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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 Tovie Leon Rome B.A., Uni v e r s i t y of 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 of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1961 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of y£ /<? S The University of B r i t i s h Columbia, Vancouver 8, Canada. Date ( i ) ABSTRACT In studying the diamagnetism of free electrons i n a uniform magnetic f i e l d i t was found that reducing the f i e l d to zero i n the wavefunction did not y i e l d the experimentally i n d i c a t e d free p a r t i c l e plane wave wavefunction. However, sol v i n g the Schroedinger Equation r e s u l t i n g from setting the f i e l d equal to zero i n the o r i g i n a l equation did y i e l d a plane wave wavefunction. This paradox was not found to he p e c u l i a r .to. the case of a charged p a r t i c l e i n a uniform magnetic f i e l d but was found to occur i n a number of other systems. In order to gain an understanding of t h i s unexpected behaviour s the following systems were analyzed: the one-dimensional square well 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 ; a one-dimensional harmonic o s c i l l a t o r ; and a charged, spinless p a r t i c l e i n a uniform magnetic f i e l d . From these studies •if the following were obtained: conditions f o r determining the r e s u l t of reducing 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 of a system may be switched o f f while maintaining the energy of the system constant; the r e l a t i o n s h i p between the r e s u l t ' o f p h y s i c a l l y switching o f f a p o t e n t i a l , the r e s u l t of reducing i t i n the wavefunction, and the s o l u t i o n of the Schroedinger Equation obtained by decreas-in 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 equation; and a general property of any wavefunction with respect to reducing any parameter within t h i s wavefunction. ( i i ) ACKNOWLEDGEMENTS I would l i k e to thank Dr. Robert Barrie, who suggested the problem i n t h i s t h e s is, f o r h i s supervision and many i n t e r e s t i n g discussions. I also wish to g r a t e f u l l y acknowledge the finan-c i a l assistance given me by the National Research Council i n the form of"a bursary. TABLE OF CONTENTS Page ABSTRACT i ACKNOWLEDGEMENTS i i CHAPTER I INTRODUCTION 1 1. General Discussion 1 2 . Boundary Conditions 3 3. S i n g u l a r i t i e s 5 4-. Terminology 7 5. Aim of Thesis 9 I I ONE DIMENSIONAL SQUARE WELL POTENTIAL 10 1. Negative Energy Solutions 10 2. P o s i t i v e Energy Solutions 15 I I I COULOMB POTENTIAL 19 1. Negative Energy Case 19 2 . P o s i t i v e Energy Solutions 24 IV UNIFORM ELECTRIC FIELD 28 1. Description of System 28 2. Wavefunction Behaviour and Block Diagram 28 3. Physical Analysis 36 V THE HARMONIC OSCILLATOR 39 1. Time Independent Treatment 39 2. Time Dependent Treatment 41 CHAPTER Page VI UNIFORM MAGNETIC FIELD 51 1. Quantum Mechanical Treatment 51 2 . C l a s s i c a l Analysis of Switching Off 53 3 . Wavefunction E s s e n t i a l S i n g u l a r i t y f o r Zero F i e l d 58 VII CONCLUSION 61 1. Discussion of Switching Off Processes 61 2. Reduction of the P o t e n t i a l i n Wave Function 64 3 . Decreasing the P o t e n t i a l i n the Wave Equation 70 4 . Discussion of the O r i g i n a l Problem and Paradox of an E l e c t r o n i n a Uniform Magnetic F i e l d 73 5. A Comment on the Uniform E l e c t r i c F i e l d Case 75 6. Summary of Thesis Conclusions 75 APPENDIX: 80 SYSTEMS CONTAINED WITHIN A PHYSICAL CONTAINER ' BIBLIOGRAPHY 84 CHAPTER I INTRODUCTION 1. GENERAL DISCUSSION One of the problems which arises i n studying the magnetic properties of s o l i d s i s that of the o r b i t a l d i a -magnetism of free electrons. This can be t r e a t e d 1 by solving the Schroedinger Equation f o r an electron i n a uniform mag-n e t i c f i e l d . Since the case of a weak f i e l d i s of i n t e r e s t , the r e s u l t of reducing the f i e l d to zero i n the so l u t i o n of t h i s Schroedinger Equation was investigated. The apparent experimental r e s u l t of switching o f f the magnetic f i e l d i s that the electron d i r e c t l y and continuously goes over to a free p a r t i c l e whose eigenfunction i s a plane wave. The mathematical treatment of decreasing the f i e l d to zero i s not so straightforward. I f the f i e l d i s decreased i n the o r i g i n a l wave equation f o r the electron i n the f i e l d the r e s u l t i s an equation whose so l u t i o n i s a plane wave. However, i f the f i e l d i s reduced to zero i n the so l u t i o n of the o r i g i n a l equation a plane wave i s not obtained. Whereas the former case i s consistent with what i s expected the l a t t e r case i s incon-s i s t e n t with what appears to be experimental evidence. So the s i t u a t i o n i s that the r e s u l t i s consistent or inconsistent with 2 the apparent experimental observations depending at which stage of the mathematics the f i e l d i s decreased to zero. This paradox of not obtaining the experimentally indicated free p a r t i c l e plane wave so l u t i o n by reducing the f i e l d , or poten-t i a l , i n the eigenfunction solu t i o n of the o r i g i n a l equation i s not pe c u l i a r to the case of an electron i n a •uniform magnetic f i e l d . I t also occurs i n other systems such as a one-dimehsional harmonic o s c i l l a t o r , a p a r t i c l e i n a square well p o t e n t i a l and a charged, spinless p a r t i c l e i n either a Coulomb or uniform e l e c t r i c f i e l d . The preceding suggests the following questions regarding a p a r t i c l e experiencing an external f i e l d : (a) Under what conditions, i f any, i s the same r e s u l t obtained by (i) reducing the f i e l d to zero i n the so l u t i o n to the o r i g i n a l wave equation and by ( i i ) solving the equation obtained from the o r i g i n a l one by l e t t i n g the f i e l d go to zero? (b) What i s the meaning or s i g n i f i c a n c e of those si t u a t i o n s where the r e s u l t s are 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 equation or i n i t s solution? The s i t u a t i o n discussed above may be i l l u s t r a t e d by the block diagram 3 corner 1 > corner 3 soluti o n i f i e l d to 0 ? | f i e l d to 0 corner 2 > corner U s o l u t i o n where the corners are occupied as follows: corner one by the o r i g i n a l wave equation for the p a r t i c l e i n the f i e l d ; corner two by the free p a r t i c l e equation obtained fron the equation i n corner one by decreasing the pot e n t i a l to zero; corner three by the soluti o n of the equation i n corner one; and corner four by the soluti o n of the equation i n corner two. I f the entry i n corner four may also be obtained by reducing the p o t e n t i a l i n the wavefunction occupying corner three the block diagram i s said to be closed or completed. In terms of thi s block diagram the preceding questions may be simply stated as i n the following. (a) Under what conditions can the block diagram be completed? (b) What i s the sig n i f i c a n c e of those situations i n which the block diagram cannot be completed? 2. BOUNDARY. CONDITIONS To completely and uniquely describe a physical system i n either quantum or c l a s s i c a l mechanics boundary con-d i t i o n s must be introduced i n addition to the d i f f e r e n t i a l equation. Since the Schroedinger Equation alone i s i n s u f f i -c i e n t to f u l l y describe a physical s i t u a t i o n the block diagram, 4 as i t stands, deals with incompletely s p e c i f i e d systems. Unexpected r e s u l t s may therefore occur. I f boundary condi-t i o n s are introduced i n conjunction with the wave equations, corners one and two of the block diagram w i l l give a complete d e s c r i p t i o n of t h e i r respective physical situations and the problem w i l l be formulated i n terms of f u l l y s p e c i f i e d systems. Henceforth i n t h i s thesis the block diagram w i l l be considered only i n terms of f u l l y s p e c i f i e d systems, that i s , where corners one and two are occupied by the boundary conditions corresponding to t h e i r respective systems i n addi-t i o n to the respective wave equation. Since the t r a n s i t i o n from corner one to corner two i s made by decreasing the p o t e n t i a l to zero i n some manner, i t follows that the boundary conditions i n corner two should be obtained by decreasing the p o t e n t i a l i n the boundary conditions of corner one. With the exception of the p o s i t i v e t o t a l energy Coulomb and uniform 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 conditions being the same i n both corners one and two. These exceptions w i l l be treated i n section two of chapter three and i n chapter four. With regard to boundary conditions two categories of systems may be distinguished. These types of systems are those i n which (a) the only p o t e n t i a l or afi e l d the p a r t i c l e experiences i s that due to an external source; or 5 (b) i n addition 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 subject to geometric constraints. The only type of geometric constraint considered i n t h i s thesis i s that of a p a r t i c l e being contained i n a physical container. I n the l a t t e r case, as herein considered, the p a r t i c l e i s always bound whereas i n the former case i t may or may not be bound. The o r i g i n a l system of an electron i n a uniform magnet-i c f i e l d can be made to i l l u s t r a t e either type of system. 1 . Corresponding to (a) the system simply consists of an otherwise unconstrained electron moving i n a uniform magnetic f i e l d which f i l l s a l l space. In t h i s case the external source i s that which produces the magnetic f i e l d . An example of (b) i s an elec t r o n confined within a c r y s t a l and experiencing a constant magnetic f i e l d at a l l points within the c r y s t a l . In the main body of t h i s thesis only the f i r s t type o f system w i l l be analysed. In the appendix the second type of system w i l l be discussed. The only type.of geometric constraint to be treated i n the appendix i s that of a physical container which i s mathematically described by an abrupt, i n f i n i t e wall p o t e n t i a l . 3. SINGULARITIES Since a d i f f e r e n t i a l equation i s characterized by the number and type of 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 of the d i f f e r e n t i a l equations i n corners one and two of the block diagram w i l l be studied. The s i t u a t i o n the block diagram 6 represents i s that of comparing the r e s u l t of applying a given procedure to the sol u t i o n of an i n i t i a l equation with the s o l u t i o n of a derived equation where the derived equation i s obtained by applying the same procedure to the i n i t i a l equation. In e f f e c t the solutions of two d i f f e r e n t i a l equations, an i n i t i a l and a derived one, are being compared. I f the i n i t i a l and derived equations have d i f f e r e n t types of s i n g u l a r i t i e s these equations are from d i f f e r e n t classes and i t may not be reasonable to a p r i o r i expect the i n i t i a l s o l u t i o n to go over to the sol u t i o n of the derived equation by a p p l i c a t i o n of the same procedure. With the exception of the square well case, i n a l l the cases herein considered the derived equation d i f f e r s from the i n i t i a l one with regard 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 following table: S i n g u l a r i t y S i n g u l a r i t y D i f f e r e n t i a l Eauation at origin. at i n f i n i t y type order Free p a r t i c l e none i r r e g u l a r fourth Square well p o t e n t i a l none. i r r e g u l a r fourth Coulomb f i e l d regular i r r e g u l a r fourth Uniform e l e c t r i c f i e l d none i r r e g u l a r f i f t h Harmonic o s c i l l a t o r none i r r e g u l a r s i x t h Uniform magnetic f i e l d none i r r e g u l a r s i x t h 7 Since the wave equation for both the free p a r t i c l e and the square well p o t e n t i a l have the same s i n g u l a r i t y pattern the square well p o t e n t i a l w i l l be treated f i r s t . The Coulomb equation i s treated next since i n addition to i t s regular s i n g u l a r i t y at the o r i g i n i t has the same type of s i n g u l a r i t y at i n f i n i t y as does the free p a r t i c l e equation. The cases of negative and p o s i t i v e t o t a l energy are treated separately f o r both the square well and the Coulomb p o t e n t i a l s . The uniform e l e c t r i c f i e l d , whose equation has a s i n g u l a r i t y at i n f i n i t y one order greater than has the free p a r t i c l e equation, i s treated next. Chapters s i x and seven are devoted to the harmonic o s c i l l a t o r and uniform magnetic f i e l d cases whose equations have s i n g u l a r i t i e s at i n f i n i t y two orders l a r g e r then the free p a r t i c l e equation. 4. TERMINOLOGY Before completing t h i s introduction the terminology associated with the po 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 derivations) ref e r s only to the p o t e n t i a l going to zero i n the wavefunction. That i s , "reduction" i s associated with the step from corner three to corner four i n the block diagram. This term r e f e r s to a purely mathematical procedure with no dependence on, or r e l a -t i o n to, any parameter or variable; for example, no connection with time. An example of reduction i s l i m f ( x ) ; x i s said to be "reduced" to b. 8 The step from corner one to corner two, that i s , the p o t e n t i a l going to zero i n the wave equation, w i l l not at present have any d e f i n i t e term ascribed to i t . Non-committal terms such as "the p o t e n t i a l i s decreased" or the "potential goes to zero" w i l l be used. The expression "switch o f f " and i t s derivatives r e f e r s only to the physical process of the p o t e n t i a l being diminished to zero. The physical process of "switching o f f " a p o t e n t i a l i s a time dependent process i n which the p o t e n t i a l i s a function of the time. For example, the switch o f f may be exponential with a time constant, a step function with respect to time or l i n e a r over a time i n t e r v a l . To incorporate the time dependence of the switch o f f i n the wave equation requires a tiiae dependent Hamiltonian. However the point of i n t e r e s t i n t h i s t h e s i s i s to describe the r e s u l t of switching o f f rather than to describe the behaviour of the system while the p o t e n t i a l i s being switched o f f . Hence the precise time dependence of the switch o f f i s not of i n t e r e s t and a l l the Hamiltonians w i l l be independent of time regardless of whether the time dependent or independent wave equation i s used. In chapter f i v e a d i s t i n c t i o n w i l l be drawn between two d i f f e r e n t types of switch o f f . 