e n t i a l .  t h a t i t be l e s s  continuous accuracy  Since  o f steps i f the  s w i t c h i n g o f f may  be a p p r o x i m a t e d w i t h  conclude  that i n a continuous  potential,  t h e p o t e n t i a l must t a k e  The f i n a l  total  arbitrary  2.  P O S I T I V E ENERGY SOLUTIONS  from  time  such  a  to  reach  switch  zero.  The p h y s i c a l s i t u a t i o n w h i c h t h i s o f the scattering  Coulomb f o r c e . constant  It  may  switching o f f ,  an i n f i n i t e  energy r e s u l t i n g  i s again  being  that a  f o r intermediate values o f the  off  as  r e q u i r e d o f Aj_  t h a n A and i n v i e w o f t h e f a c t  the c r i t e r i a  and  f o r intermediate  a l l t h a t was  satisfying  that  he  by a d i s c r e t e m e t h o d o f s w i t c h i n g o f f , one  therefore  zero.  cannot  of a spinless,  In this  case  the t o t a l  case d e s c r i b e s i s  charged  path.  to consider the i n c i d e n t  d e s c r i b e d by a p l a n e  by a  energy i s p o s i t i v e  f o r a l l points o f the p a r t i c l e ' s  i s convenient  particle  particle  wave i n t h e z - d i r e c t i o n and  5 t o work i n p a r a b o l i c c o o r d i n a t e s . form  y=  e * f  I s sought f o r the e q u a t i o n  ik  (3.10)  where  A*lm2e  l  Hence a s o l u t i o n o f t h e  k »imE., l  ^ f  + ^ ^ v O - c  ft A  .  2,  i s the p o t e n t i a l  and e i s  25  the charge on the incident p a r t i c l e . V F + 2«k  by F i s  z  2£  a*  -  AL  -  The equation s a t i s f i e d o.  r  A t t h i s stage the transformation to parabolic coordinates i s made.  Due to the a x i a l symmetry of the system and the  separating out of the incident plane wave, the solution w i l l j* = r- z  depend on f o r F.  only.  F(V-H)  Hence  i s substituted  After multiplying through by r the r e s u l t i n g  equation i n terms of By introducing  j  X ikjf  i n x i s obtained.  becomes a confluent hypergeometric equation  a  Hence  F ,F, =  > 'i '^J*^.  When normalized to u n i t f l u x the t o t a l wavefunction i s  (3.11)  T M = e&  r(i.^L) e' .R (-A ki  ,.  ,»ior).  I f the p o t e n t i a l Is reduced to zero i n the above wavefunction, that i s A reduced to zero, a l l the terms i n v o l v i n g A go to unity and the wavefunction becomes the plane wave  Q  Since the Coulomb p o t e n t i a l i s a long range one wi th the same value at  z = + <=© and z ° - «x>  the boundary  c o n d i t i o n at i n f i n i t y behaves i n a d i f f e r e n t manner f o r t h i s p o t e n t i a l than f o r the square well p o t e n t i a l .  In this  case the parameter A e x p l i c i t l y appears i n the asymptotic expression f o r the wavefunction.  This and the uniform  26 electric  field  a r e t h e o n l y cases i n t h i s t h e s i s i n which t h e  p o t e n t i a l parameter i s e x p l i c i t l y i n v o l v e d i n t h e boundary condition. in  which  The boundary c o n d i t i o n accompanying t h e e q u a t i o n  thep o t e n t i a l i s zero w i l l  w i t h thei n i t i a l to  differ  from  the condition  equation to the extent that A i s set  zero i n the i n i t i a l  condition at infinity  boundary e x p r e s s i o n .  equal  The boundary  f o r t h ep o s i t i v e energy s o l u t i o n s o f  t h e Coulomb p o t e n t i a l i s  (3.12) V^tl-nA*  ) cxp (ika- [A Uj WO-z)) +  c s c 0 e x p A k r - j A lo«kK-|A lo^O-cose)  .At»_  •*-2"ii»e') .  l  When A i s d e c r e a s e d  t o z e r o i n (3.12) t h i s b o u n d a r y c o n d i t i o n  becomes  t~e'' *  (3.13)  k  The  r e s u l t o f decreasing thep o t e n t i a l  t h e wave e q u a t i o n c a n now be s t u d i e d .  t o zero i n  As i n t h e p o s i t i v e  energy case o f t h e square w e l l p o t e n t i a l , d e c r e a s i n g t h e p o t e n t i a l d o e s n o t p l a c e a n y r e s t r i c t i o n s o n E. resulting  from d e c r e a s i n g t h ep o t e n t i a l  (3.14)  V  2  f  t o z e r o i n (3.10) i s  +k f-0. z  The  g e n e r a l s o l u t i o n o f (3.14) n o r m a l i z e d  6  ~  going The  .  By imposing  to i n f i n i t y  The e q u a t i o n  to unit flux i s  t h e b o u n d a r y c o n d i t i o n (3.13) f o r r  t h e g e n e r a l s o l u t i o n becomes  same r e s u l t may be o b t a i n e d b y r e c a l l i n g  e  that  1 k  a  .  t h e i n c i d e n t -wave v e c t o r was  k = (o, o, Jc).  absence o f any p o t e n t i a l , as i s the case i n remains u n a l t e r e d .  I n the  (3.I4),  Hence the g e n e r a l s o l u t i o n  e'^"  k  again  r  •k*  becomes  e  .  I f i n a g i v e n problem the i n c i d e n t wave  v e c t o r i s n o t p a r a l l e l to an a x i s a r o t a t i o n o f the c o o r d i n a t e system i s f i r s t  c a r r i e d out such t h a t the wave  v e c t o r i s p a r a l l e l to an a x i s i n the new c o o r d i n a t e system. The problem i s then t r e a t e d as above i n the new c o o r d i n a t e system. The b l o c k diagram f o r the p o s i t i v e energy Coulomb c a s e i s o c c u p i e d as f o l l o w s : corner one by e q u a t i o n and  boundary c o n d i t i o n  (3.14) function  (3.12);  boundary c o n d i t i o n  (3.11);  (3.10)  corner two by e q u a t i o n  (3.13);  and corner f o u r by  corner t h r e e by wave-  6 " 1  .  As has been  demonstrated, the r e s u l t o f r e d u c i n g the p o t e n t i a l i n the t h i r d corner i s t h e e n t r y i n the f o u r t h c o r n e r .  Hence the  b l o c k diagram i s c l o s e d . The p h y s i c a l s i t u a t i o n i s s t r a i g h t - f o r w a r d .  The  system c o n s i s t s o f a p a r t i c l e w i t h t o t a l energy, E, e x p e r i e n c i n g a Coulomb f o r c e .  The r e s u l t o f s w i t c h i n g o f f t h i s  f o r c e to zero i s a f r e e p a r t i c l e w i t h the same t o t a l T h i s t o t a l energy i s now a l l k i n e t i c  energy.  energy.  CHAPTER IV UNIFORM ELECTRIC FIELD  1.  DESCRIPTION OF SYSTEM The system under c o n s i d e r a t i o n i n t h i s chapter i s  t h a t o f a charged, s p i n l e s s p a r t i c l e , i n c i d e n t from t r a v e l l i n g towards electric field. to  t h e "^--axis  denoted by e.  1 ~ ~~ >  being r e p e l l e d by a u n i f o r m  00  This f i e l d ,  ^ = -*-oo  F  , i s chosen to be p a r a l l e l  and the charge on the p a r t i c l e i s I n t h i s chapter e i s assumed to be p o s i t i v e .  However t h e arguments  and r e s u l t s are. e q u a l l y a p p l i c a b l e to  a n e g a t i v e l y charged p a r t i c l e when t h e d i r e c t i o n s a r e reversed. tial to  I n t h i s system t h e p a r t i c l e e x p e r i e n c e s a poten-  -©F^+C.  S i n c e C i s a r b i t r a r y i t i s chosen  be zero thus making the zero o f p o t e n t i a l a t the o r i g i n .  The p a r t i c l e ' s t o t a l energy i s denoted by £ .  E i s the  t o t a l energy a s s o c i a t e d w i t h the motion p a r a l l e l to the ^ -axis 2.  and i s a p o s i t i v e o r n e g a t i v e c o n s t a n t .  WAVEFUNCTION BEHAVIOUR M D  BLOCK DIAGRAM  The S c h r o e d i n g e r E q u a t i o n f o r t h i s system i s  29  Y  i s expressed as  [ x, y> y) - X(x)  Y(y)  (})  and  t h r e e o r d i n a r y d i f f e r e n t i a l equations are o b t a i n e d -which k  i n v o l v e the c o n s t a n t s  k * + ku  '  x  1  Ut  s  y  ZmC  *  X  k ^ and  ;  k£  .  x  kj  where  kJ*" a r e im  and  ft  1  t i m e s the energy a s s o c i a t e d w i t h the motion i n the x and y d i r e c t i o n s r e s p e c t i v e l y and as such a r e p o s i t i v e . 1 mE  and i s p o s i t i v e o r n e g a t i v e as  equations f o r X  k  e q u a t i o n f o r Z(})  dj  y i e l d the f r e e p a r t i c l e plane wave  X  31  ft*  +  Y(y) = e ^ * 1  and  fifai^.Uj)Z  +  kt.) y  V  :  a  7  V *«•  1  the changes i n v a r i a b l e s  \T = * ( Intfx  i s . The  is  £ Z  (4.2) If  Y  X(JO = e ' *  solutions The  and  E  is  ~L.  o.  ~ J ^ssFy  + k£ W  and then  t  l  V  _  are  XmeF  made i n ( 4 « 2 ) the r e s u l t i n g e q u a t i o n f o r W l ^ & J L  U =^  f  + U ^  + (U  l  _ l )  W  is  = 0  .  T h i s i s a B e s s e l E q u a t i o n and i t s g e n e r a l s o l u t i o n i s  W = A X/3(u) for  +  B J-y, (u).  The g e n e r a l e x p r e s s i o n  Z.(}) i s t h e r e f o r e  Z  where A and B w i l l be  30 chosen to s a t i s f y the boundary c o n d i t i o n s . As was asymptotic  seen i n the p o s i t i v e energy Coulomb case,  behaviour o f the w a v e f u n c t i o n f o r a l o n g  p o t e n t i a l i s not exponential  simply  Since t h i s uniform  f i e l d p o t e n t i a l i s a l o n g range one,  electric  a l l t h a t w i l l be s p e c i -  the boundary c o n d i t i o n s i s t h a t the wavefunc-  t i o n goes to zero e x p o n e n t i a l l y as and  range  a t r i g o n o m e t r i c o r an imaginary  type o f f u n c t i o n .  f i e d regarding  the  t h a t i t o s c i l l a t e s as  ^  J  approaches  boundary c o n d i t i o n s to be s a t i s f i e d  — °o  approaches <*> .  The  by  q u a n t i t a t i v e l y s t a t e d as f o l l o w s : tends to exponentially with  To — oO  for  s a t i s f y the boundary c o n d i t i o n f o r  A i s equal  to  >  o s c i l l a t e s with J .  implies  U.4)  by  decreasing  zero  -Be'""' , 3  ^  tending  This r e l a t i o n i s obtained  a p p l y i n g the f o l l o w i n g procedure:  ^  T-meFjjIii 7 as being p o s i t i v e ; the r e l a t i o n s  tending  recognized  to  -«o  =e  '  to  i s used and  1  is  are used;  31  and A i s expressed i n terms o f B such t h a t , f o r to  - oo  where  ,  ^(j)  i s p r o p o r t i o n a l to the  K„(x)  8  =TT(L„(X)  -  I  V  U ) )  mined by u s i n g t h e asymptotic forms  X^W  (4.5) and  -~  parallel  + oo  where  to t h e 3--axis.  v  By use o f ^ ( 4.6)  1^1  function  .  B i s deter-  9  v|4*| J— l r  for Z  i s the magnitude o f the  velocity  B i s then found t o be __L  j3eF  With A and B thus s p e c i f i e d for  tending  co*(x + l i F - E )  the n o r m a l i z a t i o n c o n d i t i o n  tending to  Kp  y  t h e asymptotic forms o f  t e n d i n g t o i n f i n i t y can be w r i t t e n down.  Ky(x)^/liL  e~X  i t i s seen t h a t f o r  tending t o - ao  Z V - f i T / a n — l ^ e ^ ^ x p f - f e s ^ ^ l .  ( 4 . 6 ) goes t o zero e x p o n e n t i a l l y as ^  goes t o - <*» . '  By use o f ( 4 . 5 ) t h e asymptotic form o f  as  + 00  tends t o  i s seen t o be  (  -^Hlte^ -?)]} • ( 4 . 7 ) o s c i l l a t e s with varying  %.  Hence w i t h the above  c h o i c e s f o r A and B t h e boundary c o n d i t i o n s a r e s a t i s f i e d .  32  s o l u t i o n o f ( 4 . 2 ) which i s n o r m a l i z e d t o u n i t f l u x a t  The ^  -  and which s a t i s f i e s the boundary c o n d i t i o n s ( 4 . 3 )  00  and ( 4 . 4 ) i s t h e r e f o r e (4.8)  Z(0-±./  Xm(eF E)' K  H J 3 T ?  0  - e  iTr/3  ( I . \(l~ef} + l~>t) ] ih  I  3 m e F t i  3L  J  Jj. I" d ^ f i > 2 ^ £ )  The r e s u l t o f r e d u c i n g the p o t e n t i a l i n the wavefunction (4.8) will  be s t u d i e d i n c o n j u n c t i o n w i t h the e f f e c t  o f d e c r e a s i n g t h e p o t e n t i a l to zero i n t h e boundary c o n d i tions.  T h i s i s done i n o r d e r to determine whether t h e r e s u l t  o f r e d u c i n g the p o t e n t i a l i n the w a v e f u n c t i o n s a t i s f i e s the c o n d i t i o n s o n t h e w a v e f u n c t i o n o b t a i n e d by d e c r e a s i n g t h e p o t e n t i a l i n t h e boundary c o n d i t i o n s . and n e g a t i v e  The cases o f p o s i t i v e  E must be d i s t i n g u i s h e d .  I n r e d u c i n g the p o t e n t i a l I n t h e w a v e f u n c t i o n (4*8) f o r the case o f p o s i t i v e E , F  approximations  and a r e employed. becomes  JL-(  equations (4«5) and the s m a l l  j "™*^  *  7  s  ^ \ ^rc ~  k, y +  As F i s reduced to zero the wavefunction (4*8)  With  E  p o s i t i v e F w i l l be decreased  c o n d i t i o n s (4-3)  and (4-4-) •  As F i s decreased  zero e x p o n e n t i a l l y w i t h d e c r e a s i n g  J .  (4-.4-)  ^*  the l a t t e r l s the  o n l y c o n d i t i o n imposed on the w a v e f u n c t i o n  oscillates  for a l l j  consistent  E  by t h e boundary  S i n c e (4-.9)  to zero.  i t i s c o n s i s t e n t w i t h t h e above con-  d i t i o n on the wavefunction. (4-.8) when  becomes  oscillates  As the former i s meaningless  c o n d i t i o n s when F i s decreased  (4-3) goes  $ >>-oo  the condition that f o r with  to zero  j«~oo  becomes the c o n d i t i o n t h a t f o r to  to zero i n the boundary-  Hence r e d u c i n g the p o t e n t i a l i n  i s p o s i t i v e y i e l d s an a c c e p t a b l e and  result.  When the p r e c e d i n g approximations f o r s m a l l F and 7 8 10 T i/ t h e p r e v i o u s formulas ' ' for 1 +i and Ki a r e used, the wavefunction  U  .io)  Z(  ,) *  as F i s reduced.  (4-8)  e ' "  A  f o r negative E  H2  /.«.)*  becomes  e x [ I^I j - y v  F w i l l now be decreased  boundary c o n d i t i o n s w i t h condition that f o r  E  1  P  negative.  t o zero i n the  (4-3)  < j « oo  becomes t h e  i s zero and (4-.4-)  becomes t h e c o n d i t i o n t h a t f o r  ^ >>oo  o s c i l l a t o r y w i t h r e s p e c t t o |,,  S i n c e the l a t t e r i s  meaningless wavefunction  Z($)  is  the former i s the o n l y c o n d i t i o n imposed on the by t h e boundary c o n d i t i o n s when F i s zero and  34 E is  i s negative.  The r e s u l t o f r e d u c i n g F to zero i n (4.10)  a zero wavefunction  fora l l ^  a c c o r d w i t h the above c o n d i t i o n . tial  to zero f o r E  l e s s than i n f i n i t y i n Hence r e d u c i n g the poten-  e i t h e r p o s i t i v e o r n e g a t i v e produces a  s a t i s f a c t o r y r e s u l t i n t h a t the c o n d i t i o n on the wavefunction is  s a t i s f i e d i n both  cases.  F o r both p o s i t i v e and n e g a t i v e  E  the r e s u l t o f  d e c r e a s i n g the p o t e n t i a l i n the wave e q u a t i o n (4.2)  + ^TL Z = o •  (4.11) If  E  is  E  i s p o s i t i v e the c o n d i t i o n on the wavefunction  be o s c i l l a t o r y w i t h  3.  for  ^ > > - « 3  that i t  i m p l i e s t h a t the  s o l u t i o n o f (4«ll) corresponding to a p a r t i c l e i n c i d e n t  ^ = + 00  is  e-' >* .  (4.12) If  E  from  ,k  i s n e g a t i v e and F i s zero the c o n d i t i o n on  t h a t i t be zero f o r  (4.11) i s  <^<<<co  m  Z  is  I n t h i s case the s o l u t i o n o f -  Z-O.  S i n c e the p o t e n t i a l i s a f u n c t i o n o f f u n c t i o n s o f , o r concerned  with,  ^-  only, only  need be c o n s i d e r e d i n  t h e b l o c k diagram. As i t has been e x p l a i n e d i n chapter one, the potent i a l i n the boundary c o n d i t i o n s must be decreased g o i n g from corner one to corner two.  to zero i n  S i n c e the boundary  35 c o n d i t i o n i n corner two i s t h e r e f o r e d i f f e r e n t f o r p o s i t i v e E  from what i t i s f o r n e g a t i v e  p o s i t i v e and n e g a t i v e  E  E  , the two cases o f  must he d i s t i n g u i s h e d . Hence a  s e p a r a t e b l o c k diagram w i l l be used f o r each o f the two energy  cases. I n the case::of p o s i t i v e  populated  with  the b l o c k diagram i s  as f o l l o w s : corner one by e q u a t i o n (4.2)  boundary c o n d i t i o n s (4«3) (4.H)  E  and (4«4)J  corner two by e q u a t i o n  and the c o n d i t i o n t h a t the wavefunction ^  f°  r  and  oscillates  ^ >> -<x> ; corner t h r e e by the wavefunction  ( 4 . 8 ) ; and corner f o u r by wavefunction c o r n e r f o u r and the e q u a t i o n (4*9)  (4.12) .  The e n t r y i n  w i t h F reduced  to zero,  w h i c h i s the r e s u l t o f r e d u c i n g the p o t e n t i a l i n the wavef u n c t i o n occupying  corner t h r e e , d i f f e r by a phase f a c t o r  but d e s c r i b e the same p h y s i c a l s i t u a t i o n .  Hence to the  e x t e n t t h a t the e n t r y i n corner f o u r and the. r e s u l t o f reduci n g the p o t e n t i a l i n the wavefunction  i n corner t h r e e are  p h y s i c a l l y i n d i s t i n g u i s h a b l e , the b l o c k diagram i s c l o s e d . I n the case o f n e g a t i v e  E  the b l o c k diagram i s  o c c u p i e d as f o l l o w s : corners one and t h r e e as i n the p o s i t i v e E  case; corner two by e q u a t i o n ( 4 » H ) and the c o n d i t i o n  that f o r by  ^<<oo  ~L.~0.  the wavefunction  i s zero; and corner f o u r  The r e s u l t o f r e d u c i n g the p o t e n t i a l to zero i n  c o r n e r t h r e e , t h a t i s , (4.10) w i t h c o r n e r f o u r a r e the same.  F = 0 , and the e n t r y i n  The b l o c k diagram i s t h e r e f o r e  completed. 3.  PHYSICAL ANALYSIS In  field  the e q u a t i o n (4.9) r e s u l t i n g from r e d u c i n g  the  to zero i n the p o s i t i v e E case, the plane wave  momentum-distance e x p r e s s i o n , t h a t i s , k^j  , i s found i n  t h e argument o f the e x p o n e n t i a l thus i n d i c a t i n g the d e s i r e d p l a n e wave r e s u l t .  However the term  i n f i n i t e phase f a c t o r as F i s reduced  \y\  corresponds  X  t;  i n t r o d u c e s an  r  to z e r o .  Since only  to a p h y s i c a l o b s e r v a b l e and phase f a c t o r s  a r e not p h y s i c a l l y o b s e r v a b l e , the presence or absence o f s u c h an i n f i n i t e phase f a c t o r would not be d e t e c t a b l e . no  Hence  p h y s i c a l e x p l a n a t i o n o f t h i s i n f i n i t e phase f a c t o r i s  possible. energy  Furthermore,  since at  ^ =-•-co  an i n f i n i t e  i s r e q u i r e d to m a i n t a i n the t o t a l energy  kinetic  constant  this  system does not p r e c i s e l y correspond to an a c t u a l p h y s i c a l situation.  Hence an u n u s u a l  and p h y s i c a l l y i n e x p l i c a b l e i t e m  s u c h as an i n f i n i t e phase f a c t o r should not be s u r p r i s i n g o r disturbing.  I t i s however s a t i s f y i n g t h a t a l l the p h y s i c a l l y  o b s e r v a b l e f e a t u r e s are w e l l behaved. The w i l l now  c l a s s i c a l motion o f the p a r t i c l e i n t h i s  be a n a l y z e d .  The  c l a s s i c a l p r o c e s s which t h i s  r e p r e s e n t s i s t h a t o f a charged, from is  +00  field  spinless particle  case  incident  being r e f l e c t e d back a t some p o i n t J . 0  This  supported i n the p r e c e d i n g mathematics by the f a c t t h a t as  ^  approaches  oo  the wavefunction  becomes ( 4 . 7 ) which  d e s c r i b e s a f r e e p a r t i c l e t r a v e l l i n g i n the The by  -j.  f a c t t h a t the p a r t i c l e i s r e f l e c t e d back i s ( 4 . 6 ) which, f o r a l l f i n i t e  E  b i l i t y f o r the p a r t i c l e being a t The two  direction. supported  , i n d i c a t e s zero ^  for  ^  proba-  tending to  -00.  cases o f p o s i t i v e and n e g a t i v e E have been  d i s t i n g u i s h e d and have g i v e n d i f f e r e n t r e s u l t s .  The p h y s i -  c a l s i g n i f i c a n c e o f the v a l u e o f the t o t a l energy, E, i s to i n d i c a t e a t what p o i n t i n space,  once the zero o f  p o t e n t i a l i s f i x e d , the p a r t i c l e i s c l a s s i c a l l y back by the p o t e n t i a l b a r r i e r .  reflected  T h i s p o i n t o f course c o r r e -  sponds to the p o s i t i o n where the p a r t i c l e has zero k i n e t i c energy.  W i t h the zero o f p o t e n t i a l a t the o r i g i n , as i s  h e r e i n chosen, p o s i t i v e E corresponds p o s i t i o n with negative corresponds sfr  ^  c o o r d i n a t e and n e g a t i v e E  to r e f l e c t i o n back a t a p o s i t i o n w i t h p o s i t i v e  coordinate.  The  exact c o o r d i n a t e a t which the  p a r t i c l e i s r e f l e c t e d back i s g i v e n by is  the v a l u e o f  Functions  ^  0  i n ( 4 . 8 ) i s z e r o . ^ > J©  argument i s r e a l ; j < J  ^  = -JL..  e  ^ <C | 0  c  implies this  i m p l i e s t h i s argument i s  corresponds  to the r e g i o n o f space  which, i n c l a s s i c a l mechanics, the p a r t i c l e may  The  |  f o r which the argument o f the B e s s e l  T±L  imaginary.  to r e f l e c t i o n a t a  never e n t e r .  e f f e c t o f the p o t e n t i a l being switched o f f on  t h e p o i n t o f r e f l e c t i o n w i l l now  be s t u d i e d .  For p o s i t i v e  38 E,  the p o i n t o f r e f l e c t i o n ,  switched o f f .  j  t f  , goes to - oo  That i s , f o r F = 0 the r e s u l t i s a p a r t i c l e  t r a v e l l i n g from  «>  to  ^=co  without being  r e f l e c t e d a t an i n t e r m e d i a t e p o s i t i o n . E  as F i s  p o s i t i v e the wavefunction f o r  Hence f o r F = 0 and  | > - oo  , that i s ,  p o i n t s , should be a p l a n e wave d e s c r i b i n g a f r e e t r a v e l l i n g from from  c  + oo  to | = - oo.  to  -i-oo  at  ^  e +  wavefunction  I f E i s negative  -fro goes  as F i s switched o f f and the p o i n t o f r e f l e c t i o n i s * .  r e f l e c t e d back a t is  particle  As can be seen  (4«9) r e d u c i n g the p o t e n t i a l i n the i n i t i a l  w i t h p o s i t i v e E gives this r e s u l t .  at a l l  I f a p a r t i c l e e n t e r i n g from | = + oo  ^=+«o  is  the r e s u l t i s t h a t the p a r t i c l e  never i n any f i n i t e r e g i o n o f space.  d e s c r i b e d by a zero w a v e f u n c t i o n f o r  This situation i s  ^ 4.+<x>. As F i s  r e d u c e d to zero i n the i n i t i a l w a v e f u n c t i o n and E i s n e g a t i v e , (4.8) zero.  becomes (4.10) which i s zero f o r a l l Hence f o r both p o s i t i v e and n e g a t i v e t o t a l  when F i s energy  r e d u c i n g the p o t e n t i a l i n the w a v e f u n c t i o n y i e l d s a r e s u l t i n a c c o r d w i t h the p h y s i c a l s i t u a t i o n a r i s i n g from s w i t c h i n g o f f the p o t e n t i a l .  CHAPTER V THE HARMONIC OSCILLATOR  1.  TIME INDEPENDENT TREATMENT The  harmonic o s c i l l a t o r wave e q u a t i o n 11 i t s s o l u t i o n a r e v e i l known. The wave e q u a t i o n i s  and  time independent  £ l  (5.1) where  dx*  r-——  +  W E - i « «  v\  y | t « o  c  I  x  1  /.v /«l*stic constant , K a  frequency  mass  4  j . the c l a s s i c a l s  j fn  associated with this o s c i l l a t o r .  The energy E takes  v a l u e s g i v e n by  (5.2) where  E = ("+i)*o) n  n  i s a non-negative  c  integer.  The n o r m a l i z a t i o n I TI  i m p l i e s the boundary c o n d i t i o n s t h a t goes t o ± o o . '  For a given  n  Y  dx a 1.  This  goes to zero as x  the normalized  solution of  (5.1) i s  (5  •  3)  r  -  w=  SS?  "" *  E  M  S i n c e t h e p o t e n t i a l energy i s  X /Z  L  /^F) •  HRT (X  m  <o* x c  1  reducing the  p o t e n t i a l t o zero i s e q u i v a l e n t to r e d u c i n g K o r Reducing  cd  c  values o f n.  to zero reduces  the wavefunction  The wavefunction  <u  to z e r o .  c  to zero f o r a l l  goes to zero as  A >  C  '  /  4  O  R  AO  The  e f f e c t o f d e c r e a s i n g the p o t e n t i a l to zero i n the  wave equation w i l l now be determined. to  I f the p o t e n t i a l goes  zero such t h a t the energy v a l u e s e x p l i c i t l y s a t i s f y ( 5 . 2 )  f o r i n t e r m e d i a t e v a l u e s o f the p o t e n t i a l the r e s u l t i n g wave equation i s (5.4)  £ 1  s  o.  •  dx* If  the p o t e n t i a l goes to zero without  explicitly  requiring  t h a t ( 5 . 2 ) be s a t i s f i e d the wave e q u a t i o n becomes £ f  (5.5)  6*  x  +  &aE  4>-o.  V-  S i n c e the boundary c o n d i t i o n s t h a t  ^  i s zero a t * =  a s s o c i a t e d w i t h both ( 5 . 4 ) and ( 5 . 5 ) 4 = 0  i s the s o l u t i o n o f  ( 5 . 4 ) and ( 5 . 5 ) w i t h these boundary c o n d i t i o n s .  both  are  This i s  t h e same as i n the n e g a t i v e energy cases o f the square w e l l and  Coulomb p o t e n t i a l s . The  b l o c k diagram f o r the time independent harmonic  o s c i l l a t o r i s o c c u p i e d as f o l l o w s : corner one by e q u a t i o n (5.1)  and boundary c o n d i t i o n s t h a t  4*  goes to zero as x goes  t o i co ; corner two by the same boundary c o n d i t i o n s and e q u a t i o n (5.4) o r ( 5 . 