"Science, Faculty of"@en . "Physics and Astronomy, Department of"@en . "DSpace"@en . "UBCV"@en . "Hsieh, William Wei"@en . "2010-02-26T01:09:24Z"@en . "1978"@en . "Master of Science - MSc"@en . "University of British Columbia"@en . "This thesis presents new interaction potentials for studying systems of nucleons and pions at intermediate energies. Using a quantum field theory approach, the fundamental dynamical variables are taken to be the Fermion and Boson creation operators, F\u00E2\u0081\u00BA and B\u00E2\u0081\u00BA.\r\nFirst, considering systems with only nucleons, a 2-body nucleon interaction potential is introduced, from which the familiar pair of coupled differential equations for the deuteron is derived.\r\nNext, turning to systems of nucleons and pions, focusing primarily on the reaction p+p \u00E2\u0086\u0092 \u00CF\u0080\u00E2\u0081\u00BA+d , we introduce unconventional, penta-linear interaction potentials of the form \"F\u00E2\u0081\u00BAF\u00E2\u0081\u00BAFFB\u00E2\u0081\u00BA + adjoint\". With these unconventional potentials, we can identify F\u00E2\u0081\u00BA10> and B\u00E2\u0081\u00BA10> with physical nucleons and physical pions\u00E2\u0080\u0094 quite unlike the conventional situation with the Chew Hamiltonian, where F\u00E2\u0081\u00BA10> cannot be identified with a physical nucleon. The differential scattering cross section for pp \u00E2\u0086\u0092 \u00CF\u0080\u00E2\u0081\u00BAd ( (with polarized incident protons) is then derived in terms of our potentials. Finally, we include a simple perturbation study of the deuteron state using our potentials."@en . "https://circle.library.ubc.ca/rest/handle/2429/20997?expand=metadata"@en . "A NEW HAMILTONIAN FOR SYSTEMS OF NUCLEONS AND PIONS by WILLIAM WEI^HSIEH B.Sc, University of B r i t i s h Columbia, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1978 \u00C2\u00A9 William Wei Hsieh, 1978 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . ( W i l l i a m W. Hsieh) Department o f P h y s i c s The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Date June 17, 1978 A b s t r a c t Th is t h e s i s p resents new i n t e r a c t i o n p o t e n t i a l s f o r s tudy ing systems of nucleons and p ions at i n t e r m e d i a t e e n e r g i e s . Us ing a quantum f i e l d theory approach, the fundamental dynamical v a r i a b l e s are taken to be the Fermion and Boson c r e a t i o n o p e r a t o r s , f and B . F i r s t , c o n s i d e r i n g systems w i t h on ly n u c l e o n s , a 2-body nuc leon i n t e r a c t i o n p o t e n t i a l i s i n t r o d u c e d , from which the f a m i l i a r p a i r of coupled d i f f e r e n t i a l equat ions f o r the deuteron i s d e r i v e d . Next , t u r n i n g to systems of nucleons and p i o n s , f o c u s i n g p r i m a r i l y on the r e a c t i o n f>+p TT++ d , we i n t r o d u c e u n c o n v e n t i o n a l , p e n t a - l i n e a r i n t e r a c t i o n p o t e n t i a l s of the form \" F F FFB + a d j o i n t \" . Wi th these unconvent iona l p o t e n t i a l s , we can i d e n t i f y F+l\u00C2\u00B0\"> and B + l\u00C2\u00B0> w i t h p h y s i c a l nucleons and p h y s i c a l p i o n s \u00E2\u0080\u0094 q u i t e u n l i k e the c o n v e n t i o n a l s i t u a t i o n w i t h the Chew H a m i l t o n i a n , where F |o> cannot be i d e n t i f i e d w i t h a p h y s i c a l n u c l e o n . The d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n f o r |'|'-^\u00C2\u00A5+c( (w i th p o l a r i z e d i n c i d e n t protons) i s then d e r i v e d i n terms of our p o t e n t i a l s . F i n a l l y , we i n c l u d e a s imple p e r t u r b a t i o n study of the deuteron s t a t e u s i n g our p o t e n t i a l s . i i i TABLE OF CONTENTS page Chapter 1 Introduction 1 Chapter 2 Mathematical Preliminaries: Operators and Their Transformation Properties 4 2 .1 Fermion and Boson Creation and Annihilation Operators . 