Dispersion theory and (d.p) reactions

by

W, K. BERTRAM

A thesis submitted for examination for the degree of Doctor of Philosophy at the Australian National University

Department of Theoretical Physics School of General Studies Australian National University

STATEMENT OF ORIGINALITY

I hereby declare that, to the best of my knowledge, the work presented in this thesis, excepting those parts which have been explicitly attributed to other authors, is entirely my own. Moreover, this thesis, or any part of it, has never been submitted by me on any previous

occasion, for a degree at this or any other university.

ACKNOWLEDGEMENTS

I wish to express my sincere gratitude to my supervisor, Dr. L.J. Tassie, for suggesting the topic for this thesis and for the guidance, the many useful suggestions and criticisms, which he provided during the time that this work was in progress. I also wish to express my appreciation to Professor H*A. Buchdahl and Dr. M. Andrews for their

interest in this work and for the many useful discussions we had on various aspects of this thesis.

My thanks also go to Dr. G. Crawley for providing me with the results of his measurements of the angular distributions for (d,p) stripping off Lead, and to Miss J. Flint, who has dope such a magnificent job of typing the manuscript with its many complicated equations* I would also like to thank Mr* McMahon and the staff of the Visual Aids Unit for their efforts in photographically reproducing many of the diagrams.

i i

CONTENTS

STATEMENT OF ORIGINALITY i

ACKNOWLEDGEMENTS i

CONTENTS i i

I . INTRODUCTION 1

R e f e r e n c e s 7

I I . DISPERSION RELATIONS AND THE PRE-ACCELERATION PROBLEM

1. The E q u a tio n o f M o tio n 8

2* C a u s a l i t y and th e S c a t t e r i n g A m p litu d e f o r L in e a r

S ystem s 10

3. S c a t t e r i n g o f L ig h t by a Bound E l e c t r o n 12

4 . D is p e r s io n R e l a t i o n s 13

R e f e r e n c e s 17

I I I . INTRODUCTION TO THE PROPAGATOR FORMALISM

1. M otion o f a S in g le P a r t i c l e 18

2. The F r e e P a r t i c l e P r o p a g a to r 19

3. The P r o p a g a to r o f a P a r t i c l e w ith I n t e r n a l

S t r u c t u r e 21

4 . The Time D is p la c e m e n t O p e ra to r 23

5 . S ystem s I n v o lv in g S e v e r a l P a r t i c l e s 24

6. The I n t e g r a l E q u a tio n f o r K 26

R e f e r e n c e s 27

IV. COLLISION THEORY

1. G e n e ra l Remarks 28

2. The P e r t u r b a t i o n T re a tm e n t o f S c a t t e r i n g 30 3 . Second O rder I n e l a s t i c S c a t t e r i n g 31

V. INTERMEDIATE PROCESSES

1. A Postulate 36

2. Connected Diagrams 37

References 40

VI. VERTEX FUNCTIONS

1. The Three-Vertex 41

2. The Vertex Function 44

3. Reduced Widths 45

4. Higher Vertices 49

5. Rules for Finding Reaction Amplitudes 50

6. Note on the Relative Importance of Different

Diagrams 51

References 54

VII. SINGULARITIES OF THE AMPLITUDES

1. Methods for Finding Singularities 55

2. The Landau-Bjorken Rules 56

3. Singularities of Triangle Graphs 58

4. Singularities of the 3-Vertex 61

References 65

VIII. DEUTERON STRIPPING REACTIONS - SINGULARITIES

1. The Landau Singularities 67

2. Singularities due to the Vertex Functions 71

3. Singularities Due to Virtual Scattering 73

(a) Scattering in the Initial and Final States 74

(b) Scattering in the Intermediate State 78

References 79

IX. DEUTERON STRIPPING AMPLITUDES

1. The Plane Wave Born Approximation 80

2. Inclusion of Initial and Final State

Interactions 83

iv

4. T h e D i s t o r t e d W a v e B o r n A p p r o x i m a t i o n 91

R e f e r e n c e s 92

X. P O L O L O G Y

1.

2. T h e P r o c e d u r e o f D u l l e m o n d a n d S c h n i t z e r 95

3. T h e R e s i d u e at t h e P o l e 98

4. S u m m a t i o n o f t h e H i g h P a r t i a l W a v e s i n D W B A 102

R e f e r e n c e s 103

XI. B E H A V I O U R O F D W B A S T R I P P I N G A M P L I T U D E S N E A R T H E

B U T L E R P O L E

1.

2. T h e D W B A A m p l i t u d e 105

3. T h e T e r m M^

m

4. C o n c l u s i o n 114

R e f e r e n c e s 114

X I I . E X T R A P O L A T I O N S

1.

2. A p p l i c a t i o n to E x p e r i m e n t a l D a t a 125

3. S u b - C o u l o m b S t r i p p i n g 128

4. C o n c l u s i o n 130

R e f e r e n c e s 131

D i a g r a m s 132

X I I I . P O L O L O G Y A N D (d,p) D I F F E R E N T I A L C R O S S S E C T I O N S

1.

2. C o n c l u s i o n s 152

D i a g r a m s 153

A P P E N D I X A: E V A L U A T I O N OF T H E C O U L O M B P O L E T E R M 159

A P P E N D I X B: M E T H O D OF E X T R A P O L A T I O N 163

I. INTRODUCTION

Dispersion relations are nothing more than relations, in the form of Cauchy integrals, between the real and imaginary parts of

scattering amplitudes. In general, the scattering amplitude, f(w),

is a function of the frequency, w, of the incident wave or, in the case of quantum mechanical scattering, the energy of the incident particles. If one allows w to become complex, dispersion relations for f(w) can be

obtained when the analytic properties of f(w) are known. Such analytic

properties can, for example, be obtained from causality conditions. If the system under consideration is causal, i.e., if the scattered particles emerge from the target after the arrival of the incident particles, the scattering amplitude has no singularities in the upper half of the complex w-plane.^

Dispersion relations were first introduced into physics by

2 3

Kramers and Kronig in 1926 in the study of the propagation of light in refractive media.

In Chapter II of this thesis we use dispersion relations, similar to the Kraraers-Kronig relations, to investigate the pre­

acceleration problem. The pre-acceleration effect arises when the

4>

Lorentz-Dirac equation is used to describe the motion of a charged

particle. In Sections 1-3 it is shown that the scattering of light

by a bound electron is then non-causal. Dispersion relations for the scattering amplitude are then derived (Section 4) which differ from

the Kramers-Kronig relations. Since these dispersion relations only

2

.

d e t e r m i n e t h e e x i s t e n c e o f p r e - a c c e l e r a t i o n i s c o n s i d e r e d i n S e c t i o n 4.

I n more r e c e n t y e a r s , d i s p e r s i o n r e l a t i o n s h av e been i n t r o d u c e d i n t o o t h e r b r a n c h e s o f p h y s i c s , n o t a b l y i n t h e f i e l d o f e l e m e n t a r y p a r t i c l e p hysics.** They w ere d e r i v e d from quantum f i e l d

6 7

t h e o r y by G o ld b e r g e r * i n 1955 t o compare them w ith e x p e r i m e n t a l d a t a

g

on p i o n - n u c l e o n s c a t t e r i n g . L a t e r i n 1958 M andelstam d e v e lo p e d a scheme f o r w r i t i n g down d i s p e r s i o n r e l a t i o n s by making c e r t a i n

a s s u m p tio n s a b o u t t h e a n a l y t i c p r o p e r t i e s o f t h e S - m a t r i x , and u s i n g t h e i n v a r i a n c e p r i n c i p l e s o f s t r o n g i n t e r a c t i o n s .

9 10

A tte m p ts h av e s i n c e b e e n made 7 t o a p p l y t h e s e m e th o d s , w hich w ere c h i e f l y d e v e lo p e d f o r h i g h e n e r g y p a r t i c l e p h y s i c s , t o

t h e t h e o r y o f low e n e r g y n u c l e a r r e a c t i o n s . T hese a t t e m p t s s u f f e r from t h e d i s a d v a n t a g e t h a t a s s u m p tio n s must be made, w hich a r e a v o i d e d i n t h e more c o n v e n t i o n a l t r e a t m e n t s o f n u c l e a r r e a c t i o n s b a s e d on

S c h r ö d i n g e r ' s e q u a t i o n . D i s p e r s i o n t h e o r y i s n e v e r t h e l e s s q u i t e u s e f u l i n low e n e rg y n u c l e a r r e a c t i o n s f o r i n v e s t i g a t i n g t h e a n a l y t i c p r o p e r t i e s o f r e a c t i o n a m p l i t u d e s .