5 ) depending on the e x p l i c i t r e g a r d i n g d e c r e a s i n g the p o t e n t i a l ; function ( 5 . 3 ) ;  assumptions  corner t h r e e by wave-  and corner f o u r by 4 = 0 .  The r e s u l t o f  r e d u c i n g the p o t e n t i a l to zero i n corner t h r e e y i e l d s the  41 e n t r y i n corner f o u r . 2.  TIME DEPENDENT The  Hence the block diagram i s completed.  TREATMENT  time dependent wave e q u a t i o n f o r a harmonic  oscillator i s  (5.6)  V= Kx  where  / fc* a* ^ .  3Wx,t)  i*  i  i s the p o t e n t i a l energy  1  S i n c e the p o t e n t i a l i s independent of  lc,»)*W,i)  a t p o s i t i o n x.  o f the time the s o l u t i o n  the time dependent e q u a t i o n may be expressed  infinite  as an  sum o f the s o l u t i o n s o f the time independent  wave  e q u a t i o n w i t h the c o e f f i c i e n t depending on the time.  Using  12 t h i s technique, the n o r m a l i z e d s o l u t i o n to ( 5 . 6 ) i s co  (5.7)  e i p f - a t - j ^ - t f ) E t e f IkWe-"  ¥(«,t) =  1  \ T T /  where  d^- m K  3 1 1 ( 1  v  i  *o  4 .  *  c e n t r e o f the wave packet. oi  a l s o go t o zero and  s  x /  x/  n|  , o u  *  the i n i t i a l p o s i t i o n o f the When K l s reduced  S^x^i)  to zero u> and  becomes z e r o .  c  Therefore  t h e r e s u l t o f r e d u c i n g the p o t e n t i a l to zero i s the same f o r t h e s o l u t i o n s o f both the time dependent and time  independent  wave e q u a t i o n s . The r e s u l t o f d e c r e a s i n g the p o t e n t i a l to zero i n t h e time dependent wave e q u a t i o n w i l l now be a n a l y z e d . E does not appear i n ( 5 . 6 )  the r e s u l t o f d e c r e a s i n g the  Since  42 p o t e n t i a l to zero i n ( 5 . 6 )  is  (5.8)  =  r e g a r d l e s s o f whether or not ( 5 . 2 )  i s satisfied for inter-  m e d i a t e v a l u e s o f the p o t e n t i a l . A  tikx  at  both  (5.8) If  .  x=*oo  F o r the wavefunction  to v a n i s h  ^  c o n s i s t e n t w i t h the boundary c o n d i t i o n s i s  O.  e  f o r intermediate values of  the p o t e n t i a l the s o l u t i o n o f ( 5 . 8 ) , c o n d i t i o n s are a p p l i e d , i s lrr\%T  is  A must be zero and hence the s o l u t i o n o f  the system s a t i s f i e s ( 5 . 2 )  kef  solution of (5.8)  •iEt/fc  6  nc  The  b e f o r e the boundary  i s a constant s i n c e E  go to zero as the p o t e n t i a l does.  and  To  satisfy  the boundary c o n d i t i o n s t h i s constant i s then z e r o . The  b l o c k diagram f o r the time dependent harmonic  o s c i l l a t o r i s p o p u l a t e d as f o l l o w s : corner one (5.6)  and boundary c o n d i t i o n s t h a t  goes to i n f i n i t y ;  corner two  and the e q u a t i o n ( 5 . 8 ) ;  i n (5.7) It  corner t h r e e by the  wavefunction  S i n c e r e d u c i n g the  poten-  should be noted t h a t f o r both the time dependent treatments o f the harmonic o s c i l l a t o r  t h e energy e i g e n v a l u e s are g i v e n by n  goes to zero as 1x1  ^  y i e l d s zero the b l o c k diagram i s c l o s e d .  and time independent  with  equation  by the same boundary c o n d i t i o n s  ( 5 . 7 ) ; and corner f o u r by ^=0. tial  by  a non-negative  integer.  E  n  = (n+i)^co  Hence i n both  t  treatments  43 it  i s apparent,  energy  frora t h i s e i g e n v a l u e equation, t h a t a l l the  e i g e n v a l u e s go to zero as the p o t e n t i a l does i f the  p o t e n t i a l i s decreased i n such a manner t h a t (5.2)  is  s a t i s f i e d f o r a l l i n t e r m e d i a t e v a l u e s o f the p o t e n t i a l . The procedure •wave e q u a t i o n may  used  to s o l v e the above time dependent  be e q u a l l y w e l l a p p l i e d to s o l v i n g the  time  dependent equations corresponding to the o t h e r p o t e n t i a l s . However, as the p r e c e d i n g has shown, the r e s u l t o f r e d u c i n g the p o t e n t i a l to zero i s the same f o r the s o l u t i o n s to both t h e time dependent and  time independent  equations.  Hence i n  s t u d y i n g the r e s u l t o f r e d u c i n g the p o t e n t i a l i t i s s u f f i c i e n t t o d e a l w i t h the s o l u t i o n o f e i t h e r the time dependent o r independent 3.  time  wave e q u a t i o n .  CLASSICAL ANALYSIS OF SWITCHING  OFF  B e f o r e d i s c u s s i n g the s w i t c h i n g o f f p r o c e s s e s c l a s s i c a l d e s c r i p t i o n o f , and o s c i l l a t o r w i l l be g i v e n .  the  c r i t e r i a f o r , a harmonic  The p h y s i c a l system c o n s i d e r e d as  a harmonic o s c i l l a t o r c o n s i s t s o f a p a r t i c l e o s c i l l a t i n g  about  an e q u i l i b r i u m p o s i t i o n under the i n f l u e n c e o f a f o r c e d i r e c t e d towards the e q u i l i b r i u m p o s i t i o n and o f magnitude p r o p o r t i o n a l to the p a r t i c l e ' s d i s t a n c e from t h i s  equilibrium  position.  c  The  The  e q u a t i o n o f motion i s  X  s  d cos co -t .  magnitude o f the v e l o c i t y a t p o s i t i o n x i s  where a i s the  amplitude.  / K^-x^  1  44 The k i n e t i c total obey  energy, E , o f a harmonic o s c i l l a t o r w i t h amplitude " a " the f o l l o w i n g T * V  (5.9)  energy, T, p o t e n t i a l energy, V, and  E  :  E  5  criteria:  constant  maximum T  3  wOf  T  to'i-m r-esp«ct t o  - maximum  ^ maximum T  '  V  0 -  3  post+> on a.r>ol  1 Ka  time  2  V * maximum V.  The f i r s t c r i t e r i o n s t a t e s t h a t c o n s e r v a t i o n o f energy h o l d s for  a l l p o s i t i o n s and time.  The second c r i t e r i o n  indicates  t h a t the t o t a l energy, E, depends o n l y on the e l a s t i c c o n s t a n t , K, and amplitude a. In  this  section a detailed classical analysis  be g i v e n o f the two b a s i c ways o f switelling in  a p h y s i c a l system.  off a potential  The harmonic o s c i l l a t o r has been  chosen f o r t h i s d e t a i l e d a n a l y s i s s i n c e t h i s u s e f u l and r e l a t i v e l y  will  simple.  system i s common,  However t h i s d i s t i n c t i o n i n  methods o f s w i t c h i n g o f f the p o t e n t i a l i s a p p l i c a b l e to a l l o t h e r systems. the  The aim o f t h i s a n a l y s i s i s to i l l u s t r a t e  two types o f s w i t c h o f f and to make c l e a r the d i s t i n c t i o n  between them. These two methods o f s w i t c h i n g o f f w i l l to  as type I and type I I .  c r i t e r i a o f the p a r t i c u l a r  A type I s w i t c h o f f i s where the system a r e s a t i s f i e d f o r a l l  i n t e r m e d i a t e v a l u e s o f the p o t e n t i a l . I  be r e f e r r e d  F o r example,  i n a type  s w i t c h o f f a harmonic o s c i l l a t o r w i t h amplitude "a" remains  a harmonic o s c i l l a t o r w i t h amplitude "a" f o r a l l i n t e r m e d i a t e  t h a t i s , (5.9)  v a l u e s o f the p o t e n t i a l ,  i n t e r m e d i a t e v a l u e s o f the p o t e n t i a l . is  i s satisfied for a l l A type I I s w i t c h o f f  one i n which the p o t e n t i a l i s switched o f f without  the  c r i t e r i a o f the system "being s a t i s f i e d f o r a l l i n t e r m e d i a t e v a l u e s o f the p o t e n t i a l .  That i s , the c h a r a c t e r i s t i c  t i o n s h i p s between the v a r i o u s parameters are not  rela-  satisfied  d u r i n g the s w i t c h i n g o f f p r o c e s s . B e f o r e examining  the s w i t c h i n g o f f p r o c e s s e s i t  s h o u l d be noted t h a t f o r a harmonic o s c i l l a t o r the o n l y t h e p o t e n t i a l may  be switched o f f without imposing  way  geometric  c o n s t r a i n t s , decreased i n the wave e q u a t i o n or reduced i n the wavefunction The  i s by the e l a s t i c  constant, K,  becoming z e r o .  type I s w i t c h o f f o f the harmonic  w i l l be s t u d i e d f i r s t .  The  oscillator  c o n d i t i o n s on the s w i t c h i n g o f f  p r o c e s s i n o r d e r t h a t the p r o c e s s be o f type I may  be formu-  l a t e d i n the form o f the f o l l o w i n g theorem. THEOREM: To s a t i s f y the c o n d i t i o n t h a t f o r a l l nonzero v a l u e s o f the p o t e n t i a l the system i s a harmonic o s c i l l a t o r w i t h amplitude a, the s w i t c h o f f must be done i n the f o l l o w i n g manner: (a) The p o t e n t i a l may be switched o f f o n l y i n d i s c r e t e decrements and these may occur o n l y w h i l e the p a r t i c l e i s a t an e x t r e m i t y . (b) The p a r t i c l e ends up a t one o f the e x t r e m i t i e s w i t h n e i t h e r k i n e t i c nor p o t e n t i a l energy r e l a t i v e to the e q u i l i b r i u m p o s i t i o n , t h a t i s , the f i n a l t o t a l energy i s z e r o .  46 Proof:  F i r s t i t w i l l be shown that the f i n a l  situation i s a  p a r t i c l e w i t h n e i t h e r k i n e t i c nor p o t e n t i a l energy to  the e q u i l i b r i u m p o s i t i o n .  L e t the p o t e n t i a l be p h y s i c a l l y  l o w e r e d by some a r b i t r a r y amount. is  relative  By (5.9)  the t o t a l  energy  lowered by the same amount as i s the maximum p o t e n t i a l  energy.  T h i s i s repeated w i t h the r e s u l t t h a t the maximum  p o t e n t i a l energy and the t o t a l energy are a g a i n lowered by i d e n t i c a l amounts.  T h i s procedure i s continued u n t i l  p o t e n t i a l reaches z e r o .  the  I t I s o b v i o u s , upon r e f e r e n c e to  ( 5 . 9 ) , t h a t as the p o t e n t i a l i s thus switched o f f the v e l o c i t y , k i n e t i c and t o t a l e n e r g i e s a l l go to z e r o . r e s u l t i s a p a r t i c l e w i t h n e i t h e r motion nor p o t e n t i a l  So the energy  w i t h r e s p e c t to the e q u i l i b r i u m p o s i t i o n . Now  c o n s i d e r the p a r t i c l e a t any p o s i t i o n  t h a n a t an e x t r e m i t y .  $  other  I t then has a p o t e n t i a l energy o f  w i l l have a v e l o c i t y such t h a t i t s k i n e t i c energy a t t h i s p o s i t i o n i s g r e a t e r than the new, ^  lower maximum p o t e n t i a l  energy and the c o n d i t i o n f o r a harmonic o s c i l l a t o r i s not satisfied.  If  the p a r t i c l e w i l l have a g r e a t e r  v e l o c i t y a t t h i s p o s i t i o n than would be the case i f i t were  u n d e r g o i n g a harmonic motion w i t h a maximum p o t e n t i a l equal t o the lower p o t e n t i a l energy. ing  x=o  then a t  x=o  I f the p a r t i c l e i s approach-  the p a r t i c l e w i l l have a v e l o c i t y  due to i t s v e l o c i t y a t ^  p l u s the v e l o c i t y a c q u i r e d i n  going from  ^  to zero under the i n f l u e n c e o f the lower  potential.  S i n c e the v e l o c i t y a t ^  i s g r e a t e r than t h a t  which would he the case i f the lower p o t e n t i a l were operat i v e d u r i n g the e n t i r e journey from x=a to x=o, the v e l o c i t y a t x=o corresponds to a k i n e t i c energy a t x=o g r e a t e r than t h e maximum o f the lower p o t e n t i a l . disobeys ( 5 . 9 ) .  Thus the p a r t i c l e a g a i n  I f the p a r t i c l e i s moving away from x=o a  s i m i l a r argument shows t h a t i t would overshoot would then no l o n g e r be a harmonic  U|=a  and  o s c i l l a t o r w i t h amplitude a  Hence the p o t e n t i a l may n o t be lowered a t any p o s i t i o n such that  )-=fc±<x.  Furthermore,  s w i t c h i n g o f f the p o t e n t i a l i n  a continuous manner w h i l e the p a r t i c l e executes i t s motion i s a l s o i n c o n s i s t e n t w i t h the s t i p u l a t i o n that the system be a harmonic  o s c i l l a t o r s i n c e i t i n v o l v e s l o w e r i n g the p o t e n t i a l  a t p o i n t s o t h e r than  ± a.  T h e r e f o r e the o n l y remaining  method, and an o b v i o u s l y a c c e p t a b l e one, i s to lower the potential i n finite  s t e p s when the p a r t i c l e i s a t - a .  