4 2 .2 Singlet and Tr i p l e t Operators A ^ , SL \u00E2\u0080\u00A2> -Sf/\u00C2\u00BB ~C0 , T m / l . 8 2 .3 The 2-Fermion Operator y 14 2.4 The 2-Fermion Transfer Operator ($Zm 15 2.5 The 2-Fermion Transfer Operator 16 Chapter 3 Systems of Nucleons 17 3 . 1 The 2-Body Potential Vte 17 3 .2 Derivation of the Deuteron Equations 22 3 .3 The Singlet Even Scattering State 25 3.4 The T r i p l e t Odd Scattering State 27 Chapter 4 Systems of Nucleons and Pions 31 4 . 1 The Potential V for Singlet-Triplet Reactions 33 4 . 2 The Potential V for T r i p l e t - T r i p l e t Reactions . . . . 37 4 . 3 The D i f f e r e n t i a l Scattering Cross Section for pp->ifd . . 39 4.4 A Perturbation Study of the Deuteron State 47 Chapter 5 Summary and Conclusion 52 Bibliography 55 Appendix A The Motion Reversal Operator [J 56 Appendix B Some Properties of the Clebsch-Gordan Coefficients . . 57 Appendix C Some Properties of the Rotational Matrices D*M,(\"0*) \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 58 Appendix D Some Properties of the Spherical Harmonics Y M^ . . . 60 i v page Appendix E Wigner 3 - j , 6 - j , 9 - j Symbols and Racah C o e f f i c i e n t s . . 62 Appendix F D e r i v a t i o n of the Deuteron Equat ions 65 Appendix G The D e n s i t y Operator W 70 V Ac knowled gement s I would l i k e to take this opportunity to thank my supervisor, Dr. J. Malcolm McMillan, for having been, through the daily labyrinth of research, a superb guide\u00E2\u0080\u0094 and at d i f f i c u l t times, a deus ex machina. I am also grateful to the National Research Council of Canada for f i n a n c i a l assistance throughout the l a s t two years i n the form of scholarships. 1 Chapter _1 I n t r o d u c t i o n For a long t i m e , r e s e a r c h e r s have been i n t e r e s t e d i n systems of i n t e r a c t i n g nucleons and p i o n s , ( i n p a r t i c u l a r , the r e a c t i o n J3+|>->n++c( and i t s r e v e r s e ) ; however, d e s p i t e the l a r g e number of t h e o r i e s proposed, none of them has been e n t i r e l y s u c c e s s f u l . Us ing quantum f i e l d theory i n which the fundamental dynamical v a r i a b l e s are the Fermion and Boson c r e a t i o n o p e r a t o r s , and B + , (see s e c t i o n 2 . 1 ) , t h i s t h e s i s t a c k l e s the o l d problem w i t h a new approach by u s i n g an unconvent iona l H a m i l t o n i a n . In the c o n v e n t i o n a l Chew(1954, 1956a, 1956b) H a m i l t o n i a n , the n u c l e o n - p i o n i n t e r a c t i o n p o t e n t i a l i s of a t r i - l i n e a r fo rm, ( F FB + a d j o i n t ) , whereas i n our unconvent iona l H a m i l t o n i a n , the n u c l e o n - p i o n i n t e r a c t i o n p o t e n t i a l i s taken to have a p e n t a - l i n e a r fo rm, ( F\" F FFB + a d j o i n t ) . The c o n v e n t i o n a l approach has a s e r i o u s drawback, namely the f a c t that F l\u00C2\u00B0> i s not an e igenket of the Chew H a m i l t o n i a n , and the cumbersome n o