The r e m a in d e r o f t h i s t h e s i s i s d e v o te d t o t h e i n v e s t i g a t i o n o f t h e s i n g u l a r i t i e s o f t h e a m p l i t u d e s f o r n u c l e a r r e a c t i o n s and t h e u s e o f t h e s e s i n g u l a r i t i e s f o r t h e a n a l y s i s o f e x p e r i m e n t a l d a t a .

f o r a s i n g l e p a r t i c l e . I n S e c t i o n 3 we g e n e r a l i s e t h e s e r e s u l t s t o t h e c a s e w here t h e p a r t i c l e may h a v e some i n t e r n a l s t r u c t u r e . The

c o n n e c t i o n b etw ee n D i r a c ' s ti m e d i s p l a c e m e n t o p e r a t o r and t h e p r o p a g a t o r i s e s t a b l i s h e d i n S e c t i o n 4. F i n a l l y , i n S e c t i o n 5, t h e p r o p a g a t o r f o r m a lis m i s e x te n d e d t o s y stem s i n v o l v i n g s e v e r a l p a r t i c l e s .

C h a p te r IV, S e c t i o n 1, i s a r e v ie w o f t h e c o n n e c t i o n b etw ee n t h e p r o p a g a t o r s and t h e S - m a t r i x . S e c t i o n 2 d e a l s w i t h t h e p e r t u r b a t i o n e x p a n s io n o f t h e s c a t t e r i n g a m p l i t u d e b a s e d on t h e i n t e g r a l e q u a t i o n o f t h e Feynman p r o p a g a t o r . I t i s t h e n shown i n S e c t i o n 3 how t h e p r o p a g a t o r method i s u s e d t o c a l c u l a t e th e seco n d o r d e r te r m o f t h e p e r t u r b a t i o n s e r i e s f o r i n e l a s t i c s c a t t e r i n g . The main d i f f e r e n c e b e tw ee n t h e t r e a t m e n t g i v e n h e r e and t h a t g iv e n e l s e w h e r e i n t h e l i t e r a t u r e i s t h a t t h e i n t e r n a l s t r u c t u r e s o f t h e c o l l i d i n g p a r t i c l e s h a v e b e e n i n c l u d e d .

More g e n e r a l n u c l e a r r e a c t i o n s , w h ich i n v o l v e t h e exchange o f p a r t i c l e s , a r e t r e a t e d i n C h a p te r V. I n S e c t i o n 1 we p o s t u l a t e t h a t t h e S - m a t r i x e le m e n t f o r a c e r t a i n r e a r r a n g e m e n t c o l l i s i o n can be

r e p r e s e n t e d by an i n f i n i t e s e r i e s o f Feynman d ia g r a m s , c o r r e s p o n d i n g t o a l l p o s s i b l e i n t e r m e d i a t e p r o c e s s e s . I n S e c t i o n 2 t h e r e s u l t s , w hich w ere d e r i v e d i n p r e v i o u s c h a p t e r s , a r e \jsed t o d e r i v e t h e a m p li­

t u d e f o r an a r b i t r a r y Feynman d ia g ra m i n te rm s o f t h e a m p l i t u d e s o f i t s com ponents.

4

.

r e l a t i o n s b etw ee n v a r i o u s q u a n t i t i e s w hich a r e of im p o r ta n c e i n n u c l e a r p h y s i c s su ch as r e d u c e d w id th s and s p e c t r o s c o p i c f a c t o r s .

T h is i s f o ll o w e d ( S e c t i o n 4) by some re m a rk s c o n c e r n i n g h i g h e r v e r t i c e s . I n S e c t i o n 5 t h e r u l e s f o r f i n d i n g t h e a m p lit u d e c o r r e s p o n d i n g t o a g iv e n Feynman d ia g ra m a r e g i v e n . T h ese r u l e s f o ll o w from t h e work done i n t h e p r e c e d i n g c h a p t e r s . The r u l e s p r e s e n t e d i n t h i s s e c t i o n have

some f e a t u r e s w h ich a r e n o t found e l s e w h e r e i n t h e l i t e r a t u r e . F i r s t l y , i n t h e momentum r e p r e s e n t a t i o n , we h av e i n c l u d e d t h e r u l e s f o r t r e a t i n g t h e p o t e n t i a l i n t e r a c t i o n s b e tw e e n t h e p a r t i c l e s ( r e a l o r v i r t u a l ) i n t h e d ia g ra m . S e c o n d l y , t h e r u l e s a r e a l s o g iv e n i n t h e c o o r d i n a t e r e p r e s e n t a t i o n . The c o n c l u d i n g s e c t i o n ( S e c t i o n 6) o f C h a p te r VI i s a n o t e on t h e r e l a t i v e im p o r ta n c e o f d i f f e r e n t Feynman d ia g r a m s .

C h a p te r V II r e v ie w s t h e s i n g u l a r i t i e s o f Feynman d ia g ra m s . 12

S e c t i o n s 1-3 d e a l w i t h t h e L a n d a u -B jo rk e n r u l e s f o r f i n d i n g t h e s i n g u l a r i t i e s o f Feynman d ia g r a m s . S e c t i o n 4 d e a l s w i t h t h e s i n g u ­ l a r i t i e s o f t h e n u c l e a r 3 - v e r t e x , w here t h e method f o ll o w e d i s t h a t o f B e r t o c c h i e t a l . ^

I n S e c t i o n s 1 and 2 o f C h a p te r V III t h e p o s i t i o n s o f t h e

s i n g u l a r i t i e s o f some Feynman d ia g ra m s a r e c a l c u l a t e d , u s i n g t h e methods d e s c r i b e d i n C h a p te r V I I , f o r some s p e c i f i c ( d , p ) s t r i p p i n g r e a c t i o n s . I n S e c t i o n 3 , u s i n g t h e method d e v e lo p e d i n C h a p t e r s IV t o VI f o r t r e a t i n g t h e p o t e n t i a l i n t e r a c t i o n s b e tw ee n t h e p a r t i c l e s i n Feynman d ia g r a m s , t h e p o s i t i o n s o f t h e s i n g u l a r i t i e s due to v i r t u a l s c a t t e r i n g a r e d e t e r m i n e d .

is virtually just Shapiro's method of obtaining the Plane Wave Born

Approximation for stripping by the diagram technique. However, our

treatment, in Section 2, of the initial and final state interactions differs from the already existing t r e a t m e n t s . ^ The main source of difference is the fact that we have employed the coordinate representa­

tion rather than the more widely used momentum representation. The

expression for the (d,p) stripping amplitude thus obtained, is found to be identical in appearance to the DWBA amplitude for this process.

In Section 3 the amplitude of a diagram which contains a virtual scattering process in the intermediate state, is shown to correspond to a term in the amplitude of a (d,p) reaction as derived from the plane

wave theory. Due to the great difficulties encountered when attempting

to evaluate terms of this type, they are usually neglected in the

calculations of stripping cross sections. Chapter IX is concluded

(Section 4) with a brief discussion of the DWBA method.

The final four chapters (X to XIII) are concerned mainly with the possible applications of the analytic properties of (d,p) stripping amplitudes to the analysis of experimental data.

In Chapter X the method for obtaining reduced widths of nuclear levels from the angular distributions of (d,p) cross sections by extrapolation is discussed. This method was investigated by Dullemond and Schnitzer (Section 1), who carried out a computer experiment to test

its reliability. However, several important errors were found in this

work so that their results were not at all conclusive. In Sections 2

and 3 the necessary modifications are given to make the computer experi­

way of calculating DWBA amplitudes by analytically summing the high partial waves.

The analytic properties of DWBA amplitudes for (d,p) and

(d,n) reactions are investigated in Chapter XI. This is done with

the aid of integral representations of the wave functions which enter into the amplitude, thus enabling us to take fully into account the Coulomb interactions, something that is very difficult in the Feynman diagram method.

The results of a computer experiment, testing the feasibility of obtaining reduced widths by extrapolation, are given in Chapter XII,

Section 1. In Sections 2 and 3 the extrapolation technique, in a

slightly modified form, is applied to several sets of experimental

data. Section 4 is a discussion of the results of these calculations.

The material in Chapters X and XII has been written into a paper entitled "Polology and (d,p) Reactions", by W.K. Bertram and

L. J. Tassie. This paper has been accepted for publication in The

Physical Review.

A paper based on the material in Chapter XI entitled "The Behaviour of DWBA Stripping Amplitudes Near the Butler Pole", by M. Andrews, W.K. Bertram and L.J. Tassie, has been submitted for publication in The Australian Journal of Physics.

In the final chapter of the thesis, Chapter XIII, differential cross sections calculated using the DWBA, which uses analytic summation of the high partial waves (denoted by DWBA WP), are compared with the

found t h a t c o n s i d e r a b l y l e s s p a r t i a l waves a r e r e q u i r e d when c r o s s s e c t i o n s a r e c a l c u l a t e d w i t h t h e DWBA WP.