T h a t t h i s method complies w i t h the s t i p u l a t i o n s o f t h i s type o f s w i t c h o f f may be seen from the f o l l o w i n g argument. the p a r t i c l e i s a t  * a  the p o t e n t i a l i s lowered.  While The  r e s u l t o f t h i s i s t h e f o l l o w i n g : the t o t a l energy and maximum p o t e n t i a l energy a r e c o r r e s p o n d i n g l y lowered; the maximum  48 k i n e t i c energy i s c o r r e s p o n d i n g l y lowered because  the f o r c e  and a c c e l e r a t i o n are l e s s over the whole d i s t a n c e from x=a x=o;  and the system remains  to  a harmonic o s c i l l a t o r w i t h  amplitude a but w i t h a lower t o t a l energy and g r e a t e r p e r i o d . Whether the p a r t i c l e e v e n t u a l l y ends up a t x=a x=  or  - a depends s o l e l y on the technique used to s w i t c h o f f the  potential.  F o r example, i f i t i s wished  end up at x=a  t h a t the p a r t i c l e  t h i s can be achieved by any type I s w i t c h o f f  w h e r e i n the l a s t step ( t o zero) o c c u r s when the p a r t i c l e i s a t x=a.  I t should a l s o be noted t h a t the p a r t i c l e  end up a t the e q u i l i b r i u m p o s i t i o n ,  cannot  q.e.d.  A type I I s w i t c h o f f o f the harmonic o s c i l l a t o r p o t e n t i a l w i l l now  be i l l u s t r a t e d .  i n which a system may  There are numerous ways  undergo a type I I s w i t c h o f f .  t h e f o l l o w i n g p a r t i c u l a r example w i l l i l l u s t r a t e  However  the  p r i n c i p l e s and the r e s u l t o f such a s w i t c h o f f . I t w i l l be assumed t h a t the p o t e n t i a l i s switched o f f d u r i n g a time I n t e r v a l v e r y s h o r t compared to the p e r i o d . The p o t e n t i a l i s switched o f f over a time i n t e r v a l c e n t r e d on a time,  t  5  At  , the l a t t e r being c a l l e d the "time  o f s w i t c h o f f . " During t h i s time i n t e r v a l the p a r t i c l e t r a v e l s a d i s t a n c e Ax . Ax "a" since  At  i s much l e s s than the  i s much l e s s than the p e r i o d .  the centre o f Ax .  Let  amplitude x  I n t h i s type I I s w i t c h o f f the  0  be  r e s u l t i s a f r e e p a r t i c l e w i t h speed w i t h i n the range  /\L(* -(x -±x) )' x  11 (a -rx.*Ax)* ) '  to  1  0  l  V m  \j m  unless  x  position.  i s within  0  |x |^4X  If  i n the range  AX o f the e q u i l i b r i u m o r the extreme then the f r e e p a r t i c l e ' s speed i s  0  to or IK  t VW  c^-CXa-dx)  or  1  I  i s g r e a t e r than  where I i s the l e s s e r o f / a. -(x +AX) l  and OL  1  d  but l e s s than a.  If  J lx l - aj 0  * AX  t h e n the f r e e p a r t i c l e ' s speed i s g r e a t e r than zero but l e s s than  J 11 ( a - C i x | l  e  -  A X )  1  ) '  .  Hence when such a type I I s w i t c h o f f i s c a r r i e d o u t t h e r e s u l t I s a p a r t i c l e w i t h a speed l e s s than a / K . v  g r e a t e r than zero and  T h i s speed depends on the p a r t i c l e ' s  posi-  m  t i o n a t the time o f s w i t c h o f f and on the d i s t a n c e covered d u r i n g the s w i t c h i n g o f f p r o c e s s .  The s i g n i f i c a n t p o i n t to  n o t i c e i n t h i s example o f a type I I s w i t c h o f f i s t h a t n o t h i n g i s s t i p u l a t e d , d i s c u s s e d o r assumed r e g a r d i n g the b e h a v i o u r o f the system d u r i n g the time i n t e r v a l A t . The  two p r e c e d i n g examples have d e a l t w i t h a system  and p r o c e s s t r e a t e d i n c l a s s i c a l terms.  I t i s now o f  i n t e r e s t to d e s c r i b e the system r e s u l t i n g from the c l a s s i c a l s w i t c h o f f i n quantum mechanical  terms.  The r e s u l t o f the  t y p e I s w i t c h o f f was a p a r t i c l e w i t h zero t o t a l  energy.  50  I n quantum mechanics t h i s p a r t i c l e would be d e s c r i b e d by a c o n s t a n t wavefunction.  I f t h i s w a v e f u n c t i o n a l s o had to  s a t i s f y the n o r m a l i z a t i o n c o n d i t i o n t h i s constant would be z e r o .  |4M d x = l  J  As w i l l be r e c a l l e d  then  from  s e c t i o n s one and two, the r e s u l t o f r e d u c i n g the p o t e n t i a l i n the w a v e f u n c t i o n f o r the harmonic o s c i l l a t o r was a l s o zero.  The system r e s u l t i n g from the type I I s w i t c h o f f i s  d e s c r i b e d by a p l a n e wave  Q  ±>itK  g r e a t e r than zero and l e s s than  where J Km  IM  has a v a l u e  a .  I t should  M be p o i n t e d o u t t h a t the "a" i n /Km  a.  i s n o t a quantum  m e c h a n i c a l q u a n t i t y but e n t e r s from the r e s t r i c t i o n o f the v e l o c i t y o f the f r e e p a r t i c l e being d e s c r i b e d .  T h i s plane  wave was n o t o b t a i n e d by r e d u c i n g the p o t e n t i a l i n the harmonic o s c i l l a t o r  wave-function.  CHAPTER VI UNIFORM MAGNETIC FIELD  The system under c o n s i d e r a t i o n i n t h i s chapter c o n s i s t s o f a s p i n l e s s p a r t i c l e -with charge e moving i n a u n i f o r m magnetic f i e l d  H  which f i l l s  a l l space.  v e l o c i t y p e r p e n d i c u l a r to the f i e l d w i l l  The  be denoted by  V .  The ^ - a x i s i s chosen i n the d i r e c t i o n o f the f i e l d . 1.  QUANTUM MECHANICAL TREATMENT When  ( -  0,0) i s chosen as t h e v e c t o r p o t e n t i a l  and when  i s chosen as the form  X  is  13  o> = c W a r e s u b s t i t u t e d i n ( 6 . 1 ) an mc e q u a t i o n f o r m a l l y i d e n t i c a l to the S c h r o e d i n g e r E q u a t i o n f o r  If  M „= - c p J  x  etr  and  a harmonicx o s c i l l a t o r i s o b t a i n e d . To n o r m a l i z e a w a v e f u n c t i o n o f the above form i t i s s u f f i c i e n t t o i n t e g r a t e from y  only.  — «o  to + ©o  w i t h r e s p e c t to  T h i s i s t h e case s i n c e the p a r t s o f t h e wavefunc-  t i o n depending on x  and ^. a r e a l r e a d y i n the form o f p l a n e  waves and i n unbounded space no f u r t h e r n o r m a l i z a t i o n i s  52  p o s s i b l e or necessary.  The n o r m a l i z a t i o n c o n d i t i o n i s t h e r e -  ay - 1 .  lore 4*  that  T h i s imposes the boundary c o n d i t i o n s  goes to zero as  goes to i n f i n i t y .  U s i n g the same techniques as i n the harmonic o s c i l l a t o r case the n o r m a l i z e d wavefunction i s  (6.2)  t=e -** ^ J«« i(k  k  \l 2  I f the f i e l d The  rate of  Hjn^irJ).  - -«»iwJf** e  X n  (n»)Yfc  H  W  i s reduced  approaching  zero i s  *  /  to zero  4^  goes to z e r o .  G  as H goes to z e r o .  T h i s i s an e s s e n t i a l s i n g u l a r i t y as H goes to zero and w i l l be studied i n section three. By a g a i n u s i n g the analogy  between (6.1)  and the wave  e q u a t i o n f o r a harmonic o s c i l l a t o r the allowed energy a r e seen to be ^ 1  (6.3) where  -  E = (n*i)fc«a +££ n  n  t o zero is,  values  i s a non-negative  co  integer.  As the f i e l d i s decreased  goes to zero and the o n l y energy i s  .  That  as H goes t o zero the energy v a l u e s a s s o c i a t e d w i t h the  t r a n s v e r s e motion a l l become zero l e a v i n g o n l y the energy a s s o c i a t e d w i t h the u n a f f e c t e d motion p a r a l l e l The  to the f i e l d .  r e s u l t o f d e c r e a s i n g H to zero i n (6.1)  is  53  X" P**X 0  (6.4)  +  =  2m 2x  where  i  ^ " -EjL  s  Zm  i  s i n c e there i s no f i e l d .  r  n  ~f£ 1m  When ( 6 . 4 ) i s combined w i t h the  boundary c o n d i t i o n s t h a t 'jC goes t o zero as infinity,  1^1  goes t o  t h e o n l y a c c e p t a b l e s o l u t i o n (as i n t h e p r e v i o u s ~ ^  cases) i s  ~0 .  The b l o c k diagram f o r t h i s system i s o c c u p i e d as f o l l o w s : corner one by e q u a t i o n ( 6 . 1 ) and the boundary c o n d i t i o n s that  4^  goes to zero as  goes to ± 00  • corner two  by the same boundary c o n d i t i o n s and e q u a t i o n ( 6 . 4 ) ; t h r e e by w a v e f u n c t i o n  (6.2);  corner  and corner f o u r by 4^*0.  As  been shown, r e d u c i n g the f i e l d i n the wavefunction ( 6 . 2 )  has  r e s u l t s i n H^=o. 2.  Hence the b l o c k diagram i s  completed.  CLASSICAL ANALYSIS OF SWITCHING OFF In  c l a s s i c a l terms the f o r c e e q u a t i o n f o r a s t a b l e  o r b i t i n a u n i f o r m magnetic f i e l d i s  .e_H  (6.5)  (TIC Stating  = Y_.  *"  t h a t a system behaves as a charged p a r t i c l e moving i n  a u n i f o r m magnetic f i e l d i m p l i e s t h a t the p a r t i c l e i s moving in  a s t a b l e o r b i t i n the p l a n e p e r p e n d i c u l a r to the f i e l d .  (6.5)  i s t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r such a  stable orbit.  Hence (6..50 may be c o n s i d e r e d as t h e c l a s s i c a l  c r i t e r i o n o f t h i s system.  54  Before processes  c o n s i d e r i n g the a c t u a l s w i t c h i n g o f f  the f o l l o w i n g p o i n t should be emphasized.  p a r t i c l e ' s speed, f i e l d ' s behaviour to  the f i e l d  V  The  , remains constant independent o f  the  because the f o r c e i s always p e r p e n d i c u l a r  and no f u r t h e r c o n s t r a i n t s may  be  introduced.  I n d i s c u s s i n g the s w i t c h i n g o f f o f the magnetic field will  the two  types must a g a i n be d i s t i n g u i s h e d .  a g a i n be s t u d i e d The  Type I  first.  type I s w i t c h o f f was  d e f i n e d as being the  i n which the system s a t i s f i e s the d e f i n i n g c r i t e r i a i n t e r m e d i a t e v a l u e s o f the p o t e n t i a l .  type  for a l l  S t a t i n g t h a t the  c r i t e r i o n o f t h i s system i s s a t i s f i e d f o r a l l i n t e r m e d i a t e f i e l d v a l u e s means t h a t the e q u a t i o n these intermediate f i e l d values. be  switched  (6.5)  T h e r e f o r e , the f i e l d must  o f f i n a manner such t h a t a t a l l i n t e r m e d i a t e  s t a g e s the p a r t i c l e , t r a v e l l i n g w i t h f i n i t e has  i s satisfied for  v e l o c i t y , "v  s u f f i c i e n t time to r e a c h the d i s t a n c e , r  satisfies  (6.5)  , which  f o r the v a r i o u s i n t e r m e d i a t e f i e l d  values.  S i n c e a continuous  s w i t c h o f f may  a r b i t r a r y accuracy  by a d i s c r e t e s w i t c h o f f , o n l y the  be approximated w i t h  need be c o n s i d e r e d even though the r e s u l t s w i l l b o t h methods. will  H  x  now  The  ,  latter  apply to  d e t a i l s o f a d i s c r e t e type I s w i t c h o f f  be  analyzed.  If  i n a given->step the f i e l d  i s lowered from H to  then the r a d i u s , r , a s s o c i a t e d w i t h % x  i s greater  than  55 r  a s s o c i a t e d w i t h H.  For (6.5)  to be s a t i s f i e d the r a d i u s  must t h e r e f o r e be rj_ when the f i e l d i s % . "C , where  g r e a t e r than  t  is  b e f o r e the step from H to 1^ T h i s requirement  r -r v  , must e l a p s e  x  can be c o n s i d e r e d  completed.  r e g a r d i n g the time i s necessary  the p a r t i c l e , with i t s f i n i t e v e l o c i t y , r e q u i r e d by ( 6 . 5 ) .  position  Hence a time  to enable  to r e a c h the  Before u n d e r t a k i n g  any g i v e n  decrement the p r e v i o u s step must o f course be completed. I n the f i n a l  step the f i e l d goes from some H r'  t h e r a d i u s goes from some f i n i t e infinite.  to  r  a  1  t o zero and  where ^o- <  As above a time g r e a t e r than  r  0  is  i s required  v i n o r d e r t h a t t h i s step be completed i n a type I manner. S i n c e the f i n a l  step must be completed b e f o r e the f i e l d can  be c o n s i d e r e d as switched o f f , t h i s i n f i n i t e time f o r the last  step shows t h a t a type I s w i t c h o f f o f t h i s  magnetic f i e l d  cannot be done i n a f i n i t e  The s t i p u l a t i o n s  uniform  time.  f o r a type I s w i t c h o f f a l s o  imply  t h a t any r e s u l t o r e f f e c t due to any g i v e n step must be i d e n t i c a l t o the r e s u l t o r e f f e c t o b t a i n e d by c a r r y i n g out t h i s same step i n an a r b i t r a r i l y l a r g e number o f a r b i t r a r i l y s m a l l , c o n s e c u t i v e type I s t e p s . an a r b i t r a r y H  a  to  step from  W\  to  t o zero w i l l be s t u d i e d .  