t i o n of \" d r e s s e d \" p a r t i c l e s has to be i n t r o d u c e d . No such c o m p l i c a t i o n s a r i s e i n our approach, s i n c e both F lo> a n d B > > are e igenkets of our unconvent iona l H a m i l t o n i a n . However, we have to pay a p r i c e f o r our p e n t a - l i n e a r p o t e n t i a l s . Being' l e s s fundamental than the t r i - l i n e a r ones , our p o t e n t i a l s cannot be u s e d , f o r i n s t a n c e , to p r e d i c t the j\u00C2\u00BB^TT + c{ c ross s e c t i o n . But we can use the f>J5-\u00C2\u00BBirfc( data to o b t a i n e x p l i c i t forms f o r our p o t e n t i a l s , and then apply them to more compl icated r e a c t i o n s \u00E2\u0080\u0094 a process analogous to tha t adopted i n low energy n u c l e a r phys ics where nuc leon i n t e r a c t i o n p o t e n t i a l s were f i r s t determined from NN->NN data before be ing a p p l i e d to other problems. We believe our approach w i l l be simpler than the conventional one i n studying pion production with complex nuc l e i . Below i s an outline of this thesis: The Fermion and Boson creation operators are defined i n section 2.1. In section 2.2, the coupling of two Fermions leads to the singlet and t r i p l e t operators, A . By further combining these with appropriate Clebsch-Gordan coef f i c i e n t s and spherical harmonics, we obtain, i n sections 2 . 3 , 2 . 4 , and 2.5, the operators > (BI(T), and J4 \u00E2\u0080\u009E^a , which w i l l be deployed i n la t e r chapters. The Fermions and Bosons are to be i d e n t i f i e d with physical nucleons and physical pions. In Chapter 3 , we consider systems with only nucleons. A 2-body nucleon interaction potential i s introduced i n section 3.1, and i s used to derive a pair of coupled d i f f e r e n t i a l equations for the deuteron state. These equations, under appropriate r e s t r i c t i o n s , reduce to the fam i l i a r deuteron equations i n Bla t t and Weisskopf(1952, p.102). In sections 3 . 3 and 3 . 4 , the singlet even and t r i p l e t odd scattering states for two interacting nucleons are presented (for use i n the following chapter). In Chapter 4 , we study systems of nucleons and pions. Two of our A unconventional penta-linear potentials, V and V are introduced i n sections 4.1 and 4.2. In section 4 . 3 the d i f f e r e n t i a l scattering cross section JSL for (=f5^>TT+o( (with the incident proton p a r t i a l l y polarized perpendicular to i t s d i r e c t i o n of motion) i s derived i n f i r s t order perturbation theory using our unconventional Hamiltonian\u00E2\u0080\u0094 with results i n agreement with those i n Mandl and Regge(1955). In section 4 . 4 , as an example i l l u s t r a t i n g other types of calculations one can perform with our penta-linear potentials 3 we g i v e a s imp le study of the deuteron s t a t e . The p e n t a - l i n e a r p o t e n t i a l s g i ve to the deuteron s t a t e a p i o n i c component which we assume to be s u f f i c i e n t l y s m a l l , so tha t we can apply f i r s t order p e r t u r b a t i o n theory i n our d i s c u s s i o n . A summary and a c o n c l u s i o n i s g i ven i n Chapter 5 . 