5. AIM OF THESIS In this thesis the various aforementioned systems are analyzed with the following intentions: (a) to obtain general c r i t e r i a for determining the wave-function obtained by reducing the potential 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 potential i n the wavefunction, decreasing the potential i n the wave equation and the method of switching off the potential. 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 In 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 at i n f i n i t y , corresponds to the physical s i t u a t i o n of 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 of i t s negative p o t e n t i a l energy i n the region where the po t e n t i a l i s non-zero. I n the region 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 sys-tem are: (2.1) E - V >/0 (2.2) 0*(E| ^ 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 conditions follow from the conserva-t i o n of energy and the fact that the k i n e t i c energy i s p o s i t i v e while 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, implies that the p a r t i c l e i s confined within the region of the p o t e n t i a l . I n 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 of the p a r t i c l e leaving the region 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 region of the 11 p o t e n t i a l i s small and decreases exponentially as the distance from the region of p o t e n t i a l increases. The square well p o t e n t i a l , V, may be described as follows: - Vo -<K < x <: a where V 0 i s p o s i t i v e . The wave equation i s then v-f-( 2 . 3 ) £ ? ^ ( E * V - J * - ° • a > M > * (2.4) & * ^ ' -° ' " ' " I To maintain continuity at x=±a the boundary conditions and <p'(-«.) = are imposed. The so l u t i o n of (2.3) i s VPBA^«noix + B sin ccx where c{ - +1 2m j£+v0) ' and i s r e a l by (2.1) . To avoid an unbounded so l u t i o n and to permit normalization the boundary conditions that <^  tends to zero as |*/ tends to i n f i n i t y are imposed. The solut i o n of (2 .4) i s then <f> = C e'p* xxx. U e x < - a and i s r e a l and p o s i t i v e . By matching the solutions at x=»±a. 2 two conditions and t h e i r corresponding solutions are obtained. These continuity conditions on the wavefunction determine the allowed values 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 (2.5) «. t a n oca. • 0 B cos «<x - a< x<a. By matching at either boundary the r e l a t i o n between the c o e f f i c i e n t s i s O B e ^ cos .ca. obtained from the normalization condition f 1^1 dx*/. + oo (b) The s o l u t i o n associated with the condition ci cot oca = - p i s (2,5') [ C e" **X x>a. -a <X<a x< - a f _ _ * A A / f / 3 * — v> ~ ^ cos oca and n " v * coslot.a + a^'oc"-^j1 sin<*a. cos a: a are obtained as before by matching at a boundary and from normalization. As V 0 i s reduced to zero, ft goes to i«. In order that the wavefunction be well behaved the energy, p and oc must be determined from the appropriate continuity condition. The f i r s t condition i s ct tan oca- (3 . An obvious s o l u t i o n when a becomes -ip i s ot=(3= E = o . The other permissible values of E are determined by substituting -i(3 13 f o r o( i n the above c o n d i t i o n and s o l v i n g the o b t a i n e d tan ( - i p a) a I . T M s i s e q u i v a l e n t to tanh|5a»-A The s o l u t i o n o f t i l l s f o r f i n i t e , p o s i t i v e a i s |S = - c o , S i n c e |3 i s d e f i n e d as being p o s i t i v e t h i s i s an un a c c e p t a b l e s o l u t i o n . Hence the o n l y a c c e p t a b l e v a l u e f o r E i s zero. The c o n d i t i o n acot«.aa-^ y i e l d s the i d e n t i c a l r e s u l t when V0 i s reduced to z e r o . Hence as V0 i s reduced to z e r o c< and ^ go to zero. Thus A, B, and C go to zero as t h e p o t e n t i a l does. Hence f o r e i t h e r c o n d i t i o n the wave-f u n c t i o n goes to zero as the p o t e n t i a l does. I t should a l s o be noted t h a t the c o n t i n u i t y c o n d i t i o n s i m p l y i n g t h a t E being z e r o i s i t s o n l y a c c e p t a b l e v a l u e i s i n ac c o r d w i t h the p o t e n t i a l going to zero c o n s i s t e n t w i t h c r i t e r i a ( 2 . 2 ) . 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 s t u d i e d i n o r d e r to determine what happens when the p o t e n t i a l goes to zero i n the wave eq u a t i o n and the ensuing wave e q u a t i o n i s s o l v e d . I f the p o t e n t i a l i s decreased such t h a t 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 c r i t e r i a (2.2) o f the system are s a t i s f i e d i t i s apparent t h a t the f i n a l 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 t o zero i s a ( f r e e ) 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 poten-t i a l energy, t h a t i s , the t o t a l energy, E, i s zero. Hence as the p o t e n t i a l goes to zero a l l the energy l e v e l s c o l l a p s e to z e r o . S i n c e E goes to zero as V does u (2.6) d l ^ = 0 i s the equation describing the r e s u l t of decreasing p o t e n t i a l i n the above manner. That i s , i t i s (2.6) [rather than the apparent + z * E T = o , ( E * o) ] dx* K l ' v ' which describes the system when the p o t e n t i a l i s decreased 3 as above. By a well known theorem regarding the so l u t i o n of Laplace's Equation with boundary conditions, the soluti o n of (2.6) with the boundary conditions that y goes to zero as I x l 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 i s decreased without e x p l i c i t l y requiring that the c r i t e r i a of-the system be s a t i s f i e d f or a l l intermediate values of the p o t e n t i a l the r e s u l t i n g equation i s (2.7) 4i£ + i-EY=o. dx1 T h i s i s the equation obtained i f the pot e n t i a l i s mathemati-c a l l y set equal to zero i n ( 2 . 3 ) . The general solu t i o n of (2 . 7 ) i s f B Ae" + Be*<rx where IT i s i n general complex with the r e a l part non-negative. The condition that 4^  =»o at x = *• 00 implies A i s zero and M-^o at x=- 00 implies B i s zero. Hence the only solu t i o n consistent with the boundary conditions i s Y=o regardless whether the c r i t e r i a ( 2 . 2 ) are e x p l i c i t l y introduced or not. The reason f o r obtaining V-O both times i s that the boundary conditions 15 are the same. That i s , even though i n the l a t t e r treatment (2.2) was not e x p l i c i t l y applied to E i t was i m p l i c i t l y applied since the boundary conditions f o r a bound system were maintained. The block diagram can now be considered. The block diagram f o r the negative energy solutions f o r the square w e l l p o t e n t i a l i s occupied as follows: corner one by equa-t i o n s (2.3) and (2.4) and the boundary conditions that the wavefunction be zero at %- ± 00 ; corner two by the same boundary conditions and by equation (2.6) or (2.7) depending on what i s st i p u l a t e d regarding decreasing the p o t e n t i a l ; corner three by wavefunction (2.5) or (2.5 f ) ; and corner four by Y=0 . As has been demonstrated the r e s u l t of reducing the p o t e n t i a l i n the wavefunction (2.5) or (2.5^) i s V » 0 . Hence the r e s u l t of reducing the p o t e n t i a l i n corner three i s the same as solving corner two and the block diagram i s closed. 2. POSITIVE ENERGY SOLUTIONS The physical s i t u a t i o n which the mathematics of t h i s section describes i s that of a p a r t i c l e f e e l i n g the e f f e c t of a square well p o t e n t i a l whose value i s such that the 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 po t e n t i a l i s assumed negative the k i n e t i c energy i s then greater than the absolute value of the po t e n t i a l energy. This i s the case of sca t t e r i n g by a square well p o t e n t i a l . 16 The procedure for treating this case i s similar to the negative energy case. However the result of reducing the potential i s different. The potential 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 that <P behaves as a sinusoidally oscillating function at X=±co. Hence f i s C sin + D cos (3X where f>- ^ j 2 " ^ ' • For 1*1 < a-the equation i s again (2.3) with the solution V= A SinotX + BeOS <*X where OL = + (2m(£ + Vo) r The functions <p and V and their f i r s t derivatives are again matched at X = ± a to produce two sets of solutions each corresponding to a different relation between c£ and f$ (a) Associated with the condition <L tan oi CL = |3 tan ficc i s the solution 5 C O S OCX /x/<a D cos 0x /x/ > a To maintain continuity of the wavefunction at |xt=-a the relation between B and D i s B COS cLQ. = 0 COS ($& . As V0 i s reduced to zero, oC approaches and, i n order that the wavefunction be well-behaved, D approaches B. Hence Bc0S|3X i s the solution everywhere when the potential i s zero. Since this solution i s applicable i n a l l space the /+*> z lV\ dx = I i s not applicable -00 and B i s arbitrary. B i s usually chosen to be unity as this normalizes the function to unit flux. (b) Corresponding to the condition ai cotaca. = (3 co t pa. i s the sol u t i o n (2.8') Asinocx |x|<a C si n /3x / x / > a . By the same argument as i n the preceding the sol u t i o n s i n p% i s found to apply at a l l points when the po 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 to zero ct goes to /S and the continuity conditions at x =* a which determine the allowed values of E become i d e n t i t i e s . Hence the end r e s u l t i s a free p a r t i c l e with 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 of the pot e n t i a l going to zero i n the wave equation w i l l now be investigated. Unlike the negative energy case, E i s not bounded by the p o t e n t i a l . Hence decreasing the p o t e n t i a l to zero does not influence E and the r e s u l t i n g wave equation i s ( 2 . 7 ) . When the boundary conditions that Y o s c i l l a t e s s i n u s o i d a l l y at x = - co are imposed the sol u t i o n normalized to u n i t f l u x i s ( 2 . 9 ) COS / 2 m E ' x ( 2 . 9 ' ) s i n jzm£' X or a l i n e a r combination of the two. The physical s i g n i f i c a n c e of decreasing, the poten-t i a l i s that the p a r t i c l e i n question i s acted upon by a diminishing force. When the po 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 the p a r t i c l e and i t i s then 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 the above r e s u l t s . The block diagram f o r the p o s i t i v e energy square v e i l case i s occupied as f o l l o w s : corner one by equations (2.3) and (2.4) and the boundary c o n d i t i o n s t h a t <p s i n u s o i d a l l y o s c i l l a t e s at X=ico ; corner two by equation (2.7) w i t h the above boundary c o n d i t i o n s on 4^  j corner three by wave-f u n c t i o n (2.8) or (2.8'); and corner f o u r by wavefunction (2.9) o r (2.9'). As has been demonstrated (SIS) reduces to (2.9), or (2.8') to (2.9')* when the p o t e n t i a l i s reduced i n the wave-f u n c t i o n (2.8) or (2.8') . Hence the block diagram i s c l o s e d . CHAPTER III COULOMB POTENTIAL 1. NEGATIVE ENERGY CASE A p a r t i c l e with negative t o t a l energy i n a Coulomb f i e l d corresponds to an a t t r a c t i v e p o t e n t i a l binding the p a r t i c l e to the source of the p o t e n t i a l as i n the example of the hydrogen atom. In t h i s Coulomb case the p o t e n t i a l i s - A where A i s a non-negative constant and r i s the distance from the source of the p o t e n t i a l to the p a r t i c l e . Corresponding to the 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 case i s bounded as follows: (3.1) -mA1 < E < 0. I n f a c t the exact allowed energy values f o r the negative energy case are (3.2) £n = -mA1 on, z,3,... The allowed energies being negative corresponds to the p a r t i c l e being bound within a f i n i t e region of apace. Hence the wavefunction s a t i s f i e s the normalization condition This normalization condition i n a l l s p a c e 20 t u r n implies that Y goes to zero as any s p a t i a l coordinate approaches i n f i n i t y . The wave equation f o r t h i s system i s (3.3) Vlf - l a ( E*A ) 4 / = 0 and i s expressed i n spherical coordinates. By introducing the quantum numbers I and m the equation i s separated i n the usual manner into angular and r a d i a l equations. The s o l u t i o n f o r a given I and m i s the product Rt (r) sjltn where ^m(^f) i s a spherical harmonic. The r a d i a l equation f o r the Goulomb f i e l d i s i _ <J M J f U Lsss\( E+A -hhMAn))= o. The independent Variable r i s replaced by PsImlEl I n terms of p the so l u t i o n i s ^ Kt (fO = C n i e'Pk Lj '* ' ( p) where L i s an associated Laguerre polynomial and C f t t i s a normalizing c o e f f i c i e n t to be determined from £ * R*< (*') f 1 <lr = / and the r e l a t i o n The normalized t o t a l wavefunction i s 21 where p = Af* A r i s used as i t i s equivalent 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. I f the po t e n t i a l i s reduced to 'zero through A going to zero then the wavefunction (3.4) also goes to zero. Furthermore by (3.2) a l l the energy eigenvalues collapse to zero. As the energy goes to zero the l i n e a r and angular momentum, and therefore tt go to zero. Hence the wave-fu n c t i o n goes to zero a: i s r\ The treatment of decreasing the po t e n t i a l i n the wave equation i s sim i l a r to that used i n the negative energy square well case. I f the po t e n t i a l i s decreased such that the c r i t e r i a (3.1) and (3.2) are 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 equation describing the r e s u l t of decreasing the p o t e n t i a l to zero i n t h i s manner i s (3.5) V l t - 0 i n analogy to (2.6). The solut i o n i s f = 0 by the same theorem^ since the boundary condition i s goes to zero as r goes to i n f i n i t y . I f the po t e n t i a l i s decreased without e x p l i c i t l y s t i p u l a t i n g that E goes to zero the wave equation becomes ( 3 . 