From t h i s p o i n t o f view H  f  and then the step from  I n l o w e r i n g the f i e l d from H;  the r a d i u s i n c r e a s e s from  *\  to  ^  .  I f a very  l a r g e number o f steps are employed the r a d i u s i n c r e a s e s as  in  the mono t o n i c sequence:  For  ( 6 . 5 ) to be s a t i s f i e d  g r e a t e r than decrease  r*+ - r i  r,-, r , r (  . . .r ~ar  X)  i  f  f o r a l l i n t e r m e d i a t e v a l u e s a time must e l a p s e b e f o r e the corresponding  K  can be undertaken.  must e l a p s e i n going from V  V  Hence the t o t a l  Hf  to  time which  i s g r e a t e r than  V  v  V  T h i s agrees w i t h the p r e v i o u s r e s u l t . t o zero w i l l now be s t u d i e d .  a  r .  i  i n the f i e l d can be c o n s i d e r e d as completed and the  n e x t decrease  H  i  The f i e l d going  I f this  from  switching o f f i s  done i n an a r b i t r a r i l y l a r g e number o f s m a l l steps the f i e l d goes through the v a l u e s o f the f o l l o w i n g m o n o t o n i c a l l y d e c r e a s i n g sequence:  H  a  ;  H  4  , H  1  .  }  . ., AH  O.  t  K ^ -r  As p r e v i o u s l y e x p l a i n e d , a time g r e a t e r than  K  t  K  must  v H  e l a p s e before the step from as  completed.  K  to  H**,  can be c o n s i d e r e d  Wa. to zero to  Hence f o r the s w i t c h o f f from  be completed i n a type I f a s h i o n , the r e q u i r e d time i s g r e a t e r than Since is  f  0  r — f"a .». r - n ^ x  x  , corresponding  f i n i t e , this  .. . •+• r  a  -  r  A t t  -  r 0  „  to zero f i e l d , i s i n f i n i t e and  time i s i n f i n i t e .  Hence, as b e f o r e , an  i n f i n i t e time i s r e q u i r e d f o r a type I s w i t c h o f f o f the magnetic field.  The p o i n t o f view t h a t the r e s u l t o f any step  must be e q u i v a l e n t to the r e s u l t o f an a r b i t r a r i l y l a r g e number o f steps between the same i n i t i a l v a l u e s emphasizes the f a c t  and f i n a l f i e l d  t h a t the s w i t c h i n g o f f p r o c e s s  cannot be c o n s i d e r e d completed u n t i l  sufficient  time has  57  e l a p s e d f o r the f i n a l  step to he  completed.  S i n c e a type I s w i t c h o f f r e q u i r e s an i n f i n i t e as demonstrated above, any time i s o f type I I .  time  s w i t c h o f f completed i n a f i n i t e  A s w i t c h o f f i n which the f i e l d goes to  z e r o i n a time comparable to the p e r i o d o f the p a r t i c l e i s o f type I I and may  orbiting  be used as an example.  The  r e s u l t o f t h i s s w i t c h o f f i s a f r e e p a r t i c l e w i t h speed V  .  The d i r e c t i o n o f the f r e e p a r t i c l e ' s v e l o c i t y  depends  upon the d e t a i l s o f the s w i t c h o f f .  I n f a c t the r e s u l t o f  any  S i n c e a type I I  type I I s w i t c h o f f i s as above.  o f f i s done i n a f i n i t e finite velocity,  time and i n view o f the  the p a r t i c l e may  f i n i t e volume f o r any p a r t i c u l a r Although  type I I s w i t c h o f f .  the d i s c u s s i o n i n t h i s s e c t i o n  terminology  types of switch o f f .  The  p a r t i c l e i s at i n f i n i t y  zero.  This i s also  from these  two  r e s u l t o f the type I s w i t c h o f f V  , at i n f i n i t y .  i t s probability  volume i s z e r o .  has been i n  to d e s c r i b e i n quantum  the systems r e s u l t i n g  a p a r t i c l e , with v e l o c i t y ,  elementary  particle's  be l o c a l i z e d w i t h i n a g i v e n  c l a s s i c a l terms o n l y , i t i s o f i n t e r e s t mechanical  switch  Since  the  o f being i n any  Hence l ^ i i s zero and  4^  the r e s u l t o b t a i n e d by reducing  finite  i s also the  f i e l d i n the wavefunction. The r e s u l t o f a type I I s w i t c h ±ik•r o f f has 6 as i t s wavefunction where k = my _^ * and of  the d i r e c t i o n o f  k  the switelling o f f p r o c e s s .  i s determined  by the  T h i s plane wave was  o b t a i n e d by r e d u c i n g the f i e l d i n the  wavefunction.  was  details not  58 B e f o r e completing t h i s s e c t i o n an apparent i n c o n s i s t e n c y between the quantum mechanical and c l a s s i c a l v a l u e s f o r the  t r a n s v e r s e energy, when the f i e l d i s zero, w i l l  out.  be p o i n t e d  As shown i n the p r e v i o u s s e c t i o n , the quantum mechanical  e x p r e s s i o n f o r the t r a n s v e r s e energy becomes zero as the f i e l d i s decreased to z e r o .  However i n both types o f  classical  s w i t c h o f f the t r a n s v e r s e v e l o c i t y , and hence t r a n s v e r s e k i n e t i c energy, remains i n c l u d i n g zero 3.  constant f o r a l l v a l u e s o f the  field.  WAVEFUNCTION ESSENTIAL SINGULARITY FOR In  ZERO FIELD  t h i s system the dominating f a c t o r i n the wave-  f u n c t i o n as the f i e l d i s reduced to zero i s is,  field  Q.  the w a v e f u n c t i o n goes to zero e x p o n e n t i a l l y as  goes t o i n f i n i t y . an expansion about  S i n c e t h i s i s an e s s e n t i a l H° 0  i s i m p o s s i b l e and  , that -pj-  singularity  therefore  p e r t u r b a t i o n t e c h n i q u e s w i l l n o t g i v e the w a v e f u n c t i o n f o r a charged p a r t i c l e i n a u n i f o r m magnetic  f i e l d f o r small  fields. A l t h o u g h the harmonic  o s c i l l a t o r and magnetic  field  wave equations and wavefunctions are f o r m a l l y the same t h e r e i s one fundamental d i f f e r e n c e and i t i s t h i s which  difference  corresponds to the v a s t l y d i f f e r e n t p h y s i c a l behaviour  between the two switched o f f .  systems w i t h r e g a r d to the p o t e n t i a l being I n the harmonic  oscillator  case the independent  v a r i a b l e i s simply x—-independent thing else.  o f a l l parameters  However i n the magnetic  of p o s i t i o n  case the  )• 3ince H i s a  "independent v a r i a b l e " i s ( 'J-^o independent  field  or any-  = d(y-y„)  and (  parameter y - <^ ) i s c  t h e n the independent v a r i a b l e o f an e q u a t i o n f o r m a l l y i d e n t i c a l to the one f o r a harmonic o s c i l l a t o r .  I n the  harmonic o s c i l l a t o r case the p o t e n t i a l going to zero does not in  any way  i n f l u e n c e the independent v a r i a b l e x whereas when  t h e magnetic (  v^- v j  D  f i e l d goes to zero the "independent  ) goes to i n f i n i t y .  b e h a v i o u r o f the magnetic  Now  field  variable"  i t s h a l l be shown how  "independent  the  variable"  m a t h e m a t i c a l l y expresses the behaviour o f the p a r t i c l e i n the field.  u  3 0  - Cp  can be i d e n t i f i e d as the  x  <-j  coordinate  C H  of  the c e n t r e o f the c i r c u l a r p a t h i n the p l a n e p e r p e n d i c u l a r  to  the f i e l d .  in  Substituting u  the e x p r e s s i o n f o r  14 Similarly  X-X  a 0  ™  c  p  x  = mv  gives Vy  -» e j ^  x  c  u - u  x  and  A =-Hu x  J  =  Cm v  x  .  can be i n t r o d u c e d where  x  e  i s i d e n t i f i e d as the x c o o r d i n a t e o f the c e n t r e o f the above c i r c u l a r path. result  r  x  =  Squaring and adding produces  (x-xj -^(w-iwJ = 1  X  goes to zero the r a d i u s  V  c* v  x  .  between the magnetic  As the  goes to i n f i n i t y  w i t h the r e s u l t o f a type I s w i t c h o f f .  the f a m i l i a r field  i n agreement  These d i f f e r e n c e s  f i e l d and the harmonic o s c i l l a t o r  namely the behaviour o f the independent v a r i a b l e s and  cases, the  e s s e n t i a l s i n g u l a r i t y i n the former, correspond to the p h y s i cal  d i f f e r e n c e t h a t i n the magnetic  field  case the v e l o c i t y  60 i s undiminished  and the p a r t i c l e must go o f f to i n f i n i t y i n  a type I s w i t c h o f f whereas i n the harmonic o s c i l l a t o r the velocity  becomes zero and the p a r t i c l e i s contained  within a finite  r e g i o n of- space i n a type I s w i t c h o f f .  case  CHAPTER VII CONCLUSION  1.  DISCUSSION OF SWITCHING OFF I n t h i s t h e s i s two  PROCESSES  types o f s w i t c h i n g o f f have been  d i s t i n g u i s h e d ; namely, type I i n which the c r i t e r i a  relating  t h e parameters o f the system are s a t i s f i e d f o r a l l i n t e r mediate v a l u e s o f the p o t e n t i a l and  type I I i n which  c r i t e r i a are not s a t i s f i e d f o r i n t e r m e d i a t e v a l u e s o f potential. l e a s t one  I n a l l the bound systems c o n s i d e r e d  the  there was  at  c l a s s i c a l r e l a t i o n o r e q u a t i o n which c h a r a c t e r i z e d  t h e system. criterion. two  these  I n a l l the unbound systems t h e r e was  no  such  Hence i t f o l l o w s t h a t d i s t i n g u i s h i n g between the  types o f s w i t c h i n g o f f i s meaningful o n l y i n the case  of  a bound system. C r i t e r i a determining system undergoes w i l l now  the type o f s w i t c h o f f a bound  be g i v e n .  S i n c e the s w i t c h i n g o f f  p r o c e s s e s have been d i s c u s s e d i n c l a s s i c a l terms the w i l l be g i v e n i n c l a s s i c a l terms.  Due  criteria  to the u n c e r t a i n t y  p r i n c i p l e these c r i t e r i a cannot be d i r e c t l y extended to quantum mechanics and  t h e r e f o r e no  s p e c i f i c quantum mechani-  c a l , c r i t e r i a f o r d i s t i n g u i s h i n g the two will  be g i v e n .  s w i t c h i n g o f f methods  Even though the c h a r a c t e r i s t i c s d i s t i n g u i s h i n g  62 the two types o f s w i t c h o f f a r e n o t g i v e n i n quantum mechanic a l terms the a c t u a l d i s t i n c t i o n i n methods i s a p p l i c a b l e to a quantum mechanical d e s c r i p t i o n o f a system.  Furthermore,  s i n c e the a c t u a l e x p e r i m e n t a l procedures used i n s w i t c h i n g o f f p o t e n t i a l s are u s u a l l y o f a c l a s s i c a l nature,  this  c l a s s i c a l d i f f e r e n t i a t i o n between t h e types o f s w i t c h i n g o f f is  a p p l i c a b l e i n d e t e r m i n i n g the type o f s w i t c h o f f used  experimentally.  S i n c e a s w i t c h o f f i s e i t h e r o f type I o r  t y p e I I , i t i s s u f f i c i e n t to g i v e t h e c r i t e r i a f o r a t y p e M s w i t c h o f f s i n c e a s w i t c h o f f i n which these c r i t e r i a a r e n o t met I s n e c e s s a r i l y o f type I I . I n any bound system t h e maximum k i n e t i c energy  ever  a t t a i n e d must be l e s s than o r e q u a l t o t h e maximum o f the a b s o l u t e v a l u e s o f t h e p o t e n t i a l energy.  I t I s therefore  apparent t h a t u n l e s s t h e v e l o c i t y i s somewhere zero the p o t e n t i a l energy cannot go to zero i n a f i n i t e number o f s t e p s w i t h o u t v i o l a t i n g t h i s energy small potential.  criterion for sufficiently  T h i s may be e a s i l y seen i n a case where the  t o t a l energy i s n e g a t i v e .  F o r example, i n the n e g a t i v e energy  Coulomb case (see i n e q u a l i t y (3.8)) where the v e l o c i t y i s nowhere zero, the p o t e n t i a l cannot be zero f o r non-zero v e l o c i t y w i t h o u t t h i s i n e q u a l i t y being d i s o b e y e d . (3.9) i t was e x p l i c i t l y  By use o f  shown t h a t the p o t e n t i a l cannot go to  z e r o i n a f i n i t e number o f s t e p s .  The harmonic o s c i l l a t o r  I l l u s t r a t e s the case i n which the p o t e n t i a l can go to zero  63 s i n c e t h e r e i s a p o s i t i o n a t which the v e l o c i t y i s zero the p o t e n t i a l  can t h e r e he lowered.  