4 Chapter 2^ Mathemat ica l P r e l i m i n a r i e s : Operators and The i r T ransformat ion In t h i s c h a p t e r , the b a s i c mathemat ica l t o o l s we need to b u i l d our theory a r e d i s c u s s e d . C r e a t i o n and a n n i h i l a t i o n o p e r a t o r s f o r Fermions and- Bosons a re f i r s t i n t r o d u c e d , and t h e i r behaviour- under v a r i o u s t r a n s f o r m a t i o n s , such as ro ta t ion , , , p a r i t y , e t c . , a re then i n v e s t i g a t e d . Next , we use these elementary o p e r a t o r s to c o n s t r u c t more compl i ca ted ones, and the t r a n s f o r m a t i o n p r o p e r t i e s of the l a t t e r a re a l s o examined. 2.1 Fermion and Boson C r e a t i o n and A n n i h i l a t i o n Operators Assuming the p h y s i c a l system to be composed of Fermions and Bosons o n l y , we choose our H i l b e r t space to be a d i r e c t product of Fock spaces f o r Fermions and f o r Bosons. We s h a l l i d e n t i f y the Fermions w i t h p h y s i c a l n u c l e o n s , and the Bosons w i t h p h y s i c a l p i o n s . The fundamental dynamical v a r i a b l e s a re taken to be the Fermion a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s , r e s p e c t i v e l y , r m / t and rm^ , and the Boson a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s , By, and \u00E2\u0080\u00A2 By d e f i n i t i o n , F m^(r) , o p e r a t i n g on the vacuum s t a t e |6)> , c r e a t e s a s i n g l e Fermion a t p o s i t i o n _r, w i t h s p i n m (p ro jec ted a long the z - a x i s ) , and i s o t o p i c s p i n jx (p ro jec ted a long the z - a x i s i n the i s o t o p i c - s p i n s p a c e ) . A n a l o g o u s l y , F,^ (Ik) c r e a t e s a Fermion w i t h momentum k, s p i n m, and i s o t o p i c s p i n ju. Fn^_(t) and F\u00E2\u0080\u009E (fe) a re r e l a t e d by a F o u r i e r t r a n s f o r m : P r o p e r t i e s ( a . l . I ) 5 S i m i l a r l y , B^(r) c r e a t e s a Boson at p o s i t i o n r_, w i t h i s o t o p i c s p i n /i, w h i l e B^fk) c r e a t e s a Boson w i t h momentum k and i s o t o p i c s p i n These two operato rs a re a l s o r e l a t e d by a F o u r i e r t r a n s f o r m , Us ing the c o n v e n t i o u a l n o t a t i o n f o r commutators and ant i - commutators ( i . e . [A,B]= AB-BA, and {A,B)= AB+BA), the f o l l o w i n g r e l a t i o n s h o l d : (a.l \u00E2\u0080\u00A23) 0= B^(t\u00C2\u00BB = EjuW|o>= (!>) w i t h the f o l l o w i n g p r o p e r t i e s : 6 J)(k)B\u00C2\u00A3(OD*(10- B^(r + k) (2.1.13) combining with (2.1.1) and (2.1.2), we also have: I>00F*rfc)D+(i) = e \" - ' U F ^ ( _ 0 ( 5 J . / f ) D(t)6^k)J) + ft) - e - l i ' U B*(A) (a . l ./S) We note i n passing that JXk) can be written i n the form: D00 = e - ^ A (Z. I . I O where P - J A k [F^(k)Fm/1(k) + B > ) & > ) ] (2.1. 17) Next, we introduce a unitary G a l i l e a n boost operator with the following c h a r a c t e r i s t i c s : Gr(^) B*(A) = B j ^ + V ) ( 2 . M 7 ) where \u00E2\u0084\u00A21F and *\">B are the masses of the Fermion and the Boson, r e s p e c t i v e l y . Using (2.1.1) and (2.1.2), we have: < W r \u00C2\u00A3 ( i ) G + ( * ) = e ^ - r / * F^(r ) (zi.zo) We now introduce a unitary r o t a t i o n operator (R( Q I mo > \mji The c o u p l i n g of two nucleons leads to s i n g l e t and t r i p l e t s p i n s t a t e s of even and odd p a r i t i e s . We use the n o t a t i o n ( j , 3213 m) to denote the Clebsch-Gordan c o e f f i c i e n t r e s u l t i n g from the c o u p l i n g of two p a r t i c l e s of angular momentum (jji, w,) and (]x,^z) to form a t o t a l angular momentum of ( 3 , ^ . These c o e f f i c i e n t s a re a l s o used to couple sp