6 ) V l f + £mE V * o i n analogy to (2.7). Since the same boundary condition i s maintained the solution i s s O by an argument e s s e n t i a l l y tlie same as the one following equation (2.7) i n chapter two. Again » -0 i s the sol u t i o n when the p o t e n t i a l i s decreased to zero i n the wave equation regardless of the e x p l i c i t conditions on E. This i s because the boundary condition ( t h a t y goes to zero as r goes to i n f i n i t y such that energy i s negative and (3 .1) i s s a t i s f i e d . The block diagram for the negative energy case of the Coulomb p o t e n t i a l i s populated as follows: corner one by equation (3.3) and the boundary condition that f goes to zero as;] Y goes to i n f i n i t y ; corner two by either equation (3.5) or (3.6)(depending on the conditions e x p l i c i t l y imposed on decreasing the potential) and the above boundary condition; corner three by wavefunction (3.4-) j a^ d- corner four by 4^  ~ 0. Since the r e s u l t of reducing the p o t e n t i a l i n (3.4) i s ^=0 the block diagram i s closed. manner such that the c r i t e r i a of the system are s a t i s f i e d f o r intermediate values o f the p o t e n t i a l w i l l now be considered. The c l a s s i c a l c r i t e r i a f o r the negative t o t a l energy Qoulomb case are the force equation \y\ dp =•1 ) implies the p a r t i c l e i s bound, the The c l a s s i c a l switching o f f of t h i s p o t e n t i a l i n a (3.7) and the i n e q u a l i t y 23 The l a t t e r follows from the conservation of energy for a bound system with zero p o t e n t i a l at i n f i n i t y . By an analysis based on (3.7) i t w i l l now be shown that the v e l o c i t y goes to zero 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 switched o f f , A goes to zero and hence v 2 f goes to zero. Since A being decreased implies the force, -A , Is decreased i n magnitude, the r e s u l t i n view of the f i n i t e tangential v e l o c i t y , v, i s that r w i l l tend to increase. Hence f o r v l r to go to zero v must go to zero with the r e s u l t that 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 zero. Hence the r e s u l t of switching o f f i n the above manner i s a p a r t i c l e without k i n e t i c or p o t e n t i a l energy, that i s with zero t o t a l energy. Before -concluding the c l a s s i c a l switching o f f of 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 be use f u l further on i n t h i s t h esis, w i l l be presented. Of the various ways of d i s c r e t e l y switching o f f the p o t e n t i a l consistent with the c r i t e r i a (3.7) and (3.3) the l e a s t favorable one i s i f a given decrease occurs instantaneously. I t i s assumed that the p o t e n t i a l parameter instantaneously goes from A to Aj. where A i < A. At the instant of decrease (3.7) i s s a t i s f i e d by A and (3.8) must be s a t i s f i e d by both A and A i . Combining (3.7) and (3.8) at t h i s instant y i e l d s ,(3.9) i *w"V - A < A, . 24 T h i s i n e q u a l i t y i m p l i e s t h a t the p o t e n t i a l cannot he s w i t c h e d o f f to:• zero i n a f i n i t e number o f steps i f the c r i t e r i a o f the system are to be 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 . S i n c e a l l t h a t was r e q u i r e d o f Aj_ was t h a t i t be l e s s than A and i n view o f the f a c t t h a t a continuous s w i t c h i n g o f f may be approximated w i t h a r b i t r a r y a c c u r a c y by a d i s c r e t e method o f s w i t c h i n g o f f , one may t h e r e f o r e conclude t h a t i n a continuous s w i t c h i n g o f f , s a t i s f y i n g the c r i t e r i a 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 p o t e n t i a l must take an i n f i n i t e time to r e a c h z e r o . The f i n a l t o t a l energy r e s u l t i n g from such a s w i t c h o f f i s a g a i n z e r o . 2. POSITIVE ENERGY SOLUTIONS The p h y s i c a l s i t u a t i o n which t h i s case d e s c r i b e s i s t h a t o f the s c a t t e r i n g o f a s p i n l e s s , charged p a r t i c l e by a Coulomb f o r c e . I n t h i s case the t o t a l energy i s p o s i t i v e and constant f o r a l l p o i n t s o f the p a r t i c l e ' s p a t h . I t i s convenient to c o n s i d e r the i n c i d e n t p a r t i c l e as being d e s c r i b e d by a p l a n e wave i n the 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 . Hence a s o l u t i o n o f the form y= e ik* f I s sought f o r the e q u a t i o n ( 3 . 1 0 ) ^ f + ^ ^ v O - c where A*lm2el k l » imE. , ft2,A . i s the p o t e n t i a l and e i s 25 the charge on the incident particle. The equation satisfied by F i s V z F + 2«k 2 £ - AL - o. a* r At this stage the transformation to parabolic coordinates i s made. Due to the axial symmetry of the system and the separating out of the incident plane wave, the solution w i l l depend on j* = r- z only. Hence F (V-H ) i s substituted for F. After multiplying through by r the resulting equation i n terms of j becomes By introducing X a i k j f a confluent hypergeometric equation i n x i s obtained. Hence F = ,F, > 'i '^J*^. When normalized to unit flux the total wavefunction i s ( 3.11) T M = e& r(i.^ L) e'k i.R (-A ,. ,»ior). I f the potential Is reduced to zero i n the above wavefunction, that i s A reduced to zero, a l l the terms involving A go to unity and the wavefunction becomes the plane wave Q Since the Coulomb potential i s a long range one wi th the same value at z = + <=© and z ° - «x> the boundary condition at i n f i n i t y behaves i n a different manner for this potential than for the square well potential. In this case the parameter A exp l i c i t l y appears i n the asymptotic expression for the wavefunction. This and the uniform 26 e l e c t r i c f i e l d a r e the o n l y cases i n t h i s t h e s i s i n w h i c h the 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 the boundary c o n d i t i o n . The boundary c o n d i t i o n accompanying t h e e q u a t i o n i n w h i c h the p o t e n t i a l i s z e r o w i l l d i f f e r from t h e c o n d i t i o n w i t h the i n i t i a l e q u a t i o n t o t h e e x t e n t t h a t A i s s e t e q u a l t o z e r o i n t h e i n i t i a l boundary e x p r e s s i o n . The boundary c o n d i t i o n a t i n f i n i t y f o r the p 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)) + .At»_ c s c l 0 expAkr-jA lo«kK-|A lo^O-cose) •*-2"ii»e') . 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 boundary c o n d i t i o n becomes (3.13) t ~ e ' ' k * The r e s u l 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 t o zer o i n t h e wave e q u a t i o n can now be s t u d i e d . As i n t h e p o s i t i v e e n e r g y case o f the square w e l l p o t e n t i a l , d e c r e a s i n g the p o t e n t i a l does n o t p l a c e any r e s t r i c t i o n s on E. 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 t o z e r o i n (3.10) i s ( 3 . 1 4 ) V 2 f + k z f - 0 . 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 t o u n i t f l u x i s 6 ~ . By i m p o s i n g t h e boundary c o n d i t i o n (3.13) f o r r g o i n g t o i n f i n i t y t h e g e n e r a l s o l u t i o n becomes e 1 k a . The same r e s u l t may be o b t a i n e d by r e c a l l i n g t h a t the incident -wave vector was k = (o, o, Jc). In the absence of any p o t e n t i a l , as i s the case i n (3.I4), k remains unaltered. Hence the general s o l u t i o n e ' ^ " r again • k * becomes e . I f i n a given problem the incident wave vector i s not p a r a l l e l to an axis a r o t a t i o n of the coordinate system i s f i r s t c a r r i e d out such that the wave vector i s p a r a l l e l to an axis i n the new coordinate system. The problem i s then treated as above i n the new coordinate system. The block diagram f o r the p o s i t i v e energy Coulomb case i s occupied as follows: corner one by equation (3.10) and boundary condition (3.12); corner two by equation (3.14) boundary condition (3.13); corner three by wave-f u n c t i o n (3.11); and corner four by 6 1 " . As has been demonstrated, the r e s u l t of reducing the p o t e n t i a l i n the t h i r d corner i s the entry i n the fourth corner. Hence the block diagram i s closed. The physical s i t u a t i o n i s straight-forward. The system consists of a p a r t i c l e with t o t a l energy, E, experi-encing a Coulomb force. The r e s u l t of switching o f f t h i s f o r c e to zero i s a free p a r t i c l e with the same t o t a l energy. T h i s t o t a l energy i s now a l l k i n e t i c energy. CHAPTER IV UNIFORM ELECTRIC FIELD 1. DESCRIPTION OF SYSTEM The system under consideration i n t h i s chapter i s that of a charged, spinless p a r t i c l e , i n c i d e n t from ^ = -*-oo t r a v e l l i n g towards 1 ~ ~~ 00> being repell e d by a uniform e l e c t r i c f i e l d . This f i e l d , F , i s chosen to be p a r a l l e l to the "^--axis and the charge on the p a r t i c l e i s denoted by e. I n t h i s chapter e i s assumed to be p o s i t i v e . However the arguments and r e s u l t s are. equally applicable to a negatively charged p a r t i c l e when the di r e c t i o n s are reversed. In t h i s system the p a r t i c l e experiences a poten-t i a l -©F^+C. Since C i s a r b i t r a r y i t i s chosen to be zero thus making the zero of po t e n t i a l at 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 associated with the motion p a r a l l e l to the ^ -axis and i s a p o s i t i v e or negative constant. 2. WAVEFUNCTION BEHAVIOUR MD BLOCK DIAGRAM The Schroedinger Equation f o r t h i s system i s 29 Y i s expressed as [ x, y> y) - X(x) Y(y) (}) and three ordinary d i f f e r e n t i a l equations are obtained -which involve the constants k X ; k ^ and kj where k x * + k u 1 Ut s ZmC . k£ and kJ*" are im ' y *x ft1 times the energy associated with the motion i n the x and y d i r e c t i o n s respectively and as such are p o s i t i v e . i s 1 m E and i s p o s i t i v e or negative as E i s . The equations f o r X and Y y i e l d the free p a r t i c l e plane wave soluti o n s X(JO = e ' k * X and Y(y) = e 1 ^ * The equation f o r Z(}) i s ( 4 . 2 ) £ Z + f i f a i ^ . U j ) Z a o . d j 1 V *«• 7 I f the changes i n var i a b l e s ~L. ~ J ^ssFy + k£ W f \T = * ( Intfx + k t . ) V : t and then U = ^ V _ are 31 ft* yl XmeF made i n ( 4 « 2 ) the r e s u l t i n g equation f o r W i s l ^ & J L + U ^ + ( U l _ l ) W = 0 . T h i s i s a Bessel Equation and i t s general s o l u t i o n i s W = A X/3(u) + B J-y, (u). The general expression f o r Z.(}) i s therefore Z where A and B w i l l be 30 chosen to s a t i s f y the boundary conditions. As was seen i n the p o s i t i v e energy Coulomb case, the asymptotic behaviour of the wavefunction f o r a long range p o t e n t i a l i s not simply a trigonometric or an imaginary exponential type of function. Since t h i s uniform e l e c t r i c f i e l d p o t e n t i a l i s a long range one, a l l that w i l l be speci-f i e d regarding the boundary conditions i s that the wavefunc-t i o n goes to zero exponentially as J approaches — °o and that i t o s c i l l a t e s as ^ approaches <*> . The boundary conditions to be s a t i s f i e d by q u a n t i t a t i v e l y stated as follows: tends to zero exponentially with decreasing > U.4) implies o s c i l l a t e s with J . — oO To s a t i s f y the boundary condition f o r ^ tending to A i s equal to -Be'""'3 , This r e l a t i o n i s obtained by applying the following procedure: f o r ^ tending to -«o i s used and T-meFjjIii 1 i s 7 recognized as being p o s i t i v e ; the r e l a t i o n s = e ' are used; 31 and A i s expressed i n terms of B such that, f o r y tending to - oo , ^(j) i s proportional to the Kp function where 8 K„(x) =TT(L„(X) - I V U ) ) . B i s deter-9 mined by using the asymptotic forms ( 4 . 5 ) X^W -~ co*(x + liF-E) and the normalization condition v | 4 *| l r J — f o r Z tending to + oo where v i s the magnitude of the v e l o c i t y p a r a l l e l to the 3--axis. B i s then found to be __L j3eF With A and B thus s p e c i f i e d the asymptotic forms of fo r 1^1 tending to i n f i n i t y can be written down. By use of Ky(x)^/liL e~X i t i s seen that f o r ^ tending to - ao ( 4.6) Z V - f i T / a n — l ^ e ^ ^ x p f - f e s ^ ^ l . ( 4 . 6 ) goes to zero exponentially as ^ goes to - <*» . ' By use of ( 4 . 5 ) the asymptotic form of as tends to + 00 i s seen to be ( -^Hlte^ -?)]} • ( 4 . 7 ) o s c i l l a t e s with varying % . Hence with the above choices for A and B the boundary conditions are s a t i s f i e d . 32 The solution of ( 4 . 2 ) which i s normalized to u n i t f l u x at ^ - 00 and which s a t i s f i e s the boundary conditions ( 4 . 3 ) and ( 4 . 4 ) i s therefore (4 . 8 ) Z ( 0 - ± . / Xm ( e F K E ) ' ( I . \(l~ef} + l~>t)ih ] 0 H J 3 T ? I 3 L 3 m e F t i J - e i T r / 3 Jj. I" d ^ f i > 2 ^ £ ) The r e s u l t of reducing the p o t e n t i a l i n the wave-f u n c t i o n ( 4 . 