R e f e r e n c e s

1. S. M an d elstam , R e p t. Prog« P h y s . , J>5, 100 (1962)

2. H.A. K ram ers, A t t i . C ongr. I n t e r n . F i s i c a , Como, 2, 545 (1927) 3 . R. K r o n ig , J . Opt. Soc. Am., JN2, 547 (1926)

4 . P.A.M. D i r a c , P r o c . Roy.. Soc. (L o n d o n ), A16 7 . 148 (1938) 5. M. G ell-M an n , M.L. G o ld b e r g e r and W, T h i r r i n g , Phys. R e v .,

1 5 , 1612 (1954)

6. M.L. G o l d b e r g e r , Phys. R e v . , _97, 508 (1955) 7. M.L. G o l d b e r g e r , Phys. R e v . , 1 9 , 979 (1955) 8. S. M an d elstam , Phys. R e v . , 112» 1344 (1958) 9. R. Amado, Phys. R e v ., 1 27, 261 (1962)

10. I . S . S h a p i r o , " S e l e c t e d T o p ic s i n N u c le a r T h e o r y " , I n t e r n a t i o n a l Atomic Energy Agency, V ie n n a , 1963

11. L. B e r t o c c h i , C. C e o l i n and M. T o n in , Nuovo C im ento, 18. 770 (1960)

12. L.D. L andau, N u cl. P h y s . , 13» 181 (1959)

8.

II. DISPERSION RELATIONS AND THE PRE-ACCELERATION PROBLEM

1. The Equation of Motion

In the classical theory of electrodynamics the motion of 1 2 3

an electron is described by the Lorentz-Dirac equation * *

= (l/m)F^ + T - ( l / c ^ u ^ u ^ ] , (2.1)

u 2 3

where r is the 4-force acting on the electron, t = -§(e /me ) and

[i

m is the rest mass of the electron. The 4-velocity is denoted by u ; the dot, when it occurs on a 4-vector, denotes differentiation with respect to the proper time, whilst on a 3-vector, it denotes differenti­

ation with respect to ordinary time. The metric is taken to be

(1,1,1,-1). When the velocity v of the electron is small compared with c, the second term in the brackets in (2.1) may be neglected and we get from the first three components of (2.1)

v = (l/m)F(t) + tv . (2.2)

When the force acting on the electron is due to an electromagnetic field only this becomes

v = (e/m)E + (e/mc)v X H + t v • (2.3)

This is in general a set of three coupled differential equations. The problem that we shall consider later cannot be solved very easily using this form of the equation of motion; it is necessary to make a further simplification by neglecting the term (e/mc)v X H which is of

order v/c anyway. Thus finally we have,

The solution of this equation is* writing a(t) = v(t) and putting

c = 1,

a(t) = et^T [a(0)-(1 /m-r) f e t ^TF(t')dt'] (2.5)

If we require a(t) to remain finite as t — 00 (this seems a reasonable

restriction), then we must have

r00 I /

a(0) » (1/nrr) / e TF(t')dt' . (2.6)

J0

After substituting (2.6) in (2.5) and writing s = t'-t we obtain

r 00

a(t) = (1/nrr)[ e S^TF(t+s)ds . (2.7)

J 0

Consider a force F(t) such that

F (t) = 0

F(t) h 0

t ^ 0

t > 0 ,

then for t < 0, (2.7) becomes

-s/t,

a(t) = (1/nrr)/ e F(t+s)ds

J -t

Integrating this equation by parts we find

a(t) = [(et/T)/m] [F(0)+tF ,(0)+t2F"(0)+___ ] .

(

2

.

8

)

Provided F and its derivatives are well behaved at 0 and « , (2.8)

shows then that

a(t) = et^T a(0) for t < 0 (2.9)

The acceleration of the particle reaches about half the value of the

acceleration at t = 0 at a time T = t seconds before the force is

-24

10

I n most t r e a t m e n t s o f t h i s p r ob le m one i n v o k e s H e i s e n b e r g ' s p r i n c i p l e t o show t h a t t h i s p r e - a c c e l e r a t i o n e f f e c t i s n o t o b s e r v a b l e and t h e r e f o r e p h y s i c a l l y q u i t e m e a n i n g l e s s . However, u s i n g some

e l e m e n t a r y t h e o r y o f d i s p e r s i o n r e l a t i o n s we c a n show t h a t p r e - a c c e l e r a ­ t i o n e f f e c t s a r e , i n p r i n c i p l e , o b s e r v a b l e .

2w C a u s a l i t y and t h e S c a t t e r i n g A m pl i t u de f o r L i n e a r Systems

C o n s i d e r t h e s c a t t e r i n g o f e l e c t r o m a g n e t i c ( p l a n e ) waves by a p a r t i c l e . I f t h e i n c i d e n t wave i s monoc hromati c we c a n w r i t e i t as

A ( z , t ) = a ( w ) e iv^ Z ^ . ( 2 . 1 0 )

We h ave c h o s e n t h e wave t o move a l o n g t h e z - a x i s . The s c a t t e r e d wave, i n t h e f o r w a r d d i r e c t i o n , i s t h e n o f t h e form

G ( z , t ) = ( l / z ) a ( w ) f ( w ) e i w ^Z ^ (2.11)

The f u n c t i o n f (w ) i s t h e f o r w a r d s c a t t e r i n g a m p l i t u d e .

I f t h e inc omi ng and o u t g o i n g waves a r e n o t mo n oc h r o m a t i c , ( 2 . 1 0 ) and ( 2 . 1 1 ) become

A ( z , t ) a ( w ) e i w ( z - t ) dw

G ( z , t ) = ( 1 / z ) / a ( w ) f ( w ) e * w^z t ^dw (2.12)

I f we assume t h a t ,

( a ) t h e i n t e n s i t y o f t h e i nc omi ng wave i s f i n i t e , i . e . ,

(b) t h e i n c i d e n t wave r e a c h e s t h e s c a t t e r e r ( a t z = 0) a t t i me t = 0,

A ( 0 , t ) = 0 t < 0 ,

t h e n , i f we a l l o w w t o become complex, c o n d i t i o n s ( a ) and (b) e n s u r e t h a t a(w) h a s no p o l e s i n t h e u p p e r h a l f o f t h e complex w - p l a n e . Let g(w) d e n o t e t h e f u n c t i o n f ( w ) a ( w ) t h e n , i f g(w) d e c r e a s e s s u f f i c i e n t l y r a p i d l y as |w| — 00 we c a n w r i t e f o r z - t > 0 ,

G ( z , t ) =

f

J

wher e C i s t h e c o n t o u r shown i n F i g . l .

Re w

F i g , 1. A c o n t o u r i n t h e complex w - p l a n e

Thu s, i f f (w) h a s no p o l e s i n t h e u p p e r h a l f p l a n e ,

G ( z , t ) = 0 z - t > 0 ,

w h i l s t , i f f(w) d o e s h a v e p o l e s t h e r e , we h a v e i n g e n e r a l ,

12

.

We s e e t h u s t h a t c a u s a l i t y , o r l a c k o f i t , i s c o n t a i n e d i n t h e a n a l y t i c p r o p e r t i e s o f t h e s c a t t e r i n g a m p l i t u d e .

3. S c a t t e r i n g o f L i g h t by a Bound E l e c t r o n

We now c o n s i d e r t h e s c a t t e r i n g o f a p l a n e p o l a r i z e d beam o f m onochrom atic e l e c t r o m a g n e t i c r a d i a t i o n by a bound e l e c t r o n . We s h a l l

o n ly c o n s i d e r t h e n o n - r e l a t i v i s t i c a p p r o x i m a t i o n i n w hich t h e v e l o c i t y o f t h e e l e c t r o n i s s m a l l compared w i t h c and t h e f o r c e a c t i n g on i t i s due t o t h e e l e c t r i c component o f t h e e l e c t r o m a g n e t i c wave o n l y .

We s u p p o se t h a t t h e e l e c t r o n i s o s c i l l a t i n g a b o u t t h e o r i g i n w i t h a n a t u r a l f r e q u e n c y w^, i n a p l a n e p e r p e n d i c u l a r t o t h e z - a x i s .