ins and 9 isotopic spins. For properties of these c o e f f i c i e n t s , see Appendix B and Rose(1957). l We define a new creation operator resulting from the coupling of two nucleons as follows: C^R)=Z ^ a i ^ ^ h - V i ^ A l ^ l i / R H r ^ (B-ir) (2.2.1) where R i s the coordinate of the centre of mass of the two nucleons, and r , the r e l a t i v e separation between them. The coupling produces a state of spin s and isotopic spinO\", with m and ju , respectively, their components along the z-axis i n the spin and isotopic spin spaces. Correspondingly, we define It can be shown, with the transformation r e l a t i o n (2.1.1), that By (B.2) i n Appendix B, s can only take the values 0 or 1, and (T, also only the values 0 or 1. We introduce here four new symbols to denote A s < r the operator Am\u00E2\u0080\u009E when s and i are r e f e r r e d to a s , r e s p e c t i v e l y , the s i n g l e t odd o p e r a t o r , the s i n g l e t even o p e r a t o r , the t r i p l e t even o p e r a t o r , and the t r i p l e t odd o p e r a t o r . We now l o o k a t the or thogona l r e l a t i o n s between these o p e r a t o r s , 'cr' \"t Consider A* y (r', R') A^* (r,R)|o> . By d e f i n i t i o n , ASv(r'Jg/)A!,l(r,R)|o>=E, ^ l i ( i i K \" i | s W X i i / ^ | ( r y)(\u00C2\u00B1i \u00C2\u00AB l W l | s \u00C2\u00AB X i i / ' . ^ l ^ x F^(8 ' - ir ' ) Fm i > ; (Mr')F^/RHD F ^ ( E - tr)|o> Using ( 2 . 1 . 8 ) r e p e a t e d l y , and w i t h ( 2 . 1 . 3 ) , we get = V ^ *(*' + *r'- 5 \" ir)S(R'-ir'- B +\" S/./i V ; S(R'+^ r'-B + ir)srR'-ir'-B-^0|o> X [ 2 L i(44 - . \" . | \u00C2\u00AB , \u00C2\u00BB , X i i \u00C2\u00AB 1 ^ | s n ) ] [ Z ( U / v M ^ X \u00C2\u00B1 V . ^ l 7 0 > > With the use of (B.5b) and ( B . 6 a ) , t h i s s i m p l i f i e s to - (-i)S+ A n a l o g o u s l y , one can show t h a t : 11 - ( - 0 5 + < r \u00C2\u00A7 ( i K ' + k ' - i K + k ) S ( i K ,-V - ^ - k ) 5 l o > (a.a.st) I n t e g r a t i n g over d3R, i n ( 2 . 2 . 5 a ) , and over o(3f< i n ( 2 . 2 . 5 b ) , we have where we have in t roduced the n o t a t i o n Z ) to denote e i t h e r or We emphasize, i n p a r t i c u l a r , the f o l l o w i n g o r thogona l r e l a t i o n s from ( 2 . 2 . 5 ) : o= s;s:>>= s:.s$o>= S;T;>>= T ; O S ; V - S;,T;V> = T;S;>> = S . - tH l\u00C2\u00B0> = T0 S o |o> = S o 0 Tf 1 0 = T m / So0 |o> = T \u00E2\u0080\u009E , o 7 ^ |o> = T v T , o I \u00C2\u00B0 > (2-2.7) where the arguments of these operators have been omit ted f o r b r e v i t y . We now examine the t r a n s f o r m a t i o n p r o p e r t i e s of the s i n g l e t and t r i p l e t o p e r a t o r s . Under d i s p l a c e m e n t s , ( 2 . 1 . 1 2 ) and ( 2 . 2 . 1 ) g i ve With ( 2 . 2 . 3 ) , t h i s can be t ransformed i n t o Wk)A\u00C2\u00A3 = As\u00E2\u0080\u009E;+a.K) (*\u00E2\u0080\u00A2*\u00E2\u0080\u00A2\u00E2\u0080\u00A2>) S i m i l a r l y , under G a l i l e a n b o o s t s , we have and the cor responding r e s u l t 12 ^A^V*)^) = e I a ^ s * A # r , \u00C2\u00A3 ) (z-2.ll) Let us examine how these s i n g l e t and t r i p l e t o p e r a t o r s t rans fo rm under r o t a t i o n s . Us ing ( 2 . 1 . 2 2 ) , we have Us ing (C.5) from Appendix C, we have S i m i l a r l y , one can show that * W A \u00C2\u00A3 M M ' V ) = 5 W ^ ) A S m > a . , , ( 2 . 2 . ^ ) For the s i n g l e t o p e r a t o r s , s=0 and i s j u s t the i d e n t i t y . Hence, the t r a n s f o r m a t i o n equat ions s i m p l i f y to Under p a r i t y t r a n s f o r m a t i o n , ( 2 . 