system s t i p u l a t e s  that  I f a c r i t e r i o n of a  the p a r t i c l e must be a t i n f i n i t y  o r d e r to s a t i s f y t h i s c r i t e r i o n when the p o t e n t i a l t h e n the p o t e n t i a l  chapter s i x section system i n o r d e r t h a t  two.  time.  T h i s was  Hence, the two  i s zero,  demonstrated i n  requirements  of a  a type I s w i t c h o f f to zero may  time are: f i r s t ,  l y go to i n f i n i t y  in  i n t h i s system cannot be switched o f f i n  a type I manner i n a f i n i t e  In a finite  and  the p a r t i c l e need n o t  be done necessari-  i n o r d e r to s a t i s f y the c r i t e r i a o f the  system f o r zero p o t e n t i a l ;  and  secondly, t h e r e be a  a t which the p a r t i c l e ' s k i n e t i c energy I s z e r o . w h i c h meets these requirements  position  I n a system  the p o t e n t i a l may,  i n prin-  c i p l e , be switched o f f to zero i n a f i n i t e time i n a type I manner by l o w e r i n g the p o t e n t i a l w h i l e the p a r t i c l e i s a t a p o s i t i o n o f zero v e l o c i t y . l o w e r i n g o f the p o t e n t i a l  T h i s however r e q u i r e s a i n a zero time i n t e r v a l .  finite Since  t h i s cannot be achieved a type I s w i t c h o f f to zero i n a bound system i s n o t e x p e r i m e n t a l l y f e a s i b l e . experimental  Hence  any  s w i t c h o f f to zero i n a bound system i s o f  type  II.  The p r e c e d i n g d i s c u s s i o n s w i t c h o f f i n which the p o t e n t i a l  i s concerned  i s switched o f f to z e r o .  However, to an a r b i t r a r y degree o f accuracy, ing  o f the p o t e n t i a l  z e r o f i n a l v a l u e may  w i t h a type I  a type I  from an I n i t i a l v a l u e to a lower,  lowernon-  be e x p e r i m e n t a l l y c a r r i e d out i n those  64 cases where the p o t e n t i a l may be lowered d u r i n g a f i n i t e , non-zero  time i n t e r v a l and s t i l l  be i n a c c o r d w i t h the  c o n d i t i o n s f o r a type I s w i t c h o f f d u r i n g t h i s l o w e r i n g . F o r example, i n the u n i f o r m magnetic  f i e l d and bound Coulomb  c a s e s the p o t e n t i a l o r f i e l d may be e x p e r i m e n t a l l y lowered from  some i n i t i a l v a l u e t o a non-zero  manner.  final  one i n a type I  The d e t a i l s o f the procedure may v a r y from case to  c a s e but the p o i n t i s t h a t such a type I l o w e r i n g i s e x p e r i mentally  feasible. S i n c e i n an unbound system t h e r e i s no d i s t i n c t i o n  between the two methods o f s w i t c h i n g o f f , the.two types a r e i d e n t i c a l and both correspond to any g i v e n experimental switch o f f . 2.  REDUCTION OF THE POTENTIAL IN WAVEFUNCTION In  a l l o f the systems s t u d i e d the r e s u l t o f r e d u c i n g  t h e p o t e n t i a l i n the w a v e f u n c t i o n was one o f the f o l l o w i n g two: (a)  a zero wavefunction;  (b)  an o s c i l l a t o r y wavefunction —  either a  t r i g o n o m e t r i c o r imaginary e x p o n e n t i a l function. E a c h o f the r e s u l t s (b) corresponded  to an unbound  system.  Ea^ch o f the r e s u l t s ( a ) , w i t h the e x c e p t i o n o f the n e g a t i v e t o t a l energy u n i f o r m e l e c t r i c f i e l d  case, corresponded to  65 a bound system. five.  T h i s e x c e p t i o n w i l l be t r e a t e d i n s e c t i o n  The reasons f o r t h i s correspondence between a zero  f i n a l wavefunction and a bound system w i l l be seen i n the s u c c e e d i n g paragraphs. A significant  and s a t i s f y i n g  common f e a t u r e o f those  w a v e f u n c t i o n s which went to zero, e x c e p t i n g the above e x c e p t i o n , w i l l now the  be p r e s e n t e d .  As was  previously  stated,  wavefunction went to zero o n l y i f i t d e s c r i b e d a bound  system. the  The fundamental  characteristic  o f a bound system i s  normalization condition  systems  t h i s c o n d i t i o n determines a n o r m a l i z a t i o n c o e f f i c i e n t  w h i c h causes the w a v e f u n c t i o n to obey t h i s c o n d i t i o n . the  wavefunctions under  c o n s i d e r a t i o n (see  In a l l  ('2.5), (2.5*),  (3.4), (5.3), (6.2))  i t i s this normalization coefficient  which goes to z e r o .  Except f o r the magnetic  f i e l d wavefunc-  t i o n , these wavefunctions go to zero o n l y on account o f normalization coefficient.  I f the n o r m a l i z a t i o n c o e f f i c i e n t  would have been absent i n these systems resulting  the w a v e f u n c t i o n  from r e d u c i n g the p o t e n t i a l would have been a  c o n s t a n t but not, i n g e n e r a l , z e r o . of  their  t h i s w i l l now  be g i v e n .  The p h y s i c a l  I n a l l the bound  c o n s i d e r e d , except f o r the magnetic  significance  systems  f i e l d case, the t o t a l  energy goes to zero as the p o t e n t i a l does i n a type I s w i t c h off.  (The s i g n i f i c a n c e o f s p e c i f y i n g type I s w i t c h o f f w i l l  be seen f u r t h e r on i n t h i s s e c t i o n . ) a p a r t i c l e w i t h zero t o t a l energy.  Hence the end r e s u l t I s Having zero t o t a l  energy,  66 t h i s p a r t i c l e has an equal p r o b a b i l i t y o f being anywhere, that i s ,  a constant wavefunction to w i t h i n a phase  I n general,  it  is  o n l y upon a p p l i c a t i o n o f the n o r m a l i z a t i o n  condition, with i t s this  factor.  associated  constant must be z e r o .  normalization coefficient  boundary c o n d i t i o n s ,  that  I n the magnetic f i e l d case  a l s o goes to z e r o .  However,  the the  w a v e f u n c t i o n goes to zero more r a p i d l y due to a dominating exponential tial  decrease  infinity. general (a)  factor.  As shown I n chapter s i x ,  this  exponen-  to zero corresponds to the p a r t i c l e going  The p r e c e d i n g c o n s i d e r a t i o n s  suggest the  following  statements:  I n a l l bound systems i n which i t i s not i m p e r a t i v e t h a t . the p a r t i c l e go to i n f i n i t y i n a type I s w i t c h o f f , wavefunction, normalization  (b)  to  If  i n general,  goes to zero due to  the  coefficient.  the p a r t i c l e must go to i n f i n i t y i n a type I s w i t c h o f f  then the n o r m a l i z a t i o n c o e f f i c i e n t is  the  a g a i n goes to zero but  dominated by an e x p o n e n t i a l l y d e c r e a s i n g f a c t o r which  describes  the p a r t i c l e going to i n f i n i t y .  The p r e c e d i n g has shown how the c h a r a c t e r i s t i c p r o p e r t y o f a bound system d i r e c t l y determines the r e s u l t o f r e d u c i n g the p o t e n t i a l i n the wavefunction o f  need not be s a t i s f i e d .  such a system.  Furthermore, there are no  conditions  67  by which the p o t e n t i a l r e s t r i c t s the t o t a l energy.  Hence  r e d u c i n g the p o t e n t i a l i n the w a v e f u n c t i o n f o r an unbound system y i e l d s a f r e e p a r t i c l e wavefunction which s a t i s f i e s t h e boundary c o n d i t i o n s .  Thus f o r an unbound system the  r e s u l t o f r e d u c i n g the p o t e n t i a l i n the w a v e f u n c t i o n i s t h a t expected from e x p e r i m e n t a l o b s e r v a t i o n s . The r e l a t i o n between the r e s u l t o f r e d u c i n g the p o t e n t i a l i n the w a v e f u n c t i o n and the r e s u l t s o f the two types of  s w i t c h i n g o f f w i l l now be d i s c u s s e d .  Bound systems w i l l  a g a i n be d i s c u s s e d f i r s t . As demonstrated  i n the examples o f s w i t c h i n g o f f i n  t h e p r e v i o u s c h a p t e r s , the system r e s u l t i n g from a type I s w i t c h o f f i s quantum m e c h a n i c a l l y d e s c r i b e d by the r e s u l t o f r e d u c i n g the p o t e n t i a l i n the w a v e f u n c t i o n o f the o r i g i n a l system, The  t h a t i s , by a zero w a v e f u n c t i o n f o r a bound  reason f o r t h i s correspondence  system.  between the r e s u l t s o f a  t y p e I s w i t c h o f f , and r e d u c i n g the p o t e n t i a l i n the wavefunct i o n , w i l l become apparent when the p r o p e r t i e s o f a type I s w i t c h o f f and o f a w a v e f u n c t i o n a r e compared. In  a d d i t i o n to o t h e r parameters  and v a r i a b l e s , the  w a v e f u n c t i o n i s a f u n c t i o n o f the p o t e n t i a l o f a system and f u l l y d e s c r i b e s the system i n terms o f the p o t e n t i a l and t h e s e o t h e r parameters  and v a r i a b l e s .  t h e r e i s a one to one correspondence s p e c i f i c s e t o f parameters  F o r any s p e c i f i c  system  between the system w i t h a  and a p a r t i c u l a r wavefunction.  68 C o n s i d e r a w a v e f u n c t i o n d e s c r i b i n g any p a r t i c u l a r  system.  If  the  p o t e n t i a l parameter w i t h i n the w a v e f u n c t i o n i s changed,  the  r e s u l t i s a w a v e f u n c t i o n d e s c r i b i n g a system w i t h the  same c h a r a c t e r i s t i c s tial  energy.  and c r i t e r i a but w i t h a d i f f e r e n t poten-  That i s , the w a v e f u n c t i o n now d e s c r i b e s the  same k i n d o f system which has a d j u s t e d i t s e l f such as to satisfy i t s characteristic e q u a l to i t s new v a l u e .  c r i t e r i a when the p o t e n t i a l i s  Now  consider a p a r t i c u l a r  d e s c r i b e d by a p a r t i c u l a r w a v e f u n c t i o n . of  system  L e t the p o t e n t i a l  t h i s system be switched o f f and c o n s i d e r the system as the  potential  i s being lowered.  S i n c e the s w i t c h i n g o f f p r o c e s s  I s n o t being d e s c r i b e d , i t i s unnecessary to s t i p u l a t e whether the p r o c e s s i s c l a s s i c a l o r quantum. system  As l o n g as the  s a t i s f i e s the c r i t e r i a o f the o r i g i n a l system, the  s w i t c h o f f i s o f type I and the o r i g i n a l wavefunction, w i t h the  potential  any p a r t i c u l a r the  reduced, may be used to d e s c r i b e the system a t stage.  However, as soon as the c r i t e r i a o f  o r i g i n a l system are no l o n g e r s a t i s f i e d , the p o t e n t i a l  i s no l o n g e r being switched o f f i n the o r i g i n a l system but i n a n o t h e r , d i f f e r e n t system.  A t t h i s s t a g e , where the s w i t c h  o f f i s no l o n g e r o f type I and the p o t e n t i a l  i s being  s w i t c h e d o f f i n a d i f f e r e n t system, r e d u c i n g the p o t e n t i a l in  the o r i g i n a l w a v e f u n c t i o n no l o n g e r corresponds to the  p h y s i c a l p r o c e s s and a d i f f e r e n t w a v e f u n c t i o n d e s c r i b i n g  this  d i f f e r e n t system must now be i n t r o d u c e d and the p o t e n t i a l reduced i n t h i s l a t t e r wavefunction.  Hence i t i s seen t h a t  69 t h e i d e n t i f i c a t i o n o f the r e s u l t o f a type I s w i t c h o f f w i t h t h e r e s u l t o f r e d u c t i o n i n the w a v e f u n c t i o n f o l l o w s from the p r i m a r y p r o p e r t y o f a wavefunction and a type I s w i t c h o f f . As shown i n s e c t i o n one, any experimental s w i t c h o f f to zero i n a bound system i s n e c e s s a r i l y o f type I I . j u s t been demonstrated tial I  I t has  t h a t the r e s u l t o f r e d u c i n g the poten-  t o zero i n a w a v e f u n c t i o n d e s c r i b e s the r e s u l t o f a type  switch o f f .  Hence r e d u c i n g the p o t e n t i a l to zero i n a wave-  f u n c t i o n d e s c r i b i n g a bound system does n o t correspond to an e x p e r i m e n t a l l y f e a s i b l e method o f s w i t c h i n g o f f the p o t e n t i a l in  t h i s system.  T h i s i s the r e a s o n the r e s u l t o f r e d u c i n g  t h e p o t e n t i a l t o zero i n the w a v e f u n c t i o n o f a bound  system  does n o t y i e l d the p l a n e wave w a v e f u n c t i o n i n d i c a t e d by experimental observations.  However a type I s w i t c h o f f to a  non-zero v a l u e i s p o s s i b l e i n some bound systems.  