I f w i s t h e f r e q u e n c y o f t h e in co m in g wave t h e e q u a t i o n o f m o tio n o f t h e e l e c t r o n i s by ( 2 . 4 )

• • •

T ( 1 / T ) r - (wQ2/ T ) r (e/ro O E ^ e iWt ( 2 .1 3 )

Suppose t h a t t h e beam was t u r n e d on i n t h e re m o te p a s t so t h a t a l l t r a n s i e n t e f f e c t s i n t h e m o tio n o f t h e e l e c t r o n s h a v e d i s a p p e a r e d . The s t e a d y s t a t e s o l u t i o n o f ( 2 . 1 3 ) i s t h e n

r = (e/m ) [ 1 / (wQ2-w2-iT w 3 ) ] E Qe iw t , ( 2 .1 4 )

and t h e s c a t t e r e d wave, i n t h e f o r w a r d d i r e c t i o n , a t l a r g e d i s t a n c e s from t h e o r i g i n i s

2 2 e w

mz

1 2 2 . 3

Wq -W “ iw T

V

i w ( z - t )

( 2 .1 5 )

( e . g . , s e e r e f e r e n c e 5 ) .

f(w)

2 2

e w

₁

2 2 . 3 Wq -W -1TW

(2.16)

This function has simple poles at

where

w 1 = i [a+(1/ 3t) ]

2x,i

w. . « ± B a 2-(l/3T")]2 + i[(l/3T)-(a/2)3 , 2,3

- i l l

a = 2 3 [(A+B)3+(A-B)3] ,

(2.17)

A = <w02/t > + (2/27)(1/t3) ,

B = (w02/T)[w02+(4/27)(l/T2))i (2.18)

It can easily be verified that a ^ (2/3t). The scattering amplitude,

therefore, has a simple pole in the upper half of the complex w-plane.

Xw3

AIm w

X w2

• > R e w

Fig.2 . The positions of the singularities of f(w)

4. Dispersion Relations

In order to obtain dispersion relations it is more convenient

to use the polarization amplitude p(w),

14.

then

p(w) = ( e 2/m) [ 1 / (wQ2-w2- i T w 3 ) ]

L e t b =* a + ( 1 / 3t) t h e n , by C a u c h y ' s t h e o r e m ,

( 2 . 2 0 )

p f J a — 2 dw, = 7 r i ( w. i b ) p ( w ) . ( 2 . 2 1 )

J - 0 0

S e p a r a t i n g t h e r e a l an d i m a g i n a r y p a r t s we o b t a i n

r- oo r 00 2

Re p(w) - ---t— — / Re p ( w ' ) d w ' ---— — P/ ^ -"pW— Xm p ( w ' ) d w ' . ( 2 . 2 2 ) ir(w2+b2 ) T lV + b 2 ) J - c c w _w

The c o m pl e x r e f r a c t i v e i n d e x , n ( w ) , o f a h o mogeneous i s o t r o p i c m a t e r i a l i s r e l a t e d t o p(w) by t h e r e l a t i o n ( s e e r e f e r e n c e 5)

n( w) - 1 = 27TNp(w) . ( 2 . 2 3 )

N i s t h e number o f a to ms p e r c u b i c c e n t i m e t r e . Then ( 2 . 2 2 ) becomes

Re n( w) - 1 b

7T(w2+ b 2 )

n ( w ‘ ) - 1 ] dw' 1

2 2

TT(w + b * )

b 2-fww*

w ' - w Im n (w' ) dw' ( 2 . 2 4 )

We a l s o h a v e f r o m ( 2 . 2 0 )

n *( w ) = n ( - w ) , so t h a t we f i n a l l y o b t a i n

C 00 r 00 r

Re n( w) - ---/ [Re n ( w ' ) - l ] d w ' = 1 + ( 2 /tt) P / — “■— - Im n ( w ' ) d w ' .

7T(w + b 2 ) J 0 J 0 w , 2 -w2

( 2 . 2 5 )

I f we compare ( 2 . 2 5 ) w i t h t h e n o rm al K ra m e rs -K ro n ig r e l a t i o n ,

Re n(w)

i + (2/

P

dw.

,2 2

w -w

( 2 .2 6 )

we s e e t h a t i t d i f f e r s by h a v i n g an a d d i t i o n a l t e r m , S, on t h e l e f t ,

r CO

s =

7T(w +b ) J 0

I f we assume t h a t t h e n a t u r a l f r e q u e n c y , w^, o f an e l e c t r o n i n an +15 -1

atom i s o f t h e o r d e r o f 10 s e c we f i n d t h a t

7T(w2+b2 )

~ 10 sec,

U sing ( 2 . 1 7 ) and ( 2 . 2 0 ) t o e v a l u a t e t h e i n t e g r a l ( 2 . 2 7 ) we f i n d

[Re n ( w 1) - 1 ] dw1 = 27T2N(e2/m) (1/t) (1 / [3 a2- ( 1 /3t2 ) ]} ,

and u s i n g t h e f a c t t h a t a ^ 2 / 3t

f

J

r

J n [Re n ( w ' ) - l ] d w ‘ ^ 27T^N(e2/m)T .

I f we assume N t o be o f t h e o r d e r o f 10 ( A v o g a d r o 's number) t h e n

S <

io

.

T h e r e f o r e , from ( 2 . 2 5 ) , Re n(w) must be m e asu red t o b e t t e r t h a n 1 p a r t i n 1 0 ^ f o r t h e te rm S t o be o b s e r v a b l e .

16

.

A

--/

Fig.3 . Arrangement for measuring refractive

indices as described in the text

We have a parallel beam of light travelling from A through the diffracting

material D to a photographic plate at P. At E we have a straight edge

so that the position of its shadow on P gives us the angle of

refraction. Suppose that we take a coordinate system x on the photo­

graphic plate as shown in Fig.3 with the origin x = 0 at the position

of the geometric shadow of E. From the Fresnel theory of diffraction

the intensity of the light on P as a function of x is represented by Fig.4.

7

Fig.4 . The intensity curve for Fresnel diffraction

by a straight edge

s « [ i A / 23^

w here A i s t h e w a v e le n g th o f t h e l i g h t and i i s t h e ( o p t i c a l ) d i s t a n c e b etw ee n E and P ( i » A ) . I f we assume t h a t one c a n d e t e r m i n e t h e p o s i t i o n o f t h e g e o m e tr ic shadow ( o r p e rh a p s b e t t e r t h e p o s i t i o n o f t h e f i r s t maximum o f t h e i n t e n s i t y ) t o an a c c u r a c y o f 1$ o f s , t h e n a f t e r some e l e m e n t a r y c a l c u l a t i o n s we f i n d t h a t t h e u n c e r t a i n t y i n t h e a n g l e o f t h e r e f r a c t e d l i g h t i s

Too (A /2i)J

The u n c e r t a i n t y i n t h e o b s e r v e d r e f r a c t i v e in d e x i s a p p r o x i m a t e l y o f t h e same o r d e r o f m a g n itu d e as A0 t h e r e f o r e , from o u r p r e v i o u s r e s u l t s we would need A0 < 10 I f we c o n s i d e r t h a t t h e w a v e le n g th o f l i g h t

- 4

i n t h e v i s i b l e p a r t o f t h e s p e c tr u m i s a b o u t 10 cm. t h e n we n eed

i > 1 0 ^ cm.

We s e e t h e n t h a t t h e r e q u i r e d e x p e r i m e n t a l a c c u r a c y c a n n o t be o b t a i n e d i f t h e d i s t a n c e b etw ee n t h e p h o t o g r a p h i c p l a t e and t h e edge E i s l e s s t h a n 0.01 l i g h t y e a r s !

R e f e r e n c e s

1. P.A.M. D i r a c , P r o c . Roy. Soc. (London) A16 7 , 148 (1938)

2. J .A . W heeler and R.P. Feynman, Rev. Mod. Phys. JJ7, 157 (1945) 3 . G.N. P l a s s , Rev. Mod. P h y s . , J33, 37 (1961)

4. F. R o h r l i c h , C l a s s i c a l Charged P a r t i c l e s , S e c t i o n s 6 , 7 , A d d is o n -W e s le y , R e a d in g , Mass. (1965)

5. J . H a m ilto n , P r o g r . N u cl. P h y s . , 8, 145 (1960)

18

III. INTRODUCTION TO THE PROPAGATOR FORMALISM

1. Motion of a Single Particle

The quantum mechanical equation of motion of a single particle,

whose Hamiltonian is H, is in the Schrödinger picture:

m (3.1)

(We adopt the convention ft = c = 1.) if is the Schrödinger state of

the particle. In the Schrödinger representation if is in general a

function of space and time coordinates: if = if(r,t) •

If ifCr.j,t.j) is the wave function of the particle at some

time t^, then the wave function ^(r^jt^) at time t^ (.t^ > t.j) can be

written in the form,

if(r2, t 2 ) = K+ (r

2 ’C

(3.2)

Equation (3.2) is to be regarded as the definition of the function

K (r ,t :r ,t ), this function being known as the propagator or the

+ 2 2 1 1

1-4

Green's function of the particle. (The + on the K denotes that

it is defined for t > t^ only.)