8 ) w i l l be studied i n conjunction with the e f f e c t o f decreasing the p o t e n t i a l to zero i n the boundary condi-t i o n s . This i s done i n order to determine whether the r e s u l t o f reducing the p o t e n t i a l i n the wavefunction s a t i s f i e s the conditions on the wavefunction obtained by decreasing the p o t e n t i a l i n the boundary conditions. The cases of p o s i t i v e and negative E must be distinguished. In reducing the p o t e n t i a l In the wavefunction (4*8) f o r the case of p o s i t i v e E , equations (4«5) and the small F approximations j7"™*^ * ^ \ ^ rc ~ and J L - ( s k, y + are employed. As F i s reduced to zero the wavefunction (4*8) becomes With E p o s i t i v e F w i l l be decreased to zero i n the boundary-conditions (4-3) and (4-4-) • As F i s decreased to zero (4-3) becomes the condition that f o r j«~oo goes to zero exponentially with decreasing ^* (4-.4-) becomes the condition that f o r $ >>-oo o s c i l l a t e s w ith J . As the former i s meaningless the l a t t e r l s the only condition imposed on the wavefunction by the boundary conditions when F i s decreased to zero. Since (4-.9) o s c i l l a t e s f o r a l l j i t i s consistent with the above con-d i t i o n on the wavefunction. Hence reducing the po t e n t i a l i n (4-.8) when E i s p o s i t i v e y i e l d s an acceptable and consistent r e s u l t . When the preceding approximations f o r small F and 7 8 10 T i/ the previous formulas ' ' f o r 1 + i and Ki are used, the wavefunction (4-8) f o r negative E becomes U .io) Z( ,) * e ' " A H2 / . « . ) * e x P [ I^I j - y v 1 as F i s reduced. F w i l l now be decreased to zero i n the boundary conditions with E negative. (4-3) becomes the condition that f o r < j « oo i s zero and (4-.4-) becomes the condition that f o r ^ >>oo Z($) i s o s c i l l a t o r y with respect to |,, Since the l a t t e r i s meaningless the former i s the only condition imposed on the wavefunction by the boundary conditions when F i s zero and 34 E i s negative. The r e s u l t of reducing F to zero i n (4.10) i s a zero wavefunction f o r a l l ^ l e s s than i n f i n i t y i n accord with the above condition. Hence reducing the poten-t i a l to zero f o r E either p o s i t i v e or negative produces a s a t i s f a c t o r y r e s u l t i n that the condition on the wavefunction i s s a t i s f i e d i n both cases. For both p o s i t i v e and negative E the r e s u l t of decreasing the p o t e n t i a l i n the wave equation (4.2) i s (4.11) + ^ TLE Z = o • I f E i s p o s i t i v e the condition on the wavefunction that i t be o s c i l l a t o r y with 3 . f o r ^ > > - « 3 implies that the s o l u t i o n of (4«ll) corresponding to a p a r t i c l e incident from ^ = + 00 i s (4.12) e- ' , k >* . I f E i s negative and F i s zero the condition on Z i s that i t be zero f o r <^<<<co m In t h i s case the solut i o n of-(4.11) i s Z-O. Since the p o t e n t i a l i s a function of only, only functions of, or concerned with, -^ need be considered i n the block diagram. As i t has been explained i n chapter one, the poten-t i a l i n the boundary conditions must be decreased to zero i n going from corner one to corner two. Since the boundary 35 condition i n corner two i s therefore 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 negative E , the two cases of p o s i t i v e and negative E must he distinguished. Hence a separate block diagram w i l l be used f o r each of the two energy cases. In the case::of p o s i t i v e E the block diagram i s populated as follows: corner one by equation (4.2) and boundary conditions (4«3) and (4«4)J corner two by equation (4.H) and the condition that the wavefunction o s c i l l a t e s w ith ^ f ° r ^ >> -<x> ; corner three by the wavefunction ( 4 . 8 ) ; and corner four by wavefunction (4.12) . The entry i n corner four and the equation (4*9) with F reduced to zero, which i s the r e s u l t of reducing the p o t e n t i a l i n the wave-f u n c t i o n occupying corner three, d i f f e r by a phase factor but describe the same physical s i t u a t i o n . Hence to the extent that the entry i n corner four and the. r e s u l t of reduc-i n g the p o t e n t i a l i n the wavefunction i n corner three 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 block diagram i s closed. In the case of negative E the block diagram i s occupied as follows: corners one and three as i n the p o s i t i v e E case; corner two by equation ( 4»H) and the condition that f o r ^<<oo the wavefunction i s zero; and corner four by ~L.~0. The r e s u l t of reducing the p o t e n t i a l to zero i n corner three, that i s , (4.10) with F = 0 , and the entry i n corner four are the same. The block diagram i s therefore completed. 3. PHYSICAL ANALYSIS In the equation (4.9) r e s u l t i n g from reducing the f i e l d to zero i n the p o s i t i v e E case, the plane wave momentum-distance expression, that i s , k^j , i s found i n the argument of the exponential thus i n d i c a t i n g the desired plane wave r e s u l t . However the term t;r introduces an i n f i n i t e phase fac t o r as F i s reduced to zero. Since only \y\ X corresponds to a physical observable and phase factors are not p h y s i c a l l y observable, the presence or absence of such an i n f i n i t e phase fac t o r would not be detectable. Hence no physical explanation of t h i s i n f i n i t e phase fac t o r i s p o s s i b l e . Furthermore, since at ^ =-•-co an i n f i n i t e k i n e t i c energy i s required to maintain the t o t a l energy constant t h i s system does not p r e c i s e l y correspond to an actual physical s i t u a t i o n . Hence an unusual and p h y s i c a l l y i n e x p l i c a b l e item such as an i n f i n i t e phase factor should not be surprising or d i s t u r b i n g . I t i s however s a t i s f y i n g that a l l the p h y s i c a l l y observable features are well behaved. The c l a s s i c a l motion of the p a r t i c l e i n t h i s f i e l d w i l l now be analyzed. The c l a s s i c a l process which t h i s case represents i s that of a charged, spinless p a r t i c l e i n cident from +00 being r e f l e c t e d back at some point J 0 . This i s supported i n the preceding mathematics by the f a c t that as ^ approaches oo the wavefunction becomes (4.7) which describes a free p a r t i c l e t r a v e l l i n g i n the - j . d i r e c t i o n . The f a c t that the p a r t i c l e i s r e f l e c t e d back i s supported by (4.6) which, f o r a l l f i n i t e E , indicates zero proba-b i l i t y f o r the p a r t i c l e being at ^ f o r ^ tending to -00. The two cases of p o s i t i v e and negative E have been dist i n g u i s h e d and have given d i f f e r e n t r e s u l t s . The physi-c a l s i g n i f i c a n c e of the value of the t o t a l energy, E, i s to i n d i c a t e at what point i n space, once the zero of 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 r e f l e c t e d back by the p o t e n t i a l b a r r i e r . This point of course corre-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. With the zero of p o t e n t i a l at 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 to r e f l e c t i o n at a p o s i t i o n with negative ^ coordinate and negative E corresponds to r e f l e c t i o n back at a p o s i t i o n with p o s i t i v e sfr coordinate. The exact coordinate at which the p a r t i c l e i s r e f l e c t e d back i s given by ^ 0 = -JL. . | e i s the value of ^ f o r which the argument of the Bessel Functions T±L i n ( 4 . 8 ) i s zero. ^ > J© implies t h i s argument i s r e a l ; ^ <C | c implies t h i s argument i s imaginary. j < J0 corresponds to the region of space which, i n c l a s s i c a l mechanics, the p a r t i c l e may never enter. The e f f e c t of the p o t e n t i a l being switched o f f on the point of r e f l e c t i o n w i l l now be studied. For p o s i t i v e 38 E, the point of r e f l e c t i o n , j t f , goes to - oo as F i s switched o f f . 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 ^= c o without being r e f l e c t e d at an intermediate p o s i t i o n . Hence f o r F = 0 and E p o s i t i v e the wavefunction f o r | > - oo , that i s , at a l l points, should be a plane wave describing a free p a r t i c l e t r a v e l l i n g from c + oo to | = - oo. As can be seen from (4«9) reducing the p o t e n t i a l i n the i n i t i a l wavefunction with p o s i t i v e E gives t h i s r e s u l t . I f E i s negative -fro goes to -i-oo as F i s switched o f f and the point of r e f l e c t i o n i s at ^ e + * . I f a p a r t i c l e entering from ^ = + « o i s r e f l e c t e d back at | = + oo the r e s u l t i s that the p a r t i c l e i s never i n any f i n i t e region of space. This s i t u a t i o n i s described by a zero wavefunction f o r ^ 4.+<x>. As F i s reduced to zero i n the i n i t i a l wavefunction and E i s negative, (4 .8 ) becomes (4.10) which i s zero f o r a l l when F i s zero. Hence f o r both p o s i t i v e and negative t o t a l energy reducing the p o t e n t i a l i n the wavefunction y i e l d s a r e s u l t i n accord with the physical s i t u a t i o n a r i s i n g from switching o f f the p o t e n t i a l . CHAPTER V THE HARMONIC OSCILLATOR 1. TIME INDEPENDENT TREATMENT The time independent harmonic o s c i l l a t o r wave equation 11 and i t s sol u t i o n are v e i l known. The wave equation i s (5.1) £ l + W E - i « « c y | t « o dx* v\ x I r-—— 1 where /.va/«l*stic constant , K j . s the c l a s s i c a l 4 mass j fn frequency associated with t h i s o s c i l l a t o r . The energy E takes values given by (5.2) En = ("+i)*o)c where n i s a non-negative integer. The normalization I TI dx a 1. This imp l i e s the boundary conditions that Y goes to zero as x goes to ±oo. ' For a given n the normalized s o l u t i o n of (5.1) i s ( 5 •3) r - w = SS? E"M "X /Z*HRT (X/^F) • Since the p o t e n t i a l energy i s L m <oc* x1 reducing the p o t e n t i a l to zero i s equivalent to reducing K or <uc to zero. Reducing cd c to zero reduces the wavefunction to zero f o r a l l values of n. The wavefunction goes to zero as A > C ' / 4 O R AO The e f f e c t of decreasing the p o t e n t i a l to zero i n the wave equation w i l l now be determined. I f the po t e n t i a l goes to zero such that the energy values e x p l i c i t l y s a t i s f y ( 5 . 2 ) f o r intermediate values of 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* I f the p o t e n t i a l goes to zero without e x p l i c i t l y requiring that ( 5 . 2 ) be s a t i s f i e d the wave equation becomes ( 5 . 5 ) £ f + & a E 4>-o. 6*x V-Since the boundary conditions that ^ i s zero at * = are associated with both (5.4) and ( 5 . 5 ) 4 = 0 i s the so l u t i o n of both (5.4) and ( 5 . 5 ) with these boundary conditions. This i s the same as i n the negative energy cases of the square well and Coulomb p o t e n t i a l s . The block diagram for the time independent harmonic o s c i l l a t o r i s occupied as follows: corner one by equation ( 5 . 1 ) and boundary conditions that 4* goes to zero as x goes to i co ; corner two by the same boundary conditions and equation (5.4) or ( 5 . 5 ) depending on the e x p l i c i t assumptions regarding decreasing the p o t e n t i a l ; corner three by wave-f u n c t i o n ( 5 . 3 ) ; and corner four by 4=0. The r e s u l t of reducing the p o t e n t i a l to zero i n corner three y i e l d s the 41 entry i n corner four. Hence the block diagram i s completed. 2. TIME DEPENDENT TREATMENT The time dependent wave equation f o r a harmonic o s c i l l a t o r i s (5.6) i * 3Wx,t) / fc* a* ^ . l c , » ) * W , i ) where V= 1 Kx i i s the p o t e n t i a l energy at p o s i t i o n x. Since the p o t e n t i a l i s independent of the time the so l u t i o n o f the time dependent equation may be expressed as an i n f i n i t e sum of the solutions of the time independent wave equation with the c o e f f i c i e n t depending on the time. Using 12 t h i s technique, the normalized s o l u t i o n to (5 .6) i s co ( 5 . 7 ) ¥(«,t) = e i p f - a t - j ^ - t f ) E t e f I k We-" , o u * \ T T / 1 v i 4 . x / x / n| where d^- m K 3 1 1 ( 1 *o * s the i n i t i a l p o s i t i o n of the centre of the wave packet. When K l s reduced to zero u>c and oi also go to zero and S x^^ i) becomes zero. Therefore the r e s u l t of reducing the p o t e n t i a l to zero i s the same f o r the solutions of both the time dependent and time independent wave equations. The r e s u l t of decreasing the po t e n t i a l to zero i n the time dependent wave equation w i l l now be analyzed. Since E does not appear i n (5.6) the r e s u l t of decreasing the 42 p o t e n t i a l to zero i n ( 5 . 6 ) i s ( 5 . 8 ) = regardless of whether or not ( 5 . 2 ) i s s a t i s f i e d f o r i n t e r -mediate values of the p o t e n t i a l . The s o l u t i o n of ( 5 . 8 ) i s A t i k x • i E t / f c n c 6 . For the wavefunction to vanish at both x=*oo A must be zero and hence the s o l u t i o n of ( 5 . 8 ) consistent with the boundary conditions i s ^ e O. I f the system s a t i s f i e s ( 5 . 2 ) 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 of ( 5 . 8 ) , before the boundary conditions are applied, i s i s a constant since E and kef lrr\%T go to zero as the p o t e n t i a l does. To s a t i s f y the boundary conditions t h i s constant i s then zero. The block diagram f o r the time dependent harmonic o s c i l l a t o r i s populated as follows: corner one by equation ( 5 . 6 ) and boundary conditions that ^ goes to zero as 1x1 goes to i n f i n i t y ; corner two by the same boundary conditions and the equation ( 5 . 8 ) ; corner three by the wavefunction ( 5 . 7 ) ; and corner four by ^=0. Since reducing the poten-t i a l i n ( 5 . 7 ) y i e l d s zero the block diagram i s closed. I t should be noted that f o r both the time dependent and time independent treatments of the harmonic o s c i l l a t o r the energy eigenvalues are given by E n = ( n + i ) ^ c o t with n a non-negative integer. Hence i n both treatments 43 i t i s apparent, frora t h i s eigenvalue equation, that a l l the energy eigenvalues 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 that (5.2) i s s a t i s f i e d f o r a l l intermediate values of the p o t e n t i a l . The procedure used to solve the above time dependent •wave equation may be equally well applied to solving the time dependent equations corresponding to the other p o t e n t i a l s . However, as the preceding has shown, the r e s u l t of reducing the p o t e n t i a l to zero i s the same fo r the solutions to both the time dependent and time independent equations. Hence i n studying the r e s u l t of reducing 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 deal with the s o l u t i o n of either the time dependent or time independent wave equation. 