I n these  systems the r e s u l t o f such a l o w e r i n g to a non-zero is  value  d e s c r i b e d by the r e s u l t o f r e d u c i n g the p o t e n t i a l to t h i s  non-zero,  f i n a l v a l u e i n the o r i g i n a l  wavefunction.  I n quantum mechanical terminology, r e d u c i n g the p o t e n t i a l i n the w a v e f u n c t i o n o f a bound system d e s c r i b e s a p r o c e s s whereby the system proceeds through s u c c e s s i v e s t a t i o n a r y s t a t e s o f t h i s same system, where each  stationary  s t a t e corresponds to a lower p o t e n t i a l than the p r e v i o u s one, u n t i l the s t a t i o n a r y s t a t e corresponding to zero p o t e n t i a l is  reached.  The wavefunction r e s u l t i n g from r e d u c i n g the  70  p o t e n t i a l to any v a l u e , i n c l u d i n g zero, i s the wavefunction d e s c r i b i n g the s t a t i o n a r y s t a t e corresponding  to t h i s  reduced  v a l u e o f the p o t e n t i a l . S i n c e i n an unbound system the two types o f s w i t c h i n g off  a r e e q u i v a l e n t and r e d u c i n g the p o t e n t i a l i n the wavefunc-  t i o n d e s c r i b e s the r e s u l t o f a type I p r o c e s s , i t f o l l o w s t h a t t h e r e s u l t . o f r e d u c i n g the p o t e n t i a l to zero i n the wavefunct i o n d e s c r i b e s the r e s u l t o f p h y s i c a l l y s w i t c h i n g o f f the p o t e n t i a l i n an a r b i t r a r y manner.  T h i s i s supported  by the  examples o f unbound systems which have been analyzed i n c h a p t e r s two, t h r e e , and f o u r .  Hence i n a l l unbound systems,  t h e r e s u l t o f r e d u c i n g the p o t e n t i a l to zero i n the wavefunct i o n i s a f r e e p a r t i c l e wavefunction experimental 3.  as i s . expected  from  observations.  DECREASING THE POTENTIAL IN THE WAVE EQUATION The  r e s u l t o f d e c r e a s i n g the p o t e n t i a l i n the wave  e q u a t i o n w i l l now be d i s c u s s e d . will  The case o f a bound system  be t r e a t e d f i r s t . I f i n the wave e q u a t i o n f o r a bound system the  p o t e n t i a l i s decreased  to zero and the o t h e r parameters a r e  v a r i e d i n accord w i t h the c r i t e r i a o f the system, t h i s method of  d e c r e a s i n g the p o t e n t i a l o b v i o u s l y corresponds  switch o f f .  to a.".type I  When the r e s u l t i n g e q u a t i o n i s s o l v e d i n  c o n j u n c t i o n w i t h the boundary c o n d i t i o n s o b t a i n e d from the  71 o r i g i n a l ones by d e c r e a s i n g the p o t e n t i a l to zero the r e s u l t is  a zero wavefunction  switch o f f .  which d e s c r i b e s the r e s u l t o f a type I  I f however the p o t e n t i a l i s m a t h e m a t i c a l l y s e t  e q u a l to zero i n the e q u a t i o n without i n f l u e n c i n g any o f the o t h e r parameters t h i s then corresponds d e c r e a s e d without imposing  to the p o t e n t i a l  the c r i t e r i a o f the system.  being If  t h e r e s u l t i n g e q u a t i o n i s s o l v e d i n c o n j u n c t i o n w i t h the boundary c o n d i t i o n s d e r i v e d from the o r i g i n a l ones by d e c r e a s ing  the p o t e n t i a l to zero, the s o l u t i o n I s a g a i n a zero wave-  f u n c t i o n which a g a i n d e s c r i b e s the r e s u l t o f a type I s w i t c h off.  This at f i r s t  appears  s u r p r i s i n g s i n c e the p o t e n t i a l i n  t h e e q u a t i o n was decreased i n a manner analogous switch o f f .  to a type I I  I f , however, t h i s l a t t e r r e s u l t i n g e q u a t i o n i s  s o l v e d i n c o n j u n c t i o n w i t h d i f f e r e n t boundary c o n d i t i o n s the s o l u t i o n w i l l be a d i f f e r e n t , non-zero wavefunction.  I f these  d i f f e r e n t boundary c o n d i t i o n s a r e chosen to be those f o r a f r e e p a r t i c l e the s o l u t i o n i s a plane wave which i s the wavef u n c t i o n d e s c r i b i n g the r e s u l t o f a type I I s w i t c h o f f . The p r e c e d i n g paragraph has demonstrated t h a t the boundary c o n d i t i o n s , a s s o c i a t e d w i t h the wave e q u a t i o n r e s u l t i n g from d e c r e a s i n g the p o t e n t i a l i n the o r i g i n a l e q u a t i o n , determine  the type o f s w i t c h o f f to which d e c r e a s i n g  t h e p o t e n t i a l i n the wave e q u a t i o n corresponds. r e a s o n a b l e i f the f o l l o w i n g i s c o n s i d e r e d .  This i s  M a i n t a i n i n g the  boundary c o n d i t i o n s o f a bound system i m p l i e s t h a t the system  72 remains the same and t h a t t h e r e f o r e the c r i t e r i a o f the bound system are s a t i s f i e d . maintained  Hence i f the boundary c o n d i t i o n s a r e  f o r a l l v a l u e s o f the p o t e n t i a l i t i s apparent  t h a t the c o n d i t i o n s f o r a type I s w i t c h o f f a r e s a t i s f i e d . I f , however, the boundary c o n d i t i o n s a r e a l t e r e d w h i l e the p o t e n t i a l i s being decreased then the r e s u l t i n g e q u a t i o n i n c o n j u n c t i o n w i t h these d i f f e r e n t boundary c o n d i t i o n s d e s c r i b e s a system whose c h a r a c t e r i s t i c s d i f f e r from those o f the initial  system.  T h i s corresponds  to a type I I s w i t c h o f f .  I f i n the wave e q u a t i o n f o r an unbound system the p o t e n t i a l i s s e t equal to zero, no o t h e r parameters may be a f f e c t e d s i n c e f o r such systems the p o t e n t i a l p l a c e s no r e s t r i c t i o n s on the o t h e r parameters.  I f the r e s u l t i n g  equa-  t i o n i s s o l v e d i n c o n j u n c t i o n w i t h boundary c o n d i t i o n s o b t a i n e d from the o r i g i n a l ones by d e c r e a s i n g the p o t e n t i a l t o zero the s o l u t i o n i s an o s c i l l a t i n g f r e e p a r t i c l e wavefunction.  S i n c e the boundary c o n d i t i o n s a r e not a l t e r e d ,  w i t h t h e e x c e p t i o n o f d e c r e a s i n g the p o t e n t i a l i f i t e x p l i c i t l y appears  i n them, the system remains the same and  t h e above p r o c e s s corresponds  to a type I s w i t c h o f f which  f o r unbound systems i s i d e n t i c a l to a type I I s w i t c h o f f . Hence the s o l u t i o n o f the wave e q u a t i o n f o r an unbound system, w i t h the p o t e n t i a l decreased  to zero, i n c o n j u n c t i o n w i t h the  above boundary c o n d i t i o n s corresponds observed  result.  to the e x p e r i m e n t a l l y  73 4.  DISCUSSION OF THE ORIGINAL PROBLEM AND PARADOX OF AN ELECTRON IN A UNIFORM MAGNETIC FIELD The o r i g i n a l problem i n s e c t i o n one o f chapter one  will  now be a n a l y z e d .  u n i f o r m magnetic f i e l d  I n t h i s problem o f an e l e c t r o n i n a the system i s a bound one and the f i e l d  is  switched o f f i n a f i n i t e  of  type I I .  cannot  time.  T h i s switch o f f i s therefore  I t t h e r e f o r e f o l l o w s t h a t the experimental  result  be d e s c r i b e d by the r e s u l t o f r e d u c i n g the f i e l d i n the  wavefunction. The paradox arose i n the o r i g i n a l treatment r e d u c i n g the p o t e n t i a l i n the wavefunction  because  gave zero whereas a  p l a n e wave s o l u t i o n , which agreed w i t h experimental o b s e r v a t i o n , was o b t a i n e d by s o l v i n g the e q u a t i o n r e s u l t i n g from d e c r e a s i n g t h e p o t e n t i a l to zero i n the o r i g i n a l e q u a t i o n .  This plane  wave s o l u t i o n was o b t a i n e d because no boundary c o n d i t i o n s were a s s o c i a t e d w i t h the wave e q u a t i o n .  That i t was t h i s absence o f  accompanying boundary c o n d i t i o n s which l e d to the plane wave solution will  be shown i n the f o l l o w i n g paragraph.  S i n c e boundary c o n d i t i o n s d i d n o t accompany the wave e q u a t i o n the type o f s w i t c h o f f used was n o t s p e c i f i e d . N e g l e c t i n g the boundary c o n d i t i o n s t h a t the wavefunction to  zero a t - oo i s the same as imposing  g e n e r a l s o l u t i o n which o s c i l l a t e s a t  d i f f e r e n t ones. * oo  goes I f the  i s taken as the  s o l u t i o n to the wave e q u a t i o n w i t h zero p o t e n t i a l , the e f f e c t is  e q u i v a l e n t t o s p e c i f y i n g these d i f f e r e n t boundary c o n d i t i o n s  74  as being done f o r a f r e e p a r t i c l e . done.  As  T h i s i s what was  i n fact  shown i n the p r e v i o u s s e c t i o n , t h i s change i n  boundary c o n d i t i o n s r e s u l t s i n a wavefunction type II switch o f f .  describing a  Hence the s o l u t i o n o b t a i n e d i n t h i s  d e s c r i b e s the experimental  way  r e s u l t of a free p a r t i c l e since i n  a c t u a l i t y a type I I s w i t c h o f f i s e x p e r i m e n t a l l y c a r r i e d  out.  However, s i n c e reducing the p o t e n t i a l to zero i n the wavef u n c t i o n corresponds  to a type I s w i t c h o f f , these two mathe-  m a t i c a l d e s c r i p t i o n s o f the f i n a l  system do not agree.  t h e o r i g i n a l paradox arose because one wavefunction  Hence  was  o b t a i n e d by d e a l i n g w i t h an i n c o m p l e t e l y s p e c i f i e d bound system, and i t d e s c r i b e d the r e s u l t o f a type I I s w i t c h o f f , whereas the o t h e r wavefunction fully I  was  o b t a i n e d by d e a l i n g w i t h a  s p e c i f i e d system and i t d e s c r i b e d the r e s u l t o f a type  switch o f f . I f , however, the o r i g i n a l boundary c o n d i t i o n s had  a s s o c i a t e d w i t h the wave e q u a t i o n i n which the p o t e n t i a l decreased  been was  the s o l u t i o n o f t h i s e q u a t i o n would have been i n  agreement w i t h the r e s u l t o f reducing the p o t e n t i a l i n the wavefunction.  These i d e n t i c a l r e s u l t s would not have d e s c r i b e d the  p h y s i c a l s i t u a t i o n w i t h the p o t e n t i a l switched o f f s i n c e a l l t h e mathematics would d e s c r i b e the r e s u l t o f a type I off.  switch  75 5.  A COMMENT ON THE UNIFORM ELECTRIC FIELD CASE The u n i f o r m e l e c t r i c f i e l d  for  system i s an unbound one  both p o s i t i v e and n e g a t i v e v a l u e s o f the t o t a l energy, E .  As was shown i n chapter f o u r ,  the s i g n i f i c a n c e o f the v a l u e  o f E i s t o i n d i c a t e the p o s i t i o n a t which a p a r t i c l e i n c i d e n t from  ^ = + 00  i sclassically reflected.  Hence i n t h i s case  n e g a t i v e E does n o t i n d i c a t e a bound system. corresponds  t o r e f l e c t i o n a t <^=+oo  Negative E  when the f i e l d i s z e r o .  Hence n e g a t i v e E corresponds  to a zero p r o b a b i l i t y o f the  p a r t i c l e being i n any f i n i t e  r e g i o n o f space when the  is  zero.  Rather  field  than a n o r m a l i z i n g c o e f f i c i e n t , i t i s t h i s  i m p o s s i b i l i t y o f a p a r t i c l e w i t h n e g a t i v e E being i n a f i n i t e r e g i o n o f space when the p o t e n t i a l i s zero which accounts f o r t h e zero w a v e f u n c t i o n when the p o t e n t i a l i s reduced 6. (a)  to z e r o .  SUMMARY OF THESIS CONCLUSIONS F o r an unbound system. ( i ) The r e s u l t o f r e d u c i n g the p o t e n t i a l to zero i n the w a v e f u n c t i o n  o f the system i s an o s c i l l a t i n g  function describing  a f r e e p a r t i c l e whose k i n e t i c  energy i s equal t o the o r i g i n a l t o t a l  energy,  ( i i ) The system r e s u l t i n g from e x p e r i m e n t a l l y s w i t c h i n g off  the p o t e n t i a l i n any manner i s d e s c r i b e d by  both the r e s u l t o f r e d u c i n g the p o t e n t i a l t o zero i n the w a v e f u n c t i o n  and the s o l u t i o n o f the wave  e q u a t i o n w i t h accompanying boundary c o n d i t i o n s  76 which are o b t a i n e d by d e c r e a s i n g the p o t e n t i a l to zero i n the o r i g i n a l wave e q u a t i o n  and  boundary c o n d i t i o n s r e s p e c t i v e l y , ( i i i ) Only i n an unbound system can the p o t e n t i a l reduced  i n the wavefunction  or be  be  experimentally  switched o f f such t h a t the t o t a l energy remains constant. (b)  F o r a bound system. ( i ) The  r e s u l t o f r e d u c i n g the p o t e n t i a l to zero i n  the w a v e f u n c t i o n ( i i ) Two  i s a zero  wavefunction.  