We introduce the more convenient notation by writing

and

K+ (2:l) =

n n = YOyt^)

>

1 denotes r.,t^

1 " r l

I n t h i s n o t a t i o n ( 3 . 2 ) r e a d s

T ( 2 ) =

J

The p r o p a g a t o r K+ ( 2 : l ) s a t i s f i e s t h e e q u a t i o n

[1 S t" ‘ H( 2) ]K+ ( 2 s l ) = i&( 2 : 1 ) , ( 3 . 4 )

wh er e t h e 6 - f u n c t i o n on t h e r i g h t i s d e f i n e d as

5 ( 2 : 1 ) = 5 ( r 2- r 1) 5 ( t 2* t 1) . ( 3 . 5 )

A l s o , K ( 2 : l ) h a s t h e f o l l o w i n g p r o p e r t i e s ,

K+ ( 2 : l ) - ö G ^ - r j ) a s t 2 - , ( 3 . 6 )

K+ ( 2 : l ) = /k+ ( 2 : 3 ) K + ( 3 : 1 ) < 0 t ] < t 3 < t 2 , ( 3 . 7 )

a n d , when t h e H a m i l t o n i a n does n o t depend on t h e t i me e x p l i c i t l y ,

- i E ( t - t )

K ( 2 : 1 ) = Z ♦ n ( 2 ) * „ ( 1 ) e ( t , - t , ) e 1 2 * * * * , ( 3 . 8 )

+ n n n 2 1

wh er e 4>n (x) a r e t h e e i g e n f u n c t i o n s of t h e H a m i l t o n i a n b e l o n g i n g t o t h e e i g e n v a l u e s E^, and 0 ( t ) i s t h e s t e p f u n c t i o n .

1 2 2. The F r e e P a r t i c l e P r o p a g a t o r *

The e q u a t i o n f o r t h e f r e e p a r t i c l e p r o p a g a t o r i s

0 3 ^ + M V2 2) K+ ° ( 2 : 1 ) = i 8 ( 2 : 1 ) • ( 3 - 9)

Le t t h e F o u r i e r t r a n s f o r m o f K+^ ( 2 : l ) be d e n o t e d by K+^ ( p , E ) s uch t h a t

0 , r ” . 0 _ i ( P*72 r Et2 i ) _

20

.

where we use the notation and t ^ = - t^. We also

have

5(2:1) = 1/(27T)

i(p.r -Et )

e dpdE (3.11)

Using these in equation (3.9) we find

K+°(P,E)

Substituting this into (3.10) one gets

0 4 l~ ^P*r21 3 / e

K+ (2:1) = -[(2Mi)/(2ir) ]/e 2

-V

-iE(t2-t])

p2-2ME

This integral, however, is not defined for, in the integration with

respect to E, there is a pole in the integrand at E = p /2M. In order

to obtain the correct form of K+ ^ the singularity is moved off the

real axis by a small amount e and the limit as e — 0 is then taken. This procedure yields two different functions depending on whether

we place the singularity above or below the real axis.

Fig.5 . Contours in the complex E-plane needed to

L et be t h e i n t e g r a l

S

- i E t p -2ME±ie

>

when C i s t h e c o n t o u r shown i n F i g . 5 ( a ) ( b ) . We want t h e i n t e g r a n d t o v a n i s h o v er t h e l a r g e s e m i c i r c l e . L et E = x + i y , t h e n t h e

— j|_xt y t

e x p o n e n t i a l te r m i n t h e i n t e g r a n d y i e l d s e e , h e n c e f o r t > 0 we must u s e (b) and f o r t < 0 , u s e ( a ) . We s e e th u s t h a t

(2:1) [ ( 2 M i ) / ( 2 J r ) 4 ] / "

-l &-lt; 3 ? . r

p -2ME-i€

d pdE ( 3 . 1 3 )

We can a l s o d e f i n e a p r o p a g a t o r :

i(P'VEt2l)

K ° ( 2 : 1) = [ ( 2 M i ) / ( 2 i r ) 4 ] / -£—t--- d3 pdE . ( 3 .1 4 )

J p -2 M E + i€

K _ ^ ( 2 :1 ) i s n o n - z e r o o n ly when t ^ < t ^ .

I n f u t u r e we s h a l l o m it t h e ± s u f f i x e s on t h e K*s and w r i t e s im p ly K (2 :1 ) w h ich i s t o be u n d e r s t o o d as r e p r e s e n t i n g K when

T

t ^ > t^ and K when t ^ < t ^ .

3 . The P r o p a g a t o r o f a P a r t i c l e w i t h I n t e r n a l S t r u c t u r e

We n e x t c o n s i d e r t h e p roblem o f f i n d i n g t h e p r o p a g a t o r f o r a p a r t i c l e w h ich h a s some i n t e r n a l s t r u c t u r e . When we come t o t h e p ro b lem o f e v a l u a t i n g Feynman d ia g ra m s f o r n u c l e a r r e a c t i o n s l a t e r on we s h a l l f i n d t h a t we o n ly need t h e f r e e p a r t i c l e p r o p a g a t o r s o f t h e

22

C o n s id e r a c o m p o s ite f r e e p a r t i c l e w hich c o n s i s t s o f s e v e r a l p a r t i c l e s bound t o g e t h e r by some k in d o f n u c l e a r p o t e n t i a l s . L et t h e v a r i a b l e x d e n o t e t h e i n t e r n a l d e g r e e s o f freedom and l e t r d e n o t e t h e c o o r d i n a t e s o f t h e c e n t r e o f mass o f t h e p a r t i c l e .

The H a m i lto n i a n d e s c r i b i n g t h e e x t e r n a l m o tio n o f t h e p a r t i c l e i s d e n o te d by ( i n t h i s c a s e t h e f r e e - p a r t i c l e H a m i lto n i a n ) and f o r t h e i n t e r n a l m o tio n we h a v e H_ , t h e n t h e t o t a l H a m i l t o n i a n i s

I n

H = H _ + H

Ex I n ( 3 .1 5 )

We expand t h e wave f u n c t i o n ¥ ( r , x , t ) i n te rm s o f t h e e i g e n f u n c t i o n s \jr (r)<t> (x) o f H, w h e re ,

s t

t h e n ,

(H E x + W

^ t

0 0

= (W

V

? ) V X)

,

= & W W W ( 3 .1 6 )

U sin g t h e tim e d i s p l a c e m e n t o p e r a t o r we a l s o h av e

- i H ( t - t )

y { r 2 >x2 > t 2 ) = e Y Cr 2 Ȋc2 * 1 1) >

£ t c s t ( t 1)'l' s <^ 2 )lt' t (x 2 ) e

- i ( E s+et ) ( V t 1) _{( 3 .1 7 )}

and from ( 3 .1 6 )

( t ) = Ns *(r)<t> * ( x ) ¥ ( r , x , t ) d r d x

J s t ( 3 .1 8 )

U sing ( 3 . 1 8 ) i n ( 3 . 1 7 ) we f i n d

f - i ( E -he ) ( t - t )

s ? t V f f 1) * t * ( x l )’|,8 f f 2) * t (X2) e 3 6

K ( 2 :1 ) - Z <}>t ( x 2 )<t)t * ( x 1) e■i £ t ( t 2 ' t l )

_{? V W ?}

₁

_)e

-

1

E.<V ti)

We s e e t h e n , t h a t when t h e e i g e n f u n c t i o n s and e i g e n v a l u e s o f t h e

i n t e r n a l H a m i lto n i a n a r e <t>t (x) and r e s p e c t i v e l y , t h e n t h e p r o p a g a t o r f o r t h e p a r t i c l e i s

0 n

K ( 2 : 1 ) -

E

t t z t I C.X

0

K Ex( 2 : 1 ) i s t h e o r d i n a r y f r e e p a r t i c l e p r o p a g a t o r .

I f we w is h t o c o n s i d e r t h e c a s e i n w hich t h e p a r t i c l e p o s s e s s e s i n t r i n s i c s p i n we m ust i n c l u d e th e a p p r o p r i a t e s p i n wave f u n c t i o n s i n <Kx). The in d e x t i s t h e n u n d e r s t o o d t o i n c l u d e t h e r e l e v a n t s p i n quantum num bers.

4. The Time D is p la c e m e n t O p e r a to r

Suppose | t ^ ) and | t ^ ) a r e t h e s t a t e s ( i n t h e S c h r ö d in g e r p i c t u r e ) o f a s y s te m a t tim e s t^ and t 2 r e s p e c t i v e l y . Then t h e ti m e d i s p l a c e m e n t o p e r a t o r T i s d e f i n e d by t h e r e l a t i o n

| t 2 ) = T2 , | t 1) . ( 3 . 2 2 )

T ak in g t h e S c h r ö d in g e r r e p r e s e n t a t i o n o f t h i s e q u a t i o n and a l s o i n s e r t i n g t h e u n i t o p e r a t o r / | x ^ ) ( x ^ | d x ^ b e tw e e n T and | t ^ ) we o b t a i n

( X2

1

=

I t i^dx]

I n our n o t a t i o n t h i s i s j u s t

t ( 2 ) = / <^2 !T2 l l x 1)^( 1) d^ > t h e r e f o r e

i . e . , t h e p r o p a g a t o r K i s t h e S c h r ö d i n g e r r e p r e s e n t a t i v e o f t h e tim e d i s p l a c e m e n t o p e r a t o r .