3. CLASSICAL ANALYSIS OF SWITCHING OFF Before discussing the switching o f f processes the c l a s s i c a l d e s c r i p t i o n of, and c r i t e r i a f o r , a harmonic o s c i l l a t o r w i l l be given. The physical system considered as a harmonic o s c i l l a t o r consists of a p a r t i c l e o s c i l l a t i n g about an equilibrium p o s i t i o n under the influence of a force d i r e c t e d towards the equilibrium p o s i t i o n and of magnitude proportional to the p a r t i c l e ' s distance from t h i s equilibrium p o s i t i o n . The equation of motion i s X s d cos coc-t . The magnitude of the v e l o c i t y at p o s i t i o n x i s / K ^ - x ^ 1 where a i s the amplitude. 44 The k i n e t i c energy, T, pot e n t i a l energy, V, and t o t a l energy, E, of a harmonic o s c i l l a t o r with amplitude "a" obey the following c r i t e r i a : (5.9) T * V : E 5 constant to'i-m r-esp«ct t o post+> on a.r>ol t i m e E 3 maximum T - maximum V 3 1 K a 2 w O f T ^ maximum T ' 0 - V * maximum V. The f i r s t c r i t e r i o n states that conservation of energy holds f o r a l l positions and time. The second c r i t e r i o n indicates that the t o t a l energy, E, depends only on the e l a s t i c constant, K, and amplitude a. In t h i s section a detailed c l a s s i c a l analysis w i l l be given of the two basic ways of switelling o f f a p o t e n t i a l i n a physical system. 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 analysis since t h i s system i s common, u s e f u l and r e l a t i v e l y simple. However t h i s d i s t i n c t i o n i n methods of switching o f f the po t e n t i a l i s applicable to a l l other systems. The aim of th i s analysis i s to i l l u s t r a t e the two types of switch o f f and to make clear the d i s t i n c t i o n between them. These two methods of switching o f f w i l l be referred to as type I and type I I . A type I switch o f f i s where the c r i t e r i a of the p a r t i c u l a r system are s a t i s f i e d f o r a l l intermediate values of the p o t e n t i a l . For example, i n a type I switch o f f a harmonic o s c i l l a t o r with amplitude "a" remains a harmonic o s c i l l a t o r with amplitude "a" fo r a l l intermediate values of the p o t e n t i a l , that i s , (5.9) i s s a t i s f i e d f o r a l l intermediate values of the p o t e n t i a l . A type II switch o f f i s 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 of the system "being s a t i s f i e d f o r a l l intermediate values of 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 r e l a -t i o n s h i p s between the various parameters are not s a t i s f i e d during the switching o f f process. Before examining the switching o f f processes i t should be noted that f o r a harmonic o s c i l l a t o r the only way the p o t e n t i a l may be switched o f f without imposing geometric constraints, decreased i n the wave equation or reduced i n the wavefunction i s by the e l a s t i c constant, K, becoming zero. The type I switch o f f of the harmonic o s c i l l a t o r w i l l be studied f i r s t . The conditions on the switching o f f process i n order that the process be of type I may be formu-l a t e d i n the form of the following theorem. THEOREM: To s a t i s f y the condition that f o r a l l nonzero values of 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 with amplitude a, the switch o f f must be done i n the following manner: (a) The p o t e n t i a l may be switched o f f only i n d i s c r e t e decrements and these may occur only while the p a r t i c l e i s at an extremity. (b) The p a r t i c l e ends up at one of the extremities with neither 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 equilibrium p o s i t i o n , that i s , the f i n a l t o t a l energy i s zero. 46 Proof: F i r s t i t w i l l be shown that the f i n a l s i t u a t i o n i s a p a r t i c l e with neither 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 equilibrium p o s i t i o n . Let the p o t e n t i a l be p h y s i c a l l y lowered by some a r b i t r a r y amount. By (5.9) the t o t a l energy i s lowered by the same amount as i s the maximum po t e n t i a l energy. This i s repeated with the r e s u l t that the maximum p o t e n t i a l energy and the t o t a l energy are again lowered by i d e n t i c a l amounts. This procedure i s continued u n t i l the p o t e n t i a l reaches zero. I t Is obvious, upon reference to (5.9), that 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 energies a l l go to zero. So the r e s u l t i s a p a r t i c l e with neither motion nor p o t e n t i a l energy w i t h respect to the equilibrium p o s i t i o n . than at an extremity. I t then has a p o t e n t i a l energy of w i l l have a v e l o c i t y such that i t s k i n e t i c energy at t h i s p o s i t i o n i s greater than the new, lower maximum p o t e n t i a l ^ energy and the condition f o r a harmonic o s c i l l a t o r i s not v e l o c i t y at t h i s p o s i t i o n than would be the case i f i t were Now consider the p a r t i c l e at any p o s i t i o n $ other s a t i s f i e d . I f the p a r t i c l e w i l l have a greater undergoing a harmonic motion with a maximum p o t e n t i a l equal to the lower p o t e n t i a l energy. I f the p a r t i c l e i s approach-i n g x=o then at x=o 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 at ^ plus the v e l o c i t y acquired i n going from ^ to zero under the influence of the lower p o t e n t i a l . Since the v e l o c i t y at ^ i s greater than that which would he the case i f the lower p o t e n t i a l were opera-t i v e during the entire journey from x=a to x=o, the v e l o c i t y at x=o corresponds to a k i n e t i c energy at x=o greater than the maximum of the lower p o t e n t i a l . Thus the p a r t i c l e again disobeys ( 5 . 9 ) . 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 that i t would overshoot U | = a and would then no longer be a harmonic o s c i l l a t o r with amplitude a Hence the p o t e n t i a l may not be lowered at any p o s i t i o n such t h a t )-=fc±<x. Furthermore, switching o f f the p o t e n t i a l i n a continuous manner while the p a r t i c l e executes i t s motion i s also inconsistent with 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 since i t involves lowering the po t e n t i a l at points other than ± a . Therefore the only remaining method, and an obviously acceptable one, i s to lower the p o t e n t i a l i n f i n i t e steps when the p a r t i c l e i s at - a . That t h i s method complies with the s t i p u l a t i o n s of t h i s type of switch o f f may be seen from the following argument. While the p a r t i c l e i s at * a the p o t e n t i a l i s lowered. The r e s u l t of t h i s i s the following: the t o t a l energy and maximum p o t e n t i a l energy are correspondingly lowered; the maximum 48 k i n e t i c energy i s correspondingly lowered because the force and acceleration are l e s s over the whole distance from x=a to x=o; and the system remains a harmonic o s c i l l a t o r with amplitude a but with a lower t o t a l energy and greater period. Whether the p a r t i c l e eventually ends up at x=a or x= -a depends s o l e l y on the technique used to switch o f f the p o t e n t i a l . For example, i f i t i s wished that the p a r t i c l e end up at x=a t h i s can be achieved by any type I switch o f f wherein the l a s t step (to zero) occurs when the p a r t i c l e i s at x=a. I t should also be noted that the p a r t i c l e cannot end up at the equilibrium p o s i t i o n , q.e.d. A type II switch o f f of 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 . There are numerous ways i n which a system may undergo a type II switch o f f . However the following 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 the p r i n c i p l e s and the r e s u l t of such a switch o f f . I t w i l l be assumed that the p o t e n t i a l i s switched o f f during a time I n t e r v a l very short compared to the period. 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 At centred on a time, t 5 , the l a t t e r being c a l l e d the "time o f switch 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 distance Ax . Ax i s much l e s s than the amplitude "a" since At i s much l e s s than the period. Let x 0 be the centre of Ax . In t h i s type II switch o f f the r e s u l t i s a free p a r t i c l e with speed within the range /\L(*x-(x0-±x)1)' to 11 (a l-rx.*Ax)* )' V m \j m unless x 0 i s within AX of the equilibrium or the extreme p o s i t i o n . I f | x 0 | ^ 4 X then the free p a r t i c l e ' s speed i s i n the range t VW to or IK where I i s the l e s s e r of c ^ - C X a - d x ) 1 or / a. l-(x d+AX) 1 and OL i s greater than I but l e s s than a. I f J lx0l - aj * AX then the free p a r t i c l e ' s speed i s greater than zero but l e s s than J 11 ( a l - C i x e | - A X ) 1 ) ' . Hence when such a type II switch o f f i s ca r r i e d out the r e s u l t Is a p a r t i c l e with a speed greater than zero and l e s s than a / K . This speed depends on the p a r t i c l e ' s p o s i -v m t i o n at the time of switch o f f and on the distance covered during the switching o f f process. The s i g n i f i c a n t point to n o t i c e i n t h i s example of a type II switch o f f i s that nothing i s stipulated, discussed or assumed regarding the behaviour of the system during the time i n t e r v a l A t . The two preceding examples have dealt with a system and process treated i n c l a s s i c a l terms. I t i s now of i n t e r e s t to describe the system r e s u l t i n g from the c l a s s i c a l switch o f f i n quantum mechanical terms. The r e s u l t of the type I switch o f f was a p a r t i c l e with 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 described by a constant wavefunction. I f t h i s wavefunction also had to s a t i s f y the normalization condition J |4M dx=l then t h i s constant would be zero. As w i l l be r e c a l l e d from sections one and two, the r e s u l t of reducing the p o t e n t i a l i n the wavefunction f o r the harmonic o s c i l l a t o r was also zero. The system r e s u l t i n g from the type II switch o f f i s described by a plane wave Q±>itK where IM has a value greater than zero and l e s s than J K m a . I t should M be pointed out that the "a" i n /Km a. i s not a quantum mechanical quantity but enters from the r e s t r i c t i o n of the v e l o c i t y of the free p a r t i c l e being described. This plane wave was not obtained by reducing 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 consideration i n t h i s chapter co n s i s t s of a spinless p a r t i c l e -with charge e moving i n a uniform magnetic f i e l d H which f i l l s a l l space. The v e l o c i t y perpendicular to the f i e l d w i l l be denoted by V . The ^ - a x i s i s chosen i n the d i r e c t i o n of the f i e l d . 1 . QUANTUM MECHANICAL TREATMENT When ( - 0,0) i s chosen as the vector p o t e n t i a l and when i s chosen as the form X 1 3 i s I f M „ = - c p x and o> = c W are substituted i n ( 6 . 1 ) an J e t r mc equation formally i d e n t i c a l to the Schroedinger Equation f o r a harmonicx o s c i l l a t o r i s obtained. To normalize a wavefunction of the above form i t i s s u f f i c i e n t to integrate from — «o to + ©o with respect to y only. This i s the case since the parts of the wavefunc-t i o n depending on x and ^. are already i n the form of plane waves and i n unbounded space no further normalization i s 5 2 p o s s i b l e or necessary. The normalization condition i s there-l o r e ay - 1 . This imposes the boundary conditions that 4* goes to zero as goes to i n f i n i t y . Using the same techniques as i n the harmonic o s c i l l a t o r case the normalized wavefunction i s (6.2) t=ei(k-**k^ J«« -e-«»iwJf** Hjn^irJ). \l 2 X n ( n » ) Y f c W * / I f the f i e l d H i s reduced to zero 4^ goes to zero. The rate of approaching zero i s G as H goes to zero. This i s an es 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 again using the analogy between (6.1) and the wave equation f o r a harmonic o s c i l l a t o r the allowed energy values are seen to be 1^ -(6.3) En = (n*i)fc«a +££ where n i s a non-negative integer. As the f i e l d i s decreased to zero co goes to zero and the only energy i s . That i s , as H goes to zero the energy values associated with the transverse motion a l l become zero leaving only the energy associated with the unaffected motion p a r a l l e l to the f i e l d . The r e s u l t of decreasing H to zero i n (6.1) i s 5 3 ( 6 . 4 ) X" +P * * X =0 2 m where 2x i s ^ " -EjL ~f£ Zm i r n 1m since there i s no f i e l d . When ( 6 . 4 ) i s combined with the boundary conditions that 'jC goes to zero as 1^1 goes to i n f i n i t y , the only acceptable solu t i o n (as i n the previous cases) i s ~ ^ ~ 0 . The block diagram f o r t h i s system i s occupied as follows: corner one by equation ( 6 . 1 ) and the boundary condi-tions that 4^  goes to zero as goes to ± 00 • corner two by the same boundary conditions and equation ( 6 . 4 ) ; corner three by wavefunction ( 6 . 2 ) ; and corner four by 4^ *0. As has been shown, reducing the f i e l d i n the wavefunction ( 6 . 2 ) r e s u l t s i n H^=o. Hence the block diagram i s completed. 2 . CLASSICAL ANALYSIS OF SWITCHING OFF In c l a s s i c a l terms the force equation f o r a stable o r b i t i n a uniform magnetic f i e l d i s ( 6 . 5 ) .e_H = Y _ . (TIC *" Sta t i n g that a system behaves as a charged p a r t i c l e moving i n a uniform magnetic f i e l d implies that the p a r t i c l e i s moving i n a stable o r b i t i n the plane perpendicular to the f i e l d . ( 6 . 5 ) i s the necessary and s u f f i c i e n t condition f o r such a stable o r b i t . Hence (6..50 may be considered as the c l a s s i c a l c r i t e r i o n of t h i s system. 