methods o f s w i t c h i n g o f f must be  guished. chapter  distin-  They are d e f i n e d i n s e c t i o n three o f five.  ( i i i ) A type I s w i t c h o f f to zero i s not feasible.  The  experimentally  system r e s u l t i n g from t h i s type o f  switch o f f Is mathematically  d e s c r i b e d by  the  r e s u l t o f r e d u c i n g the p o t e n t i a l to zero i n the wavefunction  or by the s o l u t i o n o f the  equation  w i t h accompanying boundary c o n d i t i o n s which are o b t a i n e d by d e c r e a s i n g the p o t e n t i a l to zero i n the o r i g i n a l e q u a t i o n and boundary c o n d i t i o n s respectively.  Hence, the r e s u l t o f r e d u c i n g  p o t e n t i a l to zero i n the wavefunction d e s c r i b e a system r e s u l t i n g from any experimental  switch o f f .  the  does n o t feasible  77 ( i v ) I n some bound systems the p o t e n t i a l may be e x p e r i m e n t a l l y lowered type I manner.  to a non-zero v a l u e i n a  The r e s u l t o f such a l o w e r i n g i s  d e s c r i b e d by the wavefunction  o b t a i n e d by  r e d u c i n g the p o t e n t i a l i n the i n i t i a l wavefunct i o n to t h i s lower  value,  (v) The p o t e n t i a l i n a bound system can be switched o f f to zero o n l y i n a type I I f a s h i o n .  The  r e s u l t o f a type I I s w i t c h o f f cannot be d e s c r i b e d by the r e s u l t o f a l t e r i n g i n any manner the p o t e n t i a l i n the wavefunction.  The  r e s u l t o f a type I I s w i t c h o f f can be d e s c r i b e d by the s o l u t i o n o f the e q u a t i o n o b t a i n e d  from  the o r i g i n a l by d e c r e a s i n g the p o t e n t i a l to zero o n l y i f the accompanying boundary c o n d i t i o n s a r e changed t o those f o r a f r e e (c)  particle.  B l o c k diagram. ( i ) F o r any system the block diagram i s always completed i f the c o r n e r s a r e o c c u p i e d by a complete d e s c r i p t i o n o f t h e i r r e s p e c t i v e systems and i f the steps from corner one to corner two and from corner t h r e e to corner f o u r  correspond  to the same type o f p h y s i c a l s w i t c h o f f . ( i i ) I n o r d e r to be c l o s e d a b l o c k diagram must d e a l o n l y w i t h a type I s w i t c h o f f s i n c e the step  78 from corner t h r e e to corner f o u r can o n l y to t M s (iii)  correspond  process,  F o r an unbound system a c l o s e d b l o c k diagram i s concerned  w i t h the a c t u a l p h y s i c a l p r o c e s s  and  i t s f o u r t h corner d e s c r i b e s the r e s u l t o f an experimental (iv)  switch o f f .  I n a bound system i n which the p o t e n t i a l goes to  zero, a c l o s e d b l o c k diagram must d e a l w i t h  an e x p e r i m e n t a l l y i m p o s s i b l e p r o c e s s and  the  e n t r y i n the f o u r t h corner does not d e s c r i b e the r e s u l t o f an e x p e r i m e n t a l l y f e a s i b l e  procedure,  (v) An a c t u a l p h y s i c a l s w i t c h o f f to zero cannot be expressed of (vi)  as a c l o s e d b l o c k diagram f o r the  case  a bound system,  F o r some bound systems a p a r t i a l s w i t c h o f f to a non-zero v a l u e can be expressed  I n the form o f a  completed b l o c k diagram. General p r o p e r t i e s o f a  wavefunction.  ( i ) S i n c e a g i v e n wavefunction with p a r t i c u l a r c r i t e r i a ,  d e s c r i b e s a system the  wavefunction  r e s u l t i n g from a l t e r i n g the v a l u e o f any parameter  i n the i n i t i a l wavefunction  Initial  d e s c r i b e s the  system m o d i f i e d such t h a t i t has  v a l u e f o r the parameter and the o r i g i n a l  criteria.  still  the  new  satisfies a l l  79  ( i i ) Changing the value of any parameter i n the wavefunction corresponds to the r e s u l t of p h y s i c a l l y changing this parameter by the same amount i n a manner such that the c h a r a c t e r i s t i c  c r i t e r i a of  the system are s a t i s f i e d f o r the i n i t i a l ,  final  and a l l intermediate values of t h i s parameter. For any given system t h i s manner of changing the value of the parameter may or may not be experimentally f e a s i b l e .  APPENDIX SYSTEMS CONTAINED VITHIN A PHYSICAL CONTAINER A system c o n t a i n e d by a p h y s i c a l c o n t a i n e r w i l l  now  be d i s c u s s e d i n o r d e r to see the e f f e c t o f reducing the p o t e n t i a l to zero i n the w a v e f u n c t i o n d e s c r i b i n g system.  such a  I n d i s c u s s i n g r e d u c i n g the p o t e n t i a l i n the wave-  f u n c t i o n o f such a system one can p r o p e r l y d i s c u s s o n l y a system whose dimensions do not need to exceed the dimensions o f the c o n t a i n e r i n o r d e r to s a t i s f y the c r i t e r i a o f the system f o r s u f f i c i e n t l y  small p o t e n t i a l s .  I f for sufficiently  s m a l l v a l u e s o f the p o t e n t i a l the r a d i u s must exceed the l i n e a r dimensions o f any g i v e n c o n t a i n e r , then f o r these small p o t e n t i a l s the system i s n o t the one which the w a v e f u n c t i o n describes.  F o r these small p o t e n t i a l s the system has an  a l t e r e d motion due to the superimposed rebound motion.  Hence  the r e n o r m a l i z e d w a v e f u n c t i o n o f the u n c o n s t r a i n e d system no l o n g e r d e s c r i b e s the a c t u a l behaviour o f the system a t these small potentials.  T h e r e f o r e , r e d u c i n g the p o t e n t i a l i n t h i s  w a v e f u n c t i o n does not correspond to s w i t c h i n g o f f the a c t u a l system.  To d e s c r i b e such a c o n s t r a i n e d system f o r these  s m a l l p o t e n t i a l s a d i s t r i b u t i o n f u n c t i o n may  be used.  p o t e n t i a l would t h e r e f o r e have to be reduced i n t h i s t i o n function.  The distribu-  Hence t h i s appendix a p p l i e s o n l y to a system  whose dimensions need not exceed those o f the c o n t a i n e r f o r very small p o t e n t i a l s .  81  The c o n s t r a i n t imposed by the w a l l s o f t h i s c o n t a i n e r i s expressed infinity. R=0  by a p o t e n t i a l which a b r u p t l y g o e s to  L e t t h i s c o n s t r a i n i n g p o t e n t i a l be |x{  for  < a  R=<x>  and  *¥(+a)  where  f o r jx| > a .  p o t e n t i a l imposes two s i g n i f i c a n t c o n d i t i o n s become  R  This  changes: the boundary  = ¥(-<*•)  =  0  and the  n o r m a l i z a t i o n c o n d i t i o n i n one dimension becomes  Two  cases must be d i s t i n g u i s h e d .  The f i r s t i s  where the s p a t i a l l y r e s t r i c t i n g p o t e n t i a l i s imposed on an a l r e a d y bound system and the second i s where t h i s p o t e n t i a l i s imposed on an otherwise  unbound system.  Both these  c a s e s can be i l l u s t r a t e d by the harmonic o s c i l l a t o r . f i r s t w i l l be c o n s i d e r e d I n the f i r s t  first.  case the t o t a l energy i s t h a t f o r a  harmonic o s c i l l a t o r , t h a t i s ' 1  where is  $  E - (s ^-)^w  5  +  i s not i n g e n e r a l an i n t e g e r .  reduced  co  c  c  As the p o t e n t i a l  goes to zero and the r e s u l t i s a g a i n a  p a r t i c l e w i t h zero t o t a l energy. function results. t h i s constant;  The  Hence a constant wave-  There would appear to be two c h o i c e s f o r  namely,  - L 2  to s a t i s f y the n o r m a l i z a t i o n  cc  c o n d i t i o n o r zero to s a t i s f y the boundary c o n d i t i o n s . 16 Chandrasekhar  - e /z x  oC e  obtains  as the s o l u t i o n o f the f i r s t  r  '  f  v i p{i-€]J  e x c i t e d s t a t e o f a bounded  l i n e a r o s c i l l a t o r where o( and  £  are constants, X  is a  82  power s e r i e s i n  p  and  p = X /"Wc  .  Since  p  goes to zero as the p o t e n t i a l does t h i s w a v e f u n c t i o n becomes zero.  Hence the boundary c o n d i t i o n s , r a t h e r than  normaliza-  t i o n c o n d i t i o n , are s a t i s f i e d . The  second case w i l l now be c o n s i d e r e d .  The t r e a t -  ment o f t h i s case w i l l be i n c l a s s i c a l terms but the quantum m e c h a n i c a l d e s c r i p t i o n o f the end r e s u l t w i l l be g i v e n .  The  s i t u a t i o n i s that o f a p a r t i c l e constrained within a region [ - a  a  ;  ] by p e r f e c t l y r i g i d and e l a s t i c w a l l s .  "Within  t h i s r e g i o n t h e p a r t i c l e i s s u b j e c t to a f o r c e o f magnitude Kx  towards the c e n t r e ,  However the t o t a l energy o f the  p a r t i c l e i s such t h a t i t s t i l l has a f i n i t e v e l o c i t y when i t r e a c h e s the w a l l s . '/i K a  1  •*• E  0  I t s energy may t h e r e f o r e be w r i t t e n as where  ^2 K a  1  i s the t o t a l energy asso-  c i a t e d w i t h the motion under the harmonic f o r c e and  E  c  is  t h e k i n e t i c energy the p a r t i c l e has when i t reaches a w a l l . If  the p o t e n t i a l i s now switched o f f i n a type I manner  ( w h i c h corresponds to r e d u c t i o n i n the wavefunction) as i n s e c t i o n t h r e e o f chapter f o u r , the r e s u l t i s a p a r t i c l e w i t h energy  E0  bouncing between the w a l l s .  I n quantum  mechanics t h i s r e s u l t i n g system i s d e s c r i b e d f u n c t i o n f o r a free p a r t i c l e constrained  by the wave-  t o the r e g i o n  The p r e c e d i n g examples suggest the f o l l o v i n g statements: (a)  I f the e x t e r n a l source p o t e n t i a l i s reduced to zero i n the v a v e f u n c t i o n d e s c r i b i n g a bound system v h i c h i s f u r t h e r c o n s t r a i n e d by an abrupt, i n f i n i t e  potential,  the r e s u l t i s a zero as vas the case i n the absence  of  the c o n s t r a i n t p o t e n t i a l . (b)  I f the e x t e r n a l source p o t e n t i a l i s reduced to zero i n the w a v e f u n c t i o n d e s c r i b i n g an unbound system which has an i n f i n i t e  constraining potential  superimposed,  the r e s u l t i s a f r e e p a r t i c l e w a v e f u n c t i o n w i t h i n the box formed (c)  by t h i s i n f i n i t e  potential.  The boundary c o n d i t i o n s , r a t h e r than the n o r m a l i z a t i o n c o n d i t i o n , are the fundamental  c h a r a c t e r i s t i c s of a  system. These statements are c o n s i s t e n t w i t h the c o n c l u s i o n s i n chapter  seven.  0  84 BIBLIOGRAPHY 1.  S e i t z , F., The Modern Theory o f S o l i d s , McGraw-Hill, p. 583 (1940)  2.  S c h i f f , L. I . , Quantum Mechanics, McGraw-Hill, 2nd PP. 36-37 (1955).  3.  Buck, R. C . Advanced C a l c u l u s , McGraw-Hill, p . 293 (1956) .  4.  S c h i f f , L. I . , op. c i t . . pp. 80-85.  5.  Mott, N. F. and Massey, H. S., The Theory o f Atfrmic C o l l i s i o n s . Oxford, 2nd ed., p. 47  ed.,  (1949).  6.  I b i d . , p. 4 8 .  7. 8.  J e f f r e y s , H. and J e f f r e y s , B. S., Methods o f Mathematical P h v s i c s , Cambridge, p. 574 (1950). Watson, G. N., A T r e a t i s e on the Theory o f B e s s e l F u n c t i o n s . The M a c M i l l a n Company, 2nd ed., p. 78 (1945) .  9.  Ibid,., p.  199.  10.  Ibid.., P .  202.  11.  Landau, L . M. and L i f s h i t z , E . M., Quantum Mechanics N o n - R e l a t i v i s t i c Theory, Addison-Wesley, pp. 66-67 (1958).  12.  S c h i f f , L. I . , pp. c i t . , pp. 67-68.  13.  Landau, L. M.  14.  I b i d . , p.  15.  H u l l , T. E . and J u l i u s , R. S., (1956) .  16.  Chandrasekhar,  and L i f s h i t z , E. M.,  op. c i t . , p.  474.  475. Can. J . Phys., M,  S., A s t r o p h y s . J . .9_7_, 263  (1943).  914  

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