1 . 2 7 ) and ( 2 . 1 . 2 8 ) r e a d i l y show that One can r e c a s t ( 2 . 2 . 1 4 ) i n t o a more u s e f u l fo rm. Take 13 Commuting the two Fermion c r e a t i o n opera to rs by ( 2 . 1 . 5 ) , then i n t e r c h a n g i n g the dummy i n d i c e s \"i, and m2 , and u s i n g ( B . 5 b ) , we get and by s i m i l a r arguments, A:;+(-k,-K)=H) i 4 s*v:;+(i,-^ ^ . ^ ) w i t h the f a c t o r (\"0\" =+1 f o r /\ = S or T, and (-') = -I f o r Am/, = S\u00C2\u00B0 or T\u00C2\u00B0- In o ther words, Under mot ion r e v e r s a l , from ( 2 . 1 . 2 9 ) , we have Changing the summation i n d i c e s (\"\"N, i^O to ( -^i , -^) , and u s i n g ( B . 5 a ) , we o b t a i n S i m i l a r l y , from ( 2 . 1 . 3 0 ) , we can show that 14 QJm^) o r (t a n d Y?\"^ ^ s a s P n e r : L c a l harmonic . I t i s to be understood that the s p h e r i c a l harmonic Y^f]*) depends on ly on the o r i e n t a t i o n but not the magnitude of the v e c t o r \"\u00C2\u00A3 With the he lp of ( B . 6 b ) , we can e a s i l y o b t a i n the i n v e r s e r e l a t i o n , Under r o t a t i o n s , w i t h ( 2 . 2 . 1 2 ) , we have mr)i\u00C2\u00A3(t.Z)ftW= 2 \u00E2\u0080\u009E, (\" \u00C2\u00BB. I J-)Y^(f ) D ; , ( ^ ) Z J Using ( D . l ) from Appendix D , we have Th is s i m p l i f i e s by (C.5) to 15 2.4 The 2-Fermion T rans fe r Operator (Bi i^ We now c o n s t r u c t an operator 63^ which w i l l l a t e r be used i n Chapter 3 . L e t w i t h t r e s t r i c t e d to the v a l u e s 0 , 1 and 2 by ( B . 2 ) . (Qjim which conserves the number of Fermions , w i l l be c a l l e d a 2-Fermion t r a n s f e r o p e r a t o r . Other t r a n s f e r opera to rs can be c o n s t r u c t e d s i m i l a r l y u s i n g the other s i n g l e t and t r i p l e t o p e r a t o r s . Us ing the r e s u l t s of S e c t i o n 2 . 2 , and some i d e n t i t i e s from Appendices B and C, i t i s s t r a i g h t f o r w a r d to prove the t r a n s f o r m a t i o n p r o p e r t i e s of the (8gm operator as l i s t e d below: ^ ^ ( - M ; R ) ^ \u00E2\u0080\u00A2 = (-i)^d5,.ffl(^r;R) ( m ) (8jm (r,r;R) = (H)\"<8 (r'.r, R) 16 2 .5 The 2-Fermion T rans fe r Operator J^ im^ Analogous to the l a s t s e c t i o n , we now c o n s t r u c t another 2 -Fermion t r a n s f e r operator M~jimjn which w i l l be used i n Chapter 4 . L e t w i t h I r e s t r i c t e d to the v a l u e s 0 , 1 , and 2 . The reason f o r t h i s c h o i c e of momentum arguments w i l l become c l e a r i n Chapter 4 . A g a i n , i t i s s t r a i g h t f o r w a r d to d e r i v e the t r a n s f o r m a t i o n p r o p e r t i e s of t h i s operator as l i s t e d below: 17 Chapter 3^ Systems of Nucleons In this chapter, we s h a l l consider only systems of nucleons. We sh a l l write down a 2-body potential for the t r i p l e t even states and use i t to derive the familiar pair of coupled d i f f e r e n t i a l equations describing the deuteron wave function and energy. In the l a s t two sections of thi s chapter, we write down the singlet even and t r i p l e t odd scattering states which w i l l be needed i n Chapter 4. 