5 4 Before considering the actual switching o f f processes the following point should be emphasized. The p a r t i c l e ' s speed, V , remains constant independent of the f i e l d ' s behaviour because the force i s always perpendicular to the f i e l d and no further constraints may be introduced. In discussing the switching o f f of the magnetic f i e l d the two types must again be distinguished. Type I w i l l again be studied f i r s t . The type I switch o f f was defined as being the type i n which the system s a t i s f i e s the defining c r i t e r i a f o r a l l intermediate values of the p o t e n t i a l . Stating that the c r i t e r i o n of t h i s system i s s a t i s f i e d f o r a l l intermediate f i e l d values means that the equation (6.5) i s s a t i s f i e d f o r these intermediate f i e l d values. Therefore, the f i e l d must be switched o f f i n a manner such that at a l l intermediate stages the p a r t i c l e , t r a v e l l i n g with f i n i t e v e l o c i t y , "v , has s u f f i c i e n t time to reach the distance, r , which s a t i s f i e s (6.5) f o r the various intermediate f i e l d values. Since a continuous switch o f f may be approximated with a r b i t r a r y accuracy by a discrete switch o f f , only the l a t t e r need be considered even though the r e s u l t s w i l l apply to both methods. The d e t a i l s of a d i s c r e t e type I switch o f f w i l l now be analyzed. I f i n a given->step the f i e l d i s lowered from H to H x then the radius, r x , associated with % i s greater than 55 r associated with H. For ( 6 . 5 ) to be s a t i s f i e d the radius must therefore be rj_ when the f i e l d i s % . Hence a time greater than "C , where t i s rx-r , must elapse v before the step from H to 1^ can be considered completed. T h i s requirement regarding the time i s necessary to enable 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 , to reach the p o s i t i o n required by ( 6 . 5 ) . Before undertaking any given decrement the previous step must of course be completed. In the f i n a l step the f i e l d goes from some H 1 to zero and the radius goes from some f i n i t e r' to ra where r 0 i s i n f i n i t e . As above a time greater than ^ o - < i s required v i n order that t h i s step be completed i n a type I manner. Since the f i n a l step must be completed before the f i e l d can be considered as switched o f f , t h i s i n f i n i t e time f o r the l a s t step shows that a type I switch o f f of t h i s uniform magnetic f i e l d cannot be done i n a f i n i t e time. The s t i p u l a t i o n s f o r a type I switch o f f also imply that any r e s u l t or e f f e c t due to any given step must be i d e n t i c a l to the r e s u l t or e f f e c t obtained by carrying out t h i s same step i n an a r b i t r a r i l y large number of a r b i t r a r i l y small, consecutive type I steps. From t h i s point of view an a r b i t r a r y step from W\ to Hf and then the step from H a to zero w i l l be studied. In lowering the f i e l d from H; to the radius increases from *\ to ^ . I f a very l a r g e number of steps are employed the radius increases as i n the mono tonic sequence: r,-, r (, rX) . . . iri~ari r f . For (6.5) to be s a t i s f i e d for a l l intermediate values a time greater than r*+i - rK must elapse before the corresponding decrease i n the f i e l d can be considered as completed and the next decrease can be undertaken. Hence the t o t a l time which must elapse i n going from to Hf i s greater than V V V V v This agrees with the previous r e s u l t . The f i e l d going from H a to zero w i l l now be studied. I f t h i s switching o f f i s done i n an a r b i t r a r i l y large number of small steps the f i e l d goes through the values of the following monotonically decreasing sequence: H a ; H 4 , H 1 } . . . , A H t O. As previously explained, a time greater than K K ^ t - r K must v elapse before the step from H K to H**, can be considered as completed. Hence f o r the switch o f f from Wa. to zero to be completed i n a type I fashion, the required time i s greater than rx — f"a .». r x - n ^ .. . •+• r a - r A t t - r 0 - „ Since f 0 , corresponding to zero f i e l d , i s i n f i n i t e and i s f i n i t e , t h i s time i s i n f i n i t e . Hence, as before, an i n f i n i t e time i s required f o r a type I switch o f f of the mag-n e t i c f i e l d . The point of view that the r e s u l t of any step must be equivalent to the r e s u l t of an a r b i t r a r i l y large number of steps between the same i n i t i a l and f i n a l f i e l d values emphasizes the f a c t that the switching o f f process cannot be considered completed u n t i l s u f f i c i e n t time has 5 7 elapsed for the f i n a l step to he completed. Since a type I switch o f f requires an i n f i n i t e time as demonstrated above, any switch o f f completed i n a f i n i t e time i s of type I I . A switch o f f i n which the f i e l d goes to zero i n a time comparable to the period of the o r b i t i n g p a r t i c l e i s of type II and may be used as an example. The r e s u l t of t h i s switch o f f i s a free p a r t i c l e with speed V . The d i r e c t i o n of the free 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 of the switch o f f . In f a c t the r e s u l t of any type II switch o f f i s as above. Since a type II switch o f f i s done i n a f i n i t e time and i n view of the p a r t i c l e ' s f i n i t e v e l o c i t y , the p a r t i c l e may be l o c a l i z e d within a given f i n i t e volume fo r any p a r t i c u l a r type II switch o f f . Although the discussion i n t h i s section has been i n c l a s s i c a l terms only, i t i s of i n t e r e s t to describe i n quantum mechanical terminology the systems r e s u l t i n g from these two types of switch o f f . The r e s u l t of the type I switch o f f was a p a r t i c l e , with v e l o c i t y , V , at i n f i n i t y . Since the p a r t i c l e i s at i n f i n i t y i t s p r o b a b i l i t y of being i n any f i n i t e elementary volume i s zero. Hence l ^ i i s zero and 4^  i s also zero. This i s also the r e s u l t obtained by reducing the f i e l d i n the wavefunction. The r e s u l t of a type II switch ± i k • r o f f has 6 as i t s wavefunction where k = my _^ * and the d i r e c t i o n of k i s determined by the d e t a i l s of the switelling o f f process. This plane wave was not obtained by reducing the f i e l d i n the wavefunction. 58 Before completing t h i s section an apparent i n c o n s i s -tency between the quantum mechanical and c l a s s i c a l values for the transverse energy, when the f i e l d i s zero, w i l l be pointed out. As shown i n the previous section, the quantum mechanical expression f o r the transverse energy becomes zero as the f i e l d i s decreased to zero. However i n both types of c l a s s i c a l switch o f f the transverse v e l o c i t y , and hence transverse k i n e t i c energy, remains constant f o r a l l values of the f i e l d i n c l u d i n g zero f i e l d . 3. WAVEFUNCTION ESSENTIAL SINGULARITY FOR ZERO FIELD In t h i s system the dominating fa c t o r i n the wave-fu n c t i o n as the f i e l d i s reduced to zero i s Q. , that i s , the wavefunction goes to zero exponentially as -pj-goes to i n f i n i t y . Since 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 an expansion about H° 0 i s impossible and therefore perturbation techniques w i l l not give the wavefunction f o r a charged p a r t i c l e i n a uniform magnetic f i e l d f o r small f i e l d s . Although the harmonic o s c i l l a t o r and magnetic f i e l d wave equations and wavefunctions are formally the same there i s one fundamental difference and i t i s t h i s difference which corresponds to the v a s t l y d i f f e r e n t physical behaviour between the two systems with regard to the p o t e n t i a l being switched o f f . In the harmonic o s c i l l a t o r case the independent v a r i a b l e i s simply x—-independent of a l l parameters or any-thing else. However i n the magnetic f i e l d case the "independent v a r i a b l e " i s ( 'J-^o )• 3ince H i s a parameter independent of p o s i t i o n = d(y-y„) and ( y - <^ c ) i s then the independent v a r i a b l e of an equation formally 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 . In 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 i n any way influence the independent v a r i a b l e x whereas when the magnetic f i e l d goes to zero the "independent va r i a b l e " ( v^- vj D ) goes to i n f i n i t y . Now i t s h a l l be shown how the behaviour of the magnetic f i e l d "independent va r i a b l e " mathematically expresses the behaviour of the p a r t i c l e i n the f i e l d . u 0 3 - Cpx can be i d e n t i f i e d as the <-j coordinate C H o f the centre of the c i r c u l a r path i n the plane perpendicular to the f i e l d . Substituting px = mvx -» e j ^ x and A x=-Hu c J i n the expression f o r u gives u - u = C m v x . 14 S i m i l a r l y X-X 0 a c™ Vy can be introduced where x e i s i d e n t i f i e d as the x coordinate of the centre of the above c i r c u l a r path. Squaring and adding produces the f a m i l i a r r e s u l t r x = (x-xj 1-^(w-iwJ X= c* v x . As the f i e l d goes to zero the radius V goes to i n f i n i t y i n agreement w i t h the r e s u l t of a type I switch o f f . These differences between the magnetic f i e l d and the harmonic o s c i l l a t o r cases, namely the behaviour of the independent variables and 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 physi-c a l difference that i n the magnetic f i e l d 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 switch o f f whereas i n the harmonic o s c i l l a t o r case the v e l o c i t y becomes zero and the p a r t i c l e i s contained w i t h i n a f i n i t e region of- space i n a type I switch o f f . CHAPTER VII CONCLUSION 1. DISCUSSION OF SWITCHING OFF PROCESSES In t h i s thesis two types of switching o f f have been distinguished; namely, type I i n which the c r i t e r i a r e l a t i n g the parameters of the system are s a t i s f i e d f o r a l l i n t e r -mediate values of the p o t e n t i a l and type II i n which these c r i t e r i a are not s a t i s f i e d f o r intermediate values of the p o t e n t i a l . In a l l the bound systems considered there was at l e a s t one c l a s s i c a l r e l a t i o n or equation which characterized the system. In a l l the unbound systems there was no such c r i t e r i o n . Hence i t follows that d i s t i n g u i s h i n g between the two types of switching o f f i s meaningful only i n the case of a bound system. C r i t e r i a determining the type of switch o f f a bound system undergoes w i l l now be given. Since the switching o f f processes have been discussed i n c l a s s i c a l terms the c r i t e r i a w i l l be given i n c l a s s i c a l terms. Due to the uncertainty 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 therefore no s p e c i f i c quantum mechani-cal, 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 switching o f f methods w i l l be given. 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 of switch o f f are not given i n quantum mechani-c a l terms the actual d i s t i n c t i o n i n methods i s applicable to a quantum mechanical d e s c r i p t i o n of a system. Furthermore, since the actual experimental procedures used i n switching o f f p o t e n t i a l s are usually of a c l a s s i c a l nature, t h i s 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 the types of switching o f f i s applicable i n determining the type of switch o f f used experimentally. Since a switch o f f i s either of type I or type I I , i t i s s u f f i c i e n t to give the c r i t e r i a f o r a typeM switch o f f since a switch o f f i n which these c r i t e r i a are not met Is necessarily of type I I . In any bound system the maximum k i n e t i c energy ever attained must be l e s s than or equal to the maximum of the absolute values of the p o t e n t i a l energy. I t Is therefore apparent that unless the 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 of steps without v i o l a t i n g t h i s energy c r i t e r i o n 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 . This may be e a s i l y seen i n a case where the t o t a l energy i s negative. For example, i n the negative 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 without t h i s i n e q u a l i t y being disobeyed. By use of (3.9) i t was e x p l i c i t l y shown that the p o t e n t i a l cannot go to zero i n a f i n i t e number of steps. 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 since there i s a p o s i t i o n at which the v e l o c i t y i s zero and the p o t e n t i a l can there he lowered. I f a c r i t e r i o n of a system st i p u l a t e s that the p a r t i c l e must be at i n f i n i t y i n order 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 i s zero, then the p o t e n t i a l 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 time. This was demonstrated i n chapter s i x section two. Hence, the two requirements of a system i n order that a type I switch o f f to zero may be done I n a f i n i t e time are: f i r s t , the p a r t i c l e need not necessari-l y go to i n f i n i t y i n order to s a t i s f y the c r i t e r i a of the system f o r zero p o t e n t i a l ; and secondly, there be a p o s i t i o n a t which the p a r t i c l e ' s k i n e t i c energy Is zero. In a system which meets these requirements the p o t e n t i a l may, i n p r i n -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 lowering the p o t e n t i a l while the p a r t i c l e i s at a p o s i t i o n of zero v e l o c i t y . This however requires a f i n i t e lowering of the p o t e n t i a l i n a zero time i n t e r v a l . Since t h i s cannot be achieved a type I switch o f f to zero i n a bound system i s not experimentally f e a s i b l e . Hence any experimental switch o f f to zero i n a bound system i s of type I I . The preceding discussion i s concerned with a type I switch o f f i n which the p o t e n t i a l i s switched o f f to zero. However, to an a r b i t r a r y degree of accuracy, a type I lower-i n g of the p o t e n t i a l from an I n i t i a l value to a lower, non-zero f i n a l value may be experimentally ca 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 during 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 accord with the conditions f o r a type I switch o f f during t h i s lowering. For example, i n the uniform magnetic f i e l d and bound Coulomb cases the p o t e n t i a l or f i e l d may be experimentally lowered from some i n i t i a l value to a non-zero f i n a l one i n a type I manner. The d e t a i l s of the procedure may vary from case to case but the point i s that such a type I lowering i s experi-mentally f e a s i b l e . Since i n an unbound system there i s no d i s t i n c t i o n between the two methods of switching o f f , the.two types are i d e n t i c a l and both correspond to any given experimental switch o f f . 