2 4

.

1 2 5. Systems I n v o l v i n g S e v e r a l P a r t i c l e s *

I n o r d e r t o f i n d t h e p r o p a g a t o r f o r a sy ste m o f s e v e r a l p a r t i c l e s , w h ich may be m u t u a l l y i n t e r a c t i n g , i t s u f f i c e s t o c o n s i d e r s y stem s i n w hich o n ly two p a r t i c l e s a r e p r e s e n t . The r e s u l t s o b t a i n e d f o r t h i s c a s e a r e e a s i l y g e n e r a l i z e d t o c a s e s w here more t h a n two p a r t i c l e s a r e p r e s e n t .

The wave f u n c t i o n o f a two p a r t i c l e s y ste m i s w r i t t e n as ' f ( r ^ , r2* t ) w h ere r^ r e f e r s t o p a r t i c l e a and r ^ r e f e r s t o b. T h i s wave f u n c t i o n s a t i s f i e s t h e t w o - p a r t i c l e S c h r ö d in g e r e q u a t i o n

i

JL

at

w here 1 ,2 d e n o t e s t h e f a c t t h a t i n t h e s e t s o f c o o r d i n a t e s r . j , t . j and r ^ j t ^ » t^ = t ^ . As b e f o r e we c a n d e f i n e a p r o p a g a t o r by t h e r e l a t i o n

¥ ( 3 7 4 ) - j K ( 3 7 4 : 7 7 2 ) ¥ ( 7 7 2 ) d U d 2 . ( 3 .2 5 )

Then p r o c e e d i n g e x a c t l y as i n S e c t i o n 1 we f i n d t h a t K s a t i s f i e s t h e e q u a t i o n

K (7 7 4 :7 7 2 ) = i& (3 :1 ) 5 ( 4 : 2 ) . ( 3 .2 6 )

I n p a r t i c u l a r , when t h e r e i s no i n t e r a c t i o n b e tw ee n a and b t h e S c h r ö d in g e r e q u a t i o n i s s e p a r a b l e and we h av e

i — - Ha (3 ) - Hb ( 4 ) - Ha b ( 3 , 4 )

w here on t h e r i g h t we must o f c o u r s e h av e t ^ = t ^ and t^ = t ^ . We s h a l l c a l l K ( 3 , 4 : 1 , 2 ) t h e ' e q u a l ti m e s p r o p a g a t o r ' .

Suppose t h e i n t e r a c t i o n H a m i lto n i a n i s n o n - z e r o o n l y i n a f i n i t e tim e i n t e r v a l * i . e . ,

Hab(t) * 0 t 5 s t s c7 = 0 e l s e w h e r e *

Then t h e p r o p a g a t o r f o r t h i s i n t e r v a l i s K , ( 5 , 6 : 7 * 8 ) and s a t i s f i e s

a b

e q u a t i o n ( 3 . 2 6 ) .

L e t 1 , 2 , 3 e t c . , be s p a c e tim e p o i n t s w i t h t^ =» t ^ < t ^ and t^ = t ^ > t ^ . Then t h e p r o p a g a t o r Ka b ( 3 , 4 : l , 2 ) can be w r i t t e n by (3*11) and ( 3 , 2 7 ) ,

K a b ( 3 7 4 f i7 2 ) =* /Ka (3 :7 ) K b ( 4 :8 ) K a b (TT8:5T6)Ka ( 5 : l ) K b ( 6 : 2 ) d ( 5 , ^ , ? , ^ ) , ( 3 . 2 8 )

We c an now d e f i n e w h at we s h a l l c a l l t h e 'u n e q u a l t i m e s p r o p a g a t o r ' by g e n e r a l i z i n g ( 3 . 2 8 ) t o t h e c a s e w here t ^ £ t ^ and t^ £ t ^ ; t h e

p r o p a g a t o r i s t h e n d e n o te d by Ka ^ ( 3 , 4 : 1 , 2 ) . I n p a r t i c u l a r , when t ? - t , we g e t

Ka b ( 3 , 4 ; 1 , 2 ) * Ka O : DKb ( 4 : 2 ) . ( 3 .2 9 )

I t may be o f i n t e r e s t t o n o t e t h a t th e wave m e c h a n ic s f o r a many p a r t i c l e s y s te m , w hich one o b t a i n s i f one u s e s t h e s e u n e q u a l tim e p r o p a g a t o r s , i n p a r t i c u l a r e q u a t i o n ( 3 . 2 9 ) , i s c l o s e l y r e l a t e d t o t h e r e l a t i v i s t i c m e c h an ic s d e v e lo p e d by P.A.M. D i r a c i n 1 9 3 2 .3

2 6

.

e q u a t i o n ( 3 . 2 1 ) . I f t h e r e i s an i n t e r a c t i o n b etw een a and b i t i s n o t p o s s i b l e t o s e p a r a t e t h e i n t e r n a l m o tio n s o f a and b from t h e m o tio n s o f t h e i r c e n t r e s o f m ass. I n f a c t t h e p a r t i c l e s l o o s e t h e i r i d e n t i t y and t h e s y ste m (a+b) c a n o n ly be r e g a r d e d as an ( n ^ + n ^ ) - p a r t i d e s y s te m , w here n and n, a r e t h e numbers o f component p a r t i c l e s i n a and b,

a b

6. The I n t e g r a l E q u a t io n f o r K

L et K ( 2 : l ) d e n o t e t h e e q u a l tim e s p r o p a g a t o r f o r a many

p a r t i c l e s y ste m t h e H a m i lto n i a n o f w hich i s H = H^ + U, w here U r e p r e s e n t s t h e i n t e r a c t i o n b e tw e e n t h e p a r t i c l e s and i s t h e r e m a in in g p a r t o f t h e H a m i lto n i a n ( u s u a l l y t h e f r e e p a r t i c l e H a m i l t o n i a n ) . The l a b e l s 1 and 2 o f K r e p r e s e n t t h e tim e t and t h e c o o r d i n a t e s o f a l l t h e p a r t i c l e s o f t h e s y ste m . K ( 2 :1 ) t h e n s a t i s f i e s t h e e q u a t i o n :

aT - V 2) * u(2)

T h i s e q u a t i o n c a n be r e w r i t t e n i n t h e form o f an i n t e g r a l e q u a t i o n :

K ( 2 :1 ) = K ( 2 : 1 ) + / k° ( 2 :3 ) [ - i U ( 3 ) ] K ( 3 : l ) d 3 ( 3 . 3 1 )

w here K ( 2 : 1 ) s a t i s f i e s t h e e q u a t i o n

H0(2

)

- Ha(2)

d t “ 0 2

K ° ( 2 : 1 ) - U ( 2 ) K ° ( 2 :1 ) + j:1 - g j - K ° ( 2 :3 ) [ - i U ( 3 ) ] K ( 3 :1 )d3 + J 2

+ / [HQ( 2 ) - U ( 2 ) ] K ( 2 : 3 ) [ - 1 U ( 3 ) ]K ( 3 : 1 ) d3

1 8 ( 2 : 1 ) - U ( 2 ) K ° ( 2 :1 ) + U ( 2 ) K ( 2 : 1 ) - U ( 2 ) / K ° ( 2 : 3 ) I - i U ( 3 ) ] K ( 3 : 1 )d3 ,

I S ( 2 : 1 )

R e f e r e n c e s

1

.

2

,

3 . 4. 5.

R.P. Feynman and A. R. H ib b s; Quantum M echanics and P ath I n t e g r a l s , C h a p te r 4 , M cGraw-Hill (1965)

T. Wu and T. Ohmara, Quantum Theory o f S c a t t e r i n g , p . 2 9 9 , P r e n t i c e H a l l (1962)

R .P. Feynman, Rev. Mod. P h y s . , 22» 367 (1948) R .P. Feynman, P hys. R e v . , _76> 6 (1949)

2 8

.

IV. COLLISION THEORY

G e n e r a l Remarks

L et us c o n s i d e r a r e a r r a n g e m e n t c o l l i s i o n o f t h e form

A + x — B + y ( 4 . 1 )

w h ere A ,x , B ,y a r e c o m p o s ite p a r t i c l e s . I n t h e i n c i d e n t c h a n n e l t h e t o t a l H a m i lto n i a n o f t h e s y s te m may be c o n s i d e r e d as c o n s i s t i n g o f two p a r t s ,

w here may be r e g a r d e d a s a sum o f two body i n t e r a c t i o n s b etw ee n t h e p a r t i c l e s w hich c o n s t i t u t e x and t h o s e o f A. F u r th e r m o r e we assume t h a t V("r) — 0 f a s t e r t h a n 1 / |Ir| when \~r\ — 00 • S i m i l a r l y we can w r i t e t h e H a m i lto n i a n i n t h e e x i t c h a n n e l as

The p ro b lem i s now t h e f o l l o w i n g : I f i n i t i a l l y ( t — -°°) when t h e p a r t i c l e s A and x a r e i n f i n i t e l y s e p a r a t e d , t h e sys tern was fo u n d t o be i n an e i g e n s t a t e o f t h e H a m i lto n i a n H_^ , we w ish t o f i n d t h e p r o b a b i l i t y when t — 00 o f f i n d i n g t h e s y ste m i n an e i g e n s t a t e X^(x) o f H^ ( t h e p a r t i c l e s B and y b e i n g i n f i n i t e l y s e p a r a t e d ) .