3.1 The 2-Body Potential V t e We introduce the following 2-body potential Vt e(for the t r i p l e t even S13. t \u00C2\u00A3 S ) * V t e = ^ A V R g ^ ( r r ' ( R ) ^ ( r C ; R ) (3.1-1) where (f$ (\u00C2\u00A3,\u00C2\u00A3.', S.) i s t n e 2-Fermion transfer operator defined i n (2.4.1), and V. (r r' f O i s a function to be determined. From the d e f i n i t i o n of (Bj[m\u00C2\u00BB and the orthogonal properties (2.2.5) and (2.2.7), we see that Vte operating on a t r i p l e t even state recreates another t r i p l e t even state, while V t e operating on other states vanishes. We s h a l l demand this \4e be invariant under various transformations, and t h i s , i n turn, w i l l impose certain constraints on l\u00C2\u00A3m(c,E',R). F i r s t l y , we demand \4 E to be invariant under s p a t i a l displacements. Using (2.4.2), we have DU) V,eD+G>) = jVnfV/R. Z V^frr'ftQ (r,r', R+b) with appropriate change of variable. The right hand side equals V^e i f Me\u00C2\u00BB U, R_i)= W r ' \u00C2\u00A3 ' 5) > t h a t i s , must be independent of R, Second ly , we demand V t e to be i n v a r i a n t under G a l i l e a n b o o s t s . Us ing ( 2 . 4 . 3 ) , we have a u t o m a t i c a l l y , w i t h no a d d i t i o n a l c o n s t a i n t on , T h i r d l y , we r e q u i r e V t e to be i n v a r i a n t under r o t a t i o n s . Us ing ( 2 . 4 . 4 ) , we have \u00C2\u00A3f*m' which equals V e^ if 4m' where ( 3 . 1 . 3 b ) f o l l o w s from ( C . 3 ) . F o u r t h l y , we demand \4e to i n v a r i a n t under p a r i t y . Us ing ( 2 . 4 . 5 ) , we o b t a i n Z t\u00C2\u00A3 (-r -r')(B | which equals V-te i f W - r - c ' ) = ^J-r,-i') (3. F i f t h l y , we r e q u i r e Vte to be i n v a r i a n t under motion r e v e r s a l . Us ing ( 2 . 4 . 6 ) and ( 2 . 1 . 3 1 ) , we have = J W . ( V ^ Z < . h ( ^ - r ' ) f - ' / + W ^ m ( r , r ; ^ which equals V^e i f < ^ - r ' ) r - O i + ^ ^ ^ r ' ) fs.,.*) S i x t h l y , V t e must c e r t a i n l y be unchanged i f we r e p l a c e the dummy i n t e g r a t i o n v a r i a b l e t' i n ( 3 . 1 . 1 ) by - \u00C2\u00A3 . From ( 2 . 4 . 7 a ) , we see that t h i s c o n s t r a i n t i s s a t i s f i e d i f By making the same manoeuvre w i t h \u00C2\u00A3 , we get the analogous c o n d i t i o n : V\u00C2\u00A3\u00E2\u0080\u009E (-\u00C2\u00A3,\u00C2\u00A3')- V ^ ( r r ' ) (3.1-U) F i n a l l y , we r e q u i r e V t e to be H e r m i t i a n . With ( 2 . 4 . 8 ) , we have Vt \u00C2\u00BB J j s r M 3 R Z < ( r / ) f - ' ) f l ^ . J n ; r , R ) = jSr'frfR Z V \u00C2\u00BB w ( r ' j t X H r ( 0 ^ r r , \u00C2\u00A3 ' J E ) VtH - i f rfjr'r) = ^(r,r') ( 3 . 1 . 7 ) Summarizing a l l the c o n s t r a i n t s on , we have: ^ ( r , r ; R ) = ^(r,r') = vlm(-r,t')= = ^ M(-r,-r') (3./.8\u00C2\u00AB) ^ d ' . D r ( - 0^ m(r,r') ( 5 . I . H ) vfm(r,r') = (-O^^.Jr.r') (3.1.8c) 20 S ince l \u00C2\u00A3 m t ransforms under r o t a t i o n l i k e a s c a l a r , a v e c t o r , and a t e n s o r , r e s p e c t i v e l y , f o r \u00C2\u00A3 = 0 , 1 , and 2 , i t w i l l be c a l l e d , r e s p e c t i v e l y , the s c a l a r , the v e c t o r , and the tensor p o t e n t i a l f u n c t i o n . We now choose a form f o r that s a t i s f i e s these c o n s t r a i n t s . Le t ^ m ( 3 . 1 . 8 a ) i s s a t i s f i e d . A f t e r i n t e r c h a n g i n g (h, w 0 w i t h (^,\"'2) , we have by ( B . 5 b ) . For ( 3 . 1 , 8 a ) to be s a t i s f i e d , we impose, bes ides ( 3 . 1 . 1 0 ) , the a d d i t i o n a l c o n s t r a i n t : 21 Us ing (D.3) and ( B . l ) , we have a f t e r changing to ( - \u00C2\u00AB , , - i > i , ) and u s i n g ( B . 5 a ) . For ( 3 . 1 . 8 c ) to be s a t i s f i e d , we impose, b e s i d e s ( 3 . 1 . 