2. REDUCTION OF THE POTENTIAL IN WAVEFUNCTION In a l l of the systems studied the r e s u l t of reducing the p o t e n t i a l i n the wavefunction was one of the following two: (a) a zero wavefunction; (b) an o s c i l l a t o r y wavefunction — either a trigonometric or imaginary exponential function. Each of the r e s u l t s (b) corresponded to an unbound system. Ea^ch of the r e s u l t s (a), with the exception of the negative t o t a l energy uniform e l e c t r i c f i e l d case, corresponded to 65 a bound system. This exception w i l l be treated i n section f i v e . 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 succeeding paragraphs. A s i g n i f i c a n t and s a t i s f y i n g common feature of those wavefunctions which went to zero, excepting the above exception, w i l l now be presented. As was previously stated, the wavefunction went to zero only i f i t described a bound system. The fundamental c h a r a c t e r i s t i c of a bound system i s systems t h i s condition determines a normalization c o e f f i c i e n t which causes the wavefunction to obey t h i s condition. In a l l the wavefunctions under consideration (see ('2.5), (2.5*), (3.4), (5.3), (6.2)) i t i s t h i s normalization c o e f f i c i e n t which goes to zero. Except f o r the magnetic f i e l d wavefunc-t i o n , these wavefunctions go to zero only on account of t h e i r normalization c o e f f i c i e n t . I f the normalization c o e f f i c i e n t would have been absent i n these systems the wavefunction r e s u l t i n g from reducing the p o t e n t i a l would have been a constant but not, i n general, zero. The physical s i g n i f i c a n c e of t h i s w i l l now be given. In a l l the bound systems considered, except f o r the magnetic 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 switch o f f . (The s i g n i f i c a n c e of specifying type I switch o f f w i l l be seen further on i n t h i s section.) Hence the end r e s u l t Is a p a r t i c l e with zero t o t a l energy. Having zero t o t a l energy, the normalization condition 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 of being anywhere, t h a t i s , a constant wavefunction to w i th in a phase f a c t o r . I n general , i t i s only upon a p p l i c a t i o n of the normal izat ion c o n d i t i o n , with i t s associated boundary condi t ions , that t h i s constant must be zero. I n the magnetic f i e l d case the normal izat ion c o e f f i c i e n t also goes to zero. However, the wavefunction goes to zero more r a p i d l y due to a dominating exponent ia l f a c t o r . As shown In chapter s ix , t h i s exponen-t i a l decrease to zero corresponds to the p a r t i c l e going to i n f i n i t y . The preceding considerat ions suggest the fo l lowing general statements: (a) I n a l l bound systems i n which i t i s not imperative that . the p a r t i c l e go to i n f i n i t y i n a type I switch o f f , the wavefunction, i n general , goes to zero due to the normal izat ion c o e f f i c i e n t . (b) I f 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 switch o f f then the normal izat ion c o e f f i c i e n t again goes to zero but i s dominated by an exponent ia l ly decreasing fac tor which describes the p a r t i c l e going to i n f i n i t y . The preceding has shown how the c h a r a c t e r i s t i c property of a bound system d i r e c t l y determines the r e s u l t of reducing the p o t e n t i a l i n the wavefunction of such a system. need not be s a t i s f i e d . Furthermore, there are no condit ions 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 reducing the p o t e n t i a l i n the wavefunction for an unbound system y i e l d s a free p a r t i c l e wavefunction which s a t i s f i e s the boundary conditions. Thus f o r an unbound system the r e s u l t of reducing the p o t e n t i a l i n the wavefunction i s that expected from experimental observations. The r e l a t i o n between the r e s u l t of reducing the p o t e n t i a l i n the wavefunction and the r e s u l t s of the two types o f switching o f f w i l l now be discussed. Bound systems w i l l again be discussed f i r s t . As demonstrated i n the examples of switching o f f i n the previous chapters, the system r e s u l t i n g from a type I switch o f f i s quantum mechanically described by the r e s u l t of reducing the p o t e n t i a l i n the wavefunction of the o r i g i n a l system, that i s , by a zero wavefunction f o r a bound system. The reason f o r t h i s correspondence between the r e s u l t s of a type I switch o f f , and reducing the p o t e n t i a l i n the wavefunc-t i o n , w i l l become apparent when the properties of a type I switch o f f and of a wavefunction are compared. In addition to other parameters and variables, the wavefunction i s a function of the po t e n t i a l of a system and f u l l y describes the system i n terms of the p o t e n t i a l and these other parameters and va r i a b l e s . For any s p e c i f i c system there i s a one to one correspondence between the system with a s p e c i f i c set of parameters and a p a r t i c u l a r wavefunction. 68 Consider a wavefunction describing any p a r t i c u l a r system. I f the p o t e n t i a l parameter within the wavefunction i s changed, the r e s u l t i s a wavefunction describing a system with the same c h a r a c t e r i s t i c s and c r i t e r i a but with a d i f f e r e n t poten-t i a l energy. That i s , the wavefunction now describes the same kind of system which has adjusted i t s e l f such as to s a t i s f y i t s c h a r a c t e r i s t i c c r i t e r i a when the po t e n t i a l i s equal to i t s new value. Now consider a p a r t i c u l a r system described by a p a r t i c u l a r wavefunction. Let the p o t e n t i a l of t h i s system be switched o f f and consider the system as the p o t e n t i a l i s being lowered. Since the switching o f f process I s not being described, i t i s unnecessary to s t i p u l a t e whether the process i s c l a s s i c a l or quantum. As long as the system s a t i s f i e s the c r i t e r i a of the o r i g i n a l system, the switch o f f i s of type I and the o r i g i n a l wavefunction, with the p o t e n t i a l reduced, may be used to describe the system at any p a r t i c u l a r stage. However, as soon as the c r i t e r i a of the o r i g i n a l system are no longer s a t i s f i e d , the p o t e n t i a l i s no longer being switched o f f i n the o r i g i n a l system but i n another, d i f f e r e n t system. At t h i s stage, where the switch o f f i s no longer of type I and the p o t e n t i a l i s being switched o f f i n a d i f f e r e n t system, reducing the p o t e n t i a l i n the o r i g i n a l wavefunction no longer corresponds to the p h y s i c a l process and a d i f f e r e n t wavefunction describing t h i s d i f f e r e n t system must now be introduced 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 that 69 the i d e n t i f i c a t i o n of the r e s u l t of a type I switch o f f with the r e s u l t of reduction i n the wavefunction follows from the primary property of a wavefunction and a type I switch o f f . As shown i n section one, any experimental switch o f f to zero i n a bound system i s nece s s a r i l y of type I I . I t has j u s t been demonstrated that the r e s u l t of reducing the poten-t i a l to zero i n a wavefunction describes the r e s u l t of a type I switch o f f . Hence reducing the p o t e n t i a l to zero i n a wave-fun c t i o n describing a bound system does not correspond to an experimentally f e a s i b l e method of switching o f f the p o t e n t i a l i n t h i s system. This i s the reason the r e s u l t of reducing the p o t e n t i a l to zero i n the wavefunction of a bound system does not y i e l d the plane wave wavefunction indicated by experimental observations. However a type I switch o f f to a non-zero value i s possible i n some bound systems. In these systems the r e s u l t of such a lowering to a non-zero value i s described by the r e s u l t of reducing the p o t e n t i a l to t h i s non-zero, f i n a l value i n the o r i g i n a l wavefunction. In quantum mechanical terminology, reducing the p o t e n t i a l i n the wavefunction of a bound system describes a process whereby the system proceeds through successive stationary states of t h i s same system, where each stationary state corresponds to a lower p o t e n t i a l than the previous one, u n t i l the stationary state corresponding to zero p o t e n t i a l i s reached. The wavefunction r e s u l t i n g from reducing the 70 p o t e n t i a l to any value, including zero, i s the wavefunction describing the stationary state corresponding to t h i s reduced value of the p o t e n t i a l . Since i n an unbound system the two types of switching o f f are equivalent and reducing the poten t i a l i n the wavefunc-t i o n describes the r e s u l t of a type I process, i t follows that the r e s u l t . o f reducing the p o t e n t i a l to zero i n the wavefunc-t i o n describes the r e s u l t of p h y s i c a l l y switching 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. This i s supported by the examples of unbound systems which have been analyzed i n chapters two, three, and four. Hence i n a l l unbound systems, the r e s u l t of reducing the po t e n t i a l to zero i n the wavefunc-t i o n i s a free p a r t i c l e wavefunction as i s . expected from experimental observations. 3 . DECREASING THE POTENTIAL IN THE WAVE EQUATION The r e s u l t of decreasing the p o t e n t i a l i n the wave equation w i l l now be discussed. The case of a bound system w i l l be treated f i r s t . I f i n the wave equation f o r a bound system the p o t e n t i a l i s decreased to zero and the other parameters are v a r i e d i n accord with the c r i t e r i a of the system, t h i s method of decreasing the po t e n t i a l obviously corresponds to a.".type I switch o f f . When the r e s u l t i n g equation i s solved i n conjunction with the boundary conditions obtained from the 71 o r i g i n a l ones by decreasing the po t e n t i a l to zero the r e s u l t i s a zero wavefunction which describes the r e s u l t of a type I switch o f f . I f however the p o t e n t i a l i s mathematically set equal to zero i n the equation without influencing any of the other parameters t h i s then corresponds to the p o t e n t i a l being decreased without imposing the c r i t e r i a of the system. I f the r e s u l t i n g equation i s solved i n conjunction with the boundary conditions derived from the o r i g i n a l ones by decreas-i n g the po t e n t i a l to zero, the so l u t i o n Is again a zero wave-f u n c t i o n which again describes the r e s u l t of a type I switch o f f . This at f i r s t appears surprising since the p o t e n t i a l i n the equation was decreased i n a manner analogous to a type II switch o f f . I f , however, t h i s l a t t e r r e s u l t i n g equation i s solved i n conjunction with d i f f e r e n t boundary conditions 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 conditions are 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 wave-fu n c t i o n describing the r e s u l t of a type II switch o f f . The preceding paragraph has demonstrated that the boundary conditions, associated with the wave equation r e s u l t i n g from decreasing the p o t e n t i a l i n the o r i g i n a l equation, determine the type of switch o f f to which decreasing the p o t e n t i a l i n the wave equation corresponds. This i s reasonable i f the following i s considered. Maintaining the boundary conditions of a bound system implies that the system 72 remains the same and that therefore the c r i t e r i a of the bound system are s a t i s f i e d . Hence i f the boundary conditions are maintained f o r a l l values of the po t e n t i a l i t i s apparent t h a t the conditions f o r a type I switch o f f are s a t i s f i e d . I f , however, the boundary conditions are altered while 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 equation i n conjunction with these d i f f e r e n t boundary conditions describes 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 of the i n i t i a l system. This corresponds to a type II switch o f f . I f i n the wave equation f o r an unbound system the p o t e n t i a l i s set equal to zero, no other parameters may be af f e c t e d since f o r such systems the p o t e n t i a l places no r e s t r i c t i o n s on the other parameters. I f the r e s u l t i n g equa-t i o n i s solved i n conjunction with boundary conditions obtained from the o r i g i n a l ones by decreasing the p o t e n t i a l to zero the s o l u t i o n i s an o s c i l l a t i n g free p a r t i c l e wave-fu n c t i o n . Since the boundary conditions are not altered, w i t h the exception of decreasing 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 the above process corresponds to a type I switch o f f which f o r unbound systems i s i d e n t i c a l to a type II switch o f f . Hence the so l u t i o n of the wave equation f o r an unbound system, wi t h the p o t e n t i a l decreased to zero, i n conjunction with the above boundary conditions corresponds to the experimentally observed r e s u l t . 