The a s y m p t o t i c p r o p a g a t o r f o r t h e m o tio n o f t h e s y s te m i s d e n o te d by K ( 2 :T ) , and i s d e f i n e d by

H " Hi + VAx^>

H = H + V

By ( 4 . 3 )

K ( 2 :1 ) « Lira K ( 2 : l ) ( 4 . 4 )

S fi =

Jxf*(2)K(2:T)xi(1)d(<i:2)

We can also write

K(2:l) = ^ £ ( 2 :0)K(0:T)d0 , (4.6)

then the functions defined by the relations

v.(0) = /k(0 , (4.7a)

and

+f(0) = /ic(0:2)xf(2)d

(4.7b)

are solutions at t = 0 of the Schrödinger equation with the full Hamiltonian

i[(^i f)/ät] = H\|/\ f

.

(4.8)

The matrix elements S as defined above are just the elements of the usual S-matrix. This correspondence can be established from

.

4

Newton s treatment on the S-matrix if one notes that the functions and defined by equations (2.7a) and (2.7b) are just the

and \j/ ^ states normally used in S-matrix theory. In this same work a fairly comprehensive discussion can be found on the problems

associated with the S-matrix formulation of rearrangement collisions (see also reference 6).

We can now write

Sfi = &fi

+ (2n-)4iTfi

(4.9)

and

Tfi = Mfi5(V V 6(Y V ’ (4,10)

3 0

.

The r e a c t i o n a m p l i t u d e i s r e l a t e d t o t h e d i f f e r e n t i a l c r o s s s e c t i o n * .. . . 2 , 3 , 4 , 6

f o r t h e r e a c t i o n

f

- f r *

*

<*•»>

M M

w here M. = TT'77*" I s t h e r e d u c e d mass o f A and x.

Ax M.+M

A x

2. The P e r t u r b a t i o n T r e a tm e n t o f S c a t t e r i n g ^

I n t h e p a r t i c u l a r c a s e o f s c a t t e r i n g t h e p a r t i c l e s on t h e l e f t o f ( 4 . 1 ) a r e t h e same a s t h o s e on t h e r i g h t . I n t h i s c a s e we h a v e t h e n so t h a t t h e i n i t i a l and f i n a l s t a t e s X ^ and x^ a r e o r t h o g o n a l . U sing t h e i n t e g r a l e q u a t i o n o f K ( 2 :1 ) we f i n d t h a t t h e t r a n s i t i o n m a t r i x e le m e n t T - . becomes

f l

Tf . - -(2 7 r)’ 4 / x f * ( 2 ) K ° ( 2 : 3 ) V ( 3 ) K(3 : T ) Xi ( 1 ) d ( 1 , 2 , 3 ) , ( 4 . 1 2 )

and t h e t r a n s i t i o n o p e r a t o r i s

T = -(27T)“ 4 f K ° (2 :3 ) V ( 3 ) K ( 3 :T )d 3 . ( 4 . 1 3 )

( v ( x ) i s t h e i n t e r a c t i o n b e tw ee n t h e p a r t i c l e s A and x . )

From t h e s e e q u a t i o n s we c an s e e t h a t t h e s c a t t e r i n g p ro b le m may be c o n s i d e r e d s o l v e d i f t h e p r o p a g a t o r K (2 :T ) f o r t h e p e r t u r b e d H a m i lto n i a n H i s known. I n p r a c t i c e i t i s u s u a l l y i m p o s s i b l e t o f i n d K ( 2 : l ) e x a c t l y and one h a s t o b e s a t i s f i e d w i t h a p p r o x i m a t i o n s . We s h a l l c o n s i d e r two d i s t i n c t ways o f d o in g t h i s , t h e f i r s t one b e i n g t h e p e r t u r b a t i o n m ethod.

particle; the perturbing potential V is assumed to be small in the sense that it depends on some small parameter e such that V — 0 when € — 0. Now we have the integral equation for K(2:l),

K(2:1) = K°(2:l) + Jk°(2:3)[-iV(3)]K(3:1)d3

We can solve this by iteration and obtain

K(2:l) = K°(2:1) + K 1(2:l) + K2 (2:1) + .... (4.15) where

^(2:1) = /k°(2:3)[— iV(3)]K ° (3:1)d3

K2 (2:1) = Jk°(2:3)[-iV(3)]K°(3:4)[iV(4)]K°(4:1)d(3,4) , etc. (4.16)

Hence the transition operator becomes

T - T (1) + T (2) + .... (4.17)

where

[l/(27T)4 ]/K0 (2:3)V(3)K°(3:l)d3 ,

x (2) . _{[i/(2ir)4 ]/lC0 (2:3)V(3)K0(3:4)V(4)K0 (4:1)d(3,4) , etc. (4.18)}

Thus we have obtained a power series in

V

V#

3. Second Order Inelastic Scattering

We consider the scattering of a particle 'a* by a particle 1b * both of which may be composite particles. We denote the internal

a — a b — b

3 2.

H = - ( l / 2 M a )Va 2 - (l/2 M b )Vb 2 + Ha + Hb + V

H and H a r e t h e H a m i lto n i a n s f o r t h e i n t e r n a l m o tio n s o f a and b .

a b

V , i s t h e i n t e r a c t i o n b e tw ee n a and b . We l e t 4> (x ) d e n o t e t h e

ab n a

a b

e i g e n f u n c t i o n s o f b e l o n g i n g t o e i g e n v a l u e s and 0^ t h e e i g e n ­ f u n c t i o n s o f H, b e l o n g i n g t o e i g e n v a l u e s e \ The i n t e r a c t i o n V , i s

d n ab

t a k e n t o be o f t h e form:

a b —a —b.

Vab = V^X , x >r “ r ) ( 4 . 1 9 )

We s h a l l n o t c o n s i d e r s p i n - o r b i t t y p e i n t e r a c t i o n s . From ( 4 ,1 8 ) we s e e t h a t t h e t r a n s i t i o n o p e r a t o r f o r t h i s p r o c e s s i s

T(2) = [ i / ( 2 7 r ) 4 ] /K0a b( 3 , 4 : 5 7 6 ) V ( 5 , 6 ) K 0a b( 5 7 6 : 7 7 8 ) V ( 7 , 8 ) K 0a b( T 7 8 : T , 2 ) d ( 5 . 6 , 7 , 8 :

U sing ( 3 . 2 7 ) t h i s can be w r i t t e n a s

T (2) = [ i / ( 2 7 r ) 4 ] / K °a ( 3 : 5 )K 0b ( 4 : 6 ) V ( 5 , 6 ) K ° a ( 5: 7) Kb ( 6 : 8 ) V ( 7 , 8 ) K 0a (7: T) X

X K b ( 8 : 2 ) d ( 5 , 6 , 7 , 8 ) ( 4 . 2 0 )

( 4 .2 0 ) c a n be r e p r e s e n t e d by a Feynman d ia g ra m :

Ka ( 7 ; , ) --- ^

---Ka (3;7>

---K ( 3 ; 5 ) a

^

---i v(7, 8) | V( 5, 6)

- n 9 * " - “

V < 8 ; 2 ) 0 KhU( 6 ; 8 ) V ( 4 ; 6 )

F i g , 6 . A Feynman d ia g ra m r e p r e s e n t i n g a seco n d o r d e r

s c a t t e r i n g p r o c e s s i n t h e c o o r d i n a t e r e p r e s e n t a t i o n

ilc 3 P 3 -nT b b . a b ^ o\ A a , a , 1Ki *r l b , b 1Ki , r 2

0 , 2 ) = 4>s ( x ] ) e 4>t ( x 2 ) e

.i» a — a .-p b —• b

l k , r 0 . . i k , r .