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 section one of chapter one w i l l now be analyzed. In t h i s problem of an electron i n a uniform magnetic f i e l d the system i s a bound one and the f i e l d i s switched o f f i n a f i n i t e time. This switch o f f i s therefore o f type I I . I t therefore follows that the experimental r e s u l t cannot be described by the r e s u l t of reducing 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 because reducing the p o t e n t i a l i n the wavefunction gave zero whereas a plane wave solution, which agreed with experimental observation, was obtained by solving the equation r e s u l t i n g from decreasing the p o t e n t i a l to zero i n the o r i g i n a l equation. This plane wave so l u t i o n was obtained because no boundary conditions were associated with the wave equation. That i t was t h i s absence of accompanying boundary conditions which l e d to the plane wave s o l u t i o n w i l l be shown i n the following paragraph. Since boundary conditions did not accompany the wave equation the type of switch o f f used was not s p e c i f i e d . Neglecting the boundary conditions that the wavefunction goes to zero at - oo i s the same as imposing d i f f e r e n t ones. I f the general s o l u t i o n which o s c i l l a t e s at * oo i s taken as the s o l u t i o n to the wave equation with zero p o t e n t i a l , the e f f e c t i s equivalent to specifying these d i f f e r e n t boundary conditions 7 4 as being done fo r a free p a r t i c l e . This i s what was i n f a c t done. As shown i n the previous section, t h i s change i n boundary conditions r e s u l t s i n a wavefunction describing a type II switch o f f . Hence the s o l u t i o n obtained i n t h i s way describes the experimental 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 II switch o f f i s experimentally carried out. However, since reducing the p o t e n t i a l to zero i n the wave-fu n c t i o n corresponds to a type I switch o f f , these two mathe-matical descriptions of the f i n a l system do not agree. Hence the o r i g i n a l paradox arose because one wavefunction was obtained by dealing with an incompletely s p e c i f i e d bound system, and i t described the r e s u l t of a type II switch o f f , whereas the other wavefunction was obtained by dealing with a f u l l y s p e c i f i e d system and i t described the r e s u l t of a type I switch o f f . I f , however, the o r i g i n a l boundary conditions had been associated with the wave equation i n which the p o t e n t i a l was decreased the s o l u t i o n of t h i s equation would have been i n agreement with the r e s u l t of reducing the p o t e n t i a l i n the wave-function. These i d e n t i c a l r e s u l t s would not have described the p h y s i c a l s i t u a t i o n with the p o t e n t i a l switched o f f since a l l the mathematics would describe the r e s u l t of a type I switch o f f . 75 5. A COMMENT ON THE UNIFORM ELECTRIC FIELD CASE The uniform e l e c t r i c f i e l d system i s an unbound one f o r both p o s i t i v e and negative values of the t o t a l energy, E. As was shown i n chapter four, the sig n i f i c a n c e of the value o f E i s to ind i c a t e the p o s i t i o n at which a p a r t i c l e incident from ^ = + 00 i s c l a s s i c a l l y r e f l e c t e d . Hence i n t h i s case negative E does not ind i c a t e a bound system. Negative E corresponds to r e f l e c t i o n at <^=+oo when the f i e l d i s zero. Hence negative E corresponds to a zero p r o b a b i l i t y of the p a r t i c l e being i n any f i n i t e region of space when the f i e l d i s zero. Rather than a normalizing 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 of a p a r t i c l e with negative E being i n a f i n i t e r egion of space when the p o t e n t i a l i s zero which accounts f o r the zero wavefunction when the p o t e n t i a l i s reduced to zero. 6 . SUMMARY OF THESIS CONCLUSIONS (a) For an unbound system. (i) The r e s u l t of reducing the p o t e n t i a l to zero i n the wavefunction of the system i s an o s c i l l a t i n g function describing a free p a r t i c l e whose k i n e t i c energy i s equal to 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 experimentally switching o f f the p o t e n t i a l i n any manner i s described by both the r e s u l t of reducing the p o t e n t i a l to zero i n the wavefunction and the so l u t i o n of the wave equation with accompanying boundary conditions 76 which are obtained by decreasing the p o t e n t i a l to zero i n the o r i g i n a l wave equation and boundary conditions respectively, ( i i i ) Only i n an unbound system can the p o t e n t i a l be reduced i n the wavefunction or be experimentally switched o f f such that the t o t a l energy remains constant. (b) For a bound system. (i) The r e s u l t of reducing the p o t e n t i a l to zero i n the wavefunction i s a zero wavefunction. ( i i ) Two methods of switching o f f must be d i s t i n -guished. They are defined i n section three of chapter f i v e . ( i i i ) A type I switch o f f to zero i s not experimentally f e a s i b l e . The system r e s u l t i n g from t h i s type of switch o f f Is mathematically described by the r e s u l t of reducing 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 of the equation with accompanying boundary conditions which are obtained by decreasing the p o t e n t i a l to zero i n the o r i g i n a l equation and boundary conditions respectively. Hence, the r e s u l t of reducing the p o t e n t i a l to zero i n the wavefunction does not describe a system r e s u l t i n g from any f e a s i b l e experimental switch o f f . 77 (iv) In some bound systems the po t e n t i a l may be experimentally lowered to a non-zero value i n a type I manner. The r e s u l t of such a lowering i s described by the wavefunction obtained by reducing the po t e n t i a l i n the i n i t i a l wavefunc-t 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 only i n a type II fashion. The r e s u l t of a type II switch o f f cannot be described by the r e s u l t of 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 of a type II switch o f f can be described by the so l u t i o n of the equation obtained from the o r i g i n a l by decreasing the p o t e n t i a l to zero only i f the accompanying boundary conditions are changed to those for a free p a r t i c l e . (c) Block diagram. (i) For any system the block diagram i s always completed i f the corners are occupied by a complete d e s c r i p t i o n of t h e i r respective systems and i f the steps from corner one to corner two and from corner three to corner four correspond to the same type of physical switch o f f . ( i i ) I n order to be closed a block diagram must deal only with a type I switch o f f since the step 78 from corner three to corner four can correspond only to t M s process, ( i i i ) For an unbound system a closed block diagram i s concerned with the actual physical process and i t s fourth corner describes the r e s u l t of an experimental switch o f f . (iv) I n a bound system i n which the p o t e n t i a l goes to zero, a closed block diagram must deal with an experimentally impossible process and the entry i n the fourth corner does not describe the r e s u l t of an experimentally f e a s i b l e procedure, (v) An actual physical switch o f f to zero cannot be expressed as a closed block diagram f o r the case of a bound system, (vi) For some bound systems a p a r t i a l switch o f f to a non-zero value can be expressed In the form of a completed block diagram. General properties of a wavefunction. (i) Since a given wavefunction describes a system with p a r t i c u l a r c r i t e r i a , the wavefunction r e s u l t i n g from a l t e r i n g the value of any parame-ter i n the i n i t i a l wavefunction describes the I n i t i a l system modified such that i t has the new value f o r the parameter and s t i l l s a t i s f i e s a l l the o r i g i n a l c r i t e r i a . 7 9 ( i i ) Changing the value of any parameter i n the wave-function corresponds to the result of physically changing this parameter by the same amount i n a manner such that the characteristic c r i t e r i a of the system are satisfied for the i n i t i a l , f i n a l and a l l intermediate values of this parameter. For any given system this manner of changing the value of the parameter may or may not be experi-mentally feasible. APPENDIX SYSTEMS CONTAINED VITHIN A PHYSICAL CONTAINER A system contained by a physical container w i l l now be discussed i n order to see the e f f e c t of reducing the p o t e n t i a l to zero i n the wavefunction describing such a system. In discussing reducing the p o t e n t i a l i n the wave-fu n c t i o n of such a system one can properly discuss only a system whose dimensions do not need to exceed the dimensions of the container i n order to s a t i s f y the c r i t e r i a of 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 f o r s u f f i c i e n t l y small values of the p o t e n t i a l the radius must exceed the l i n e a r dimensions of any given container, then f o r these small p o t e n t i a l s the system i s not the one which the wavefunction describes. For these small po t e n t i a l s the system has an altered motion due to the superimposed rebound motion. Hence the renormalized wavefunction of the unconstrained system no longer describes the actual behaviour of the system at these small p o t e n t i a l s . Therefore, reducing the p o t e n t i a l i n t h i s wavefunction does not correspond to switching o f f the actual system. To describe such a constrained system f o r these small po t e n t i a l s a d i s t r i b u t i o n function may be used. The p o t e n t i a l would therefore have to be reduced i n t h i s d i s t r i b u -t i o n function. Hence th i s appendix applies only to a system whose dimensions need not exceed those of the container f o r very small p o t e n t i a l s . 8 1 The constraint imposed by the walls o f t h i s container i s expressed by a p o t e n t i a l which abruptly goes to i n f i n i t y . Let t h i s constraining p o t e n t i a l be R where R = 0 for |x{ < a and R=<x> for jx| > a . This p o t e n t i a l imposes two s i g n i f i c a n t changes: the boundary conditions become *¥(+a) = ¥(-<*•) = 0 and the normalization condition i n one dimension becomes Two cases must be distinguished. 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 already 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 cases 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 . The f i r s t w i l l be considered f i r s t . In the f i r s t case the t o t a l energy i s that f o r a harmonic o s c i l l a t o r , that i s 1 ' 5 E - ( s + ^ - ) ^ w c where $ i s not i n general an integer. As the p o t e n t i a l i s reduced coc goes to zero and the r e s u l t i s again a p a r t i c l e with zero t o t a l energy. Hence a constant wave-fun c t i o n r e s u l t s . There would appear to be two choices f o r t h i s constant; namely, - L to s a t i s f y the normalization 2 cc condition or zero to s a t i s f y the boundary conditions. 16 - e x/z f v i Chandrasekhar obtains oC e r ' p{i-€]J as the solution of the f i r s t excited state of a bounded l i n e a r o s c i l l a t o r where o( and £ are constants, X i s a 8 2 power series 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 wavefunction becomes zero. Hence the boundary conditions, rather than normaliza-t i o n condition, are s a t i s f i e d . The second case w i l l now be considered. The t r e a t -ment of 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 mechanical d e s c r i p t i o n of the end r e s u l t w i l l be given. The s i t u a t i o n i s that of 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 walls. "Within t h i s region the p a r t i c l e i s subject to a force of magnitude Kx towards the centre, However the t o t a l energy of the p a r t i c l e i s such that 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 reaches the walls. I t s energy may therefore be written as ' / i K a 1 •*• E 0 where 2^ Ka 1 i s the t o t a l energy asso-c i a t e d with the motion under the harmonic force and E c i s the k i n e t i c energy the p a r t i c l e has when i t reaches a wa l l . I f the p o t e n t i a l i s now switched o f f i n a type I manner (which corresponds to reduction i n the wavefunction) as i n secti o n three of chapter four, the r e s u l t i s a p a r t i c l e with energy E0 bouncing between the walls. In quantum mechanics t h i s r e s u l t i n g system i s described by the wave-f u n c t i o n f o r a free p a r t i c l e constrained to the region The preceding examples suggest the f o l l o v i n g statements: (a) I f the external source p o t e n t i a l i s reduced to zero i n the vavefunction describing a bound system vhich i s further constrained by an abrupt, i n f i n i t e p o t e n t i a l , the r e s u l t i s a zero as vas the case i n the absence of the constraint p o t e n t i a l . (b) I f the external source p o t e n t i a l i s reduced to zero i n the wavefunction describing an unbound system which has an i n f i n i t e constraining p o t e n t i a l superimposed, the r e s u l t i s a free p a r t i c l e wavefunction within the box formed by t h i s i n f i n i t e p o t e n t i a l . (c) The boundary conditions, rather than the normalization condition, are the fundamental c h a r a c t e r i s t i c s of a system. These statements are consistent with the conclusions i n chapter seven. 0 84 BIBLIOGRAPHY 1. S e i t z , F., The Modern Theory of Solids, McGraw-Hill, p. 583 (1940) 2. S c h i f f , L. I., Quantum Mechanics, McGraw-Hill, 2nd ed., PP. 36-37 (1955). 3. Buck, R. C . Advanced Calculus, 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 of Atfrmic C o l l i s i o n s . Oxford, 2nd ed., p. 47 (1949). 6. I b i d . , p. 4 8 . 7. J e f f r e y s , H. and J e f f r e y s , B. S., Methods of Mathematical Phvsics, Cambridge, p. 574 (1950). 8. Watson, G. N., A Treatise on the Theory of Bessel Functions. The MacMillan 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 Non - 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. and L i f s h i t z , E. M., op. c i t . , p. 474. 1 4 . I b i d . , p. 475. 15. H u l l , T. E. and J u l i u s , R. S., Can. J . Phys., M, 914 (1956) . 16. Chandrasekhar, S., Astrophys. J . .9_7_, 263 (1943). 

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