, a b / a A a / a \ f 3 ,4 b b. f 4

\Jrf ( 3 , 4 ) = 4>u ( x 3 ) e <>v ( x 4 ) e ( 4 . 2 1 )

On c a r r y i n g o u t t h e i n t e g r a t i o n s o v e r 1 , 2 , 3 , % we f i n d

'|ri a b ( 7 , 8 ) = /K0a b ( 7 , 8 : l , 2 H i a b ( 1 , 2 ) d ( <i, 2 )

i ( E a+ E . b+e a +e b ) t

-= ^ i a b ( % 8 ) e ( 4 . 2 1 a )

<ei b

wh er e and E^ a r e t h e k i n e t i c e n e r g i e s o f a and b r e s p e c t i v e l y . ab

We o b t a i n a s i m i l a r r e l a t i o n f o r ( 4 , 6 ) . Then on u s i n g ( 3 . 1 3 ) we o b t a i n

Tf i = [i/(27T)4 ][4mamb/(2 7 r)8 )/4lua* ( X5a )4.vb* ( x 6b ) e

.

i ( E f +Ef ) t 5 V(x5a . x 6b J 5a - ? 6b ) S * > s X aV > * . a , b i ( p . r _ _ - E t )

. . . * . - i e t c_ - i e t__ e *a 57 a 57 b b b* b a 57 ß 57 e ... ... ...

A a

ß 6

P

8

a a a

i ( V r 68 “ Ebt 5 7 ) _a h _{b — a —} a b _d. - i ( E *+E b-K *+€ b) t ?_{t i s t /} £

‘ 2

pb - 2Mb V i€

V(x7 >x 8 , r 7 " r 8 ) e

i ( 7 . a .7 _ a+C b . r b) . . , -i

. e 1 7 1 8 * s a ( x 7 a ) * t b (xg ) d ( 5 , 6 , 7 , 8 ) d J pa dEa dJ pbdEb , ( 4 . 2 2 )

wh er e r

^

( u , v | v ( r 5a - r 6b ) | a , ß )

m f * u

, r

5

- r

6

.

x ^ b (x6b)dx5adx6b ,

3 4

.

( u , v | V(p) | a , ß ) = [ 1 / (27r)3^ 2 ] j^ ( u , v | V ( r ) | a , ß ) e i p *r d3r

S u b s t i t u t i n g t h i s i n ( 4 . 2 2 ) we c a n t h e n c a r r y o u t t h e i n t e g r a t i o n o v e r r _ , r - , r _ , r 0 , t _ , t ^ and o b t a i n

5 6 7 8 5 7

r

( Pa2-2V a - i e ) ( p b2-2flbEb - i e ) l f i = 1 (2TT)6

&(-lcf a - p 1-fpa ) X

X 6 ( - k f b+ p 1+pb )S ( - P a - P 2+ k i a )5 ( “ Pb+P2+^ i b ^B ^Ef a+Ef b+€u a+€v b *

■ea a' e ßb’ Ea ' Eb) 6 ( € a a'f€ß b+Ea'fEb*Ei a- Ei b- € s a- e t b )d 3 (P a ,P b ’ Pl ’ P2 ) X

X dE dEL a b

F i n a l l y i n t e g r a t i n g o v e r p, >p,,p,.»E, > we g e t

b I 2 b

T = i f i

4M M (27T)

-r a — a —

^ v f ( u >viV(pa - k f ) | a , ß ) ( a , ß | v ( k i - p a > | s , t > 6 o£ß

J

a a a b b b

d p dE X a a

X B f f i aÄ . b-iTf a-lEf b ) 6 ( E i a+e s a+Ei b4 € t;b- E f a - € ua - E £b- 6 v b ) , ( 4 .2 3 ) where

— r a , T b — Pb = kf + k f • Pa ’

E = E / + e a + E b + € b - c a - e b - E .

b f u f v a ß a

The s c a t t e r i n g a m p l i t u d e i s j u s t T w i t h o u t t h e d e l t a f u n c t i o n s .

The e x p r e s s i o n ( 4 . 2 3 ) c a n be- r e p r e s e n t e d by a d i a g r a m ( F i g . 7) w h e r e i n t h e f a c t o r s which a p p e a r i n t h e i n t e g r a n d o f ( 4 . 2 3 ) a r e

by a s s o c i a t i n g w i t h each i n t e r n a l l i n e o f t h e d ia g ra m t h e c o r r e s p o n d i n g f a c t o r i n t h e i n t e g r a n d o f

r- a „ a , a k . , E. +€

__i---1----V

-p , E +e a cL a

•p a a a

k E +€

__ f ___

b~

I 0 - 1

I (p -2M E - i e ) < a , ß | v f E . a- i a ) | s , t > |

a a a

r Tb....„ B^Ti'

k . , E. +€ l 9 l t

<pb2-2Mb V i £ ) "'

---( u , v | V---(pa“lcf a ) | a , ß )

■^5—

—r

k . , E . +€

f f v

V

F i g . 7 . A Feynman d ia g ra m r e p r e s e n t i n g a seco n d o r d e r s c a t t e r i n g p r o c e s s i n t h e momentum r e p r e s e n t a t i o n

R e f e r e n c e s

1.

2

.

3.

4 .

5. 6

.

7.

8

.

R .P . Feynman and A.R. H ib b s , Quantum M echanics and P a th I n t e g r a l s , C h a p te r 6 , M cG raw -H ill, N.Y. (1965)

N .F. M ott and H.S.W. M assey, The T h eo ry o f Atomic C o l l i s i o n s , T h i r d E d i t i o n , C la r e n d o n P r e s s , Oxford (1965)

M.L, G o ld b e r g e r and K.M. W atson, C o l l i s i o n T h e o r y , C h a p te r 5 , J o h n W ile y and S o n s, N.Y, (1965)

R. Newton, S c a t t e r i n g T h e o ry o f Waves and P a r t i c l e s , C h a p te r 6 , M cG raw -H ill, N . Y , ' (1966)

M. G ell-M ann and M.L. G o l d b e r g e r , P hys. R e v . , jin, 398 (1953) H. E c k s t e i n , Phys. R e v ., 101, 880 (1956)

W. B r e n ig and R. Haag, F o r t s c h . d . P h y s ik , _7, 183 ( 1 9 5 9 ) ; Quantum S c a t t e r i n g T h e o ry , Ed. M. R o ss, I n d i a n a U.P. (1963)

3 6

.

V. INTERMEDIATE PROCESSES

1. A P o s t u l a t e

We h av e s e e n i n t h e p r e v i o u s c h a p t e r t h a t a Feynman d ia g ra m method f o r s im p le s c a t t e r i n g c an be e v o lv e d from t h e p e r t u r b a t i o n s e r i e s o f t h e p r o p a g a t o r . T h is d ia g ra m method may be g e n e r a l i z e d t o more

c o m p lic a te d s c a t t e r i n g p ro b lem s w hich i n v o l v e p a r t i c l e ex ch an g e.

However, t h i s g e n e r a l i z a t i o n i s b a s e d on an a s s u m p tio n which we s h a l l i n t r o d u c e h e r e as a p o s t u l a t e , t h e p o s t u l a t e b e i n g t h e f o l l o w i n g :

The t o t a l a m p l i t u d e f o r t h e t r a n s i t i o n o f a sy ste m from t h e s t a t e I i ) a t tim e t^ t o t h e s t a t e | f ) a t t i s t h e sum o f t h e a m p l i t u d e s o f a l l p o s s i b l e s u c c e s s i o n s o f i n t e r m e d i a t e p r o c e s s e s w hich y i e l d th e s t a t e i f ) a t tim e t i f a t tim e t^ t h e s y s te m was i n t h e s t a t e j i ) . Between s u c c e s s i v e i n t e r m e d i a t e p r o c e s s e s t h e c o n f i g u r a t i o n o f th e s y ste m does n o t c h a n g e , o r i n o t h e r w o rd s, t h e n u c l e i p ro d u ced by t h e p r e c e d i n g p r o c e s s , do n o t i n t e r a c t u n t i l t h e f o l l o w i n g p r o c e s s . The a m p l i t u d e f o r a p a r t i c u l a r s u c c e s s i o n o f i n t e r m e d i a t e p r o c e s s e s can t h e n be r e p r e s e n t e d by a Feynman d ia g ra m .

The a s s u m p ti o n t h a t t h e t o t a l a m p l i t u d e f o r a r e a r r a n g e m e n t r e a c t i o n can be b r o k e n up i n t h i s way h as so f a r n e v e r b een j u s t i f i e d ,

in d e e d i t i s p o s s i b l e t h a t t h i s a s s u m p ti o n i s i n c o r r e c t . I n t h e c a s e o f s t r i p p i n g r e a c t i o n s , f o r exam p le, i t i s q u i t e e a s y , b u t v e r y m essy, t o d e r i v e an e x p a n s i o n f o r t h e a m p l i t u d e , u s i n g v a r i o u s a l t e r n a t i v e forms o f t h e i n t e g r a l e q u a t i o n o f t h e p r o p a g a t o r , so t h a t t h e f i r s t te rm c o r r e s p o n d s t o t h e p o l e g ra p h ( c f . F i g . 1 3 ). However, t h e r e m a in in g te rm s a r e t h e n to o c o m p l i c a t e d to be c o n v e n i e n t l y r e p r e s e n t e d by