1 0 ) , the e x t r a c o n s t r a i n t Wi th ( D . l ) and ( G . 5 ) , ( 3 . 1 . 8 d ) i s a u t o m a t i c a l l y s a t i s f i e d w i t h no a d d i t i o n a l c o n s t r a i n t s on ^ Summarizing a l l the c o n s t r a i n t s on \"^ni^ > w e have: l ^ ^ r ' ) i s r e a l ( 3 . 1 . 1 3 a ) Vjyj (r ,r ' )^0 on ly i f both ^ and j?2 a re even (3.1-13 0 ) We note i n p a s s i n g that we cou ld have cons t ruc ted V+ e out of the M operato rs d e f i n e d i n ( 2 . 3 . 1 ) and ( 2 . 3 . 2 ) . By r e a r r a n g i n g the C l e b s c h -Gordan c o e f f i c i e n t s and the s p h e r i c a l harmonics i n ( 3 . 1 . 9 ) and ( 2 . 4 . 1 ) , we can r e w r i t e \4e i n ( 3 . 1 . 1 ) a s : w i t h ^ ( r , r ' ) = Zh)\u00E2\u0084\u00A2 (zl+i)VJ(ilx I T i ^ ^ r ' ) (3./.\s) where i s a Racah c o e f f i c i e n t . ( S e e Appendix E). With the he lp of ( E . l l ) , we can a l s o e s t a b l i s h the i n v e r s e r e l a t i o n , 22 3 .2 D e r i v a t i o n of the Deuteron Equat ions We w r i t e down the deuteron s t a t e | K , m)> as f o l l o w s : ' . _ _ t \u00E2\u0080\u00A2\u00C2\u00BB \u00C2\u00AB \u00E2\u0080\u0094 r \ / \u00E2\u0080\u0094 t \u00E2\u0080\u0094 c ( 3 . ? . | f t ) Th is can be w r i t t e n i n the f o l l o w i n g more compact fo rm: w i t h J=s=l , and UJSs ^ ^ f \u00C2\u00B0 r r ' |^ 0 otiieruiise Demanding < Is ,m | K', ^ '\"> = S (\u00C2\u00A3 _ K') , we have, by u s i n g ( 2 . 2 . 5 a ) , (D.5) and ( B . 6 a ) , the f o l l o w i n g n o r m a l i z a t i o n c o n d i t i o n : DO f dr [V(rO+ UT2(r)3 = | ( 3 - * - * ) Using ( 2 . 2 . 3 ) and ( D . 7 ) , we can a l s o w r i t e the deuteron s t a t e as 'TMs w i t h J=s=l , and OO where ^ i s a s p h e r i c a l B e s s e l f u n c t i o n . Us ing ( 2 . 3 . 1 ) , ( 3 . 2 . 3 ) becomes i K, m>=z/Au J < s (H ' y;;1k , K ) i o> (,.\u00C2\u00BB.*> w i t h J=s=l . With the momentum operator P de f ined i n ( 2 . 1 . 1 7 ) , i t i s s t r a i g h t f o r w a r d 23 to show by u s i n g ( 2 . 1 . 3 ) , ( 2 . 1 . 7 ) and ( 3 . 2 . 3 ) , that I t i s a l s o easy to show that the f o l l o w i n g ho lds under p a r i t y (by ( 2 . 2 . 1 4 ) , (2 .2 .15b) and ( 3 . 2 . 3 ) ) , and under r o t a t i o n (by ( 2 . 3 . 3 ) ) : ( P|K,W>= 1-K,\u00C2\u00AB> (3 .2 . a ) m' m Thus, these equat ions c o n f i r m that our deuteron s t a t e lK,m^> , hav ing momentum K , does have even p a r i t y and t ransforms under r o t a t i o n w i t h t o t a l angular momentum J=l and z-component m. We w r i t e down the u s u a l t ime- independent Schrodinger e q u a t i o n : H j f c . ^ s (K.+ VNN)|k,\u00C2\u00AB> = \u00C2\u00A3|k\">> O.a.7) where \u00C2\u00A3 i s the energy e i g e n v a l u e . K0 i s the n o n - r e l a t i v i s t i c k i n e t i c energy operator d e f i n e d by w i t h 1% be ing the mass of a n u c l e o n . The n u c l e o n - n u c l e o n i n t e r a c t i o n p o t e n t i a l V N N i s d e f i n e d by V N N = V t e + V ^ 0 + V s e + V s o (3.2.,) where \ft0> Vse > ^so > r e s p e c t i v e l y , the p o t e n t i a l f o r the t r i p l e t odd s t a t e s , the s i n g l e t even s t a t e s , and the s i n g l e t odd s t a t e s , are d e f i n e d s i m i l a r l y to V ^ g . However, due to the o r thogona l r e l a t i o n s ( 2 . 2 . 7 ) , Vt. | ^ > = V s e ) K , \u00E2\u0084\u00A2 > = V S 0 I M > = 0 (mo) Hence, we are l e f t w i t h 24 ( \u00C2\u00AB 0 + V f c e)|K,^>= Elt^y 0-2.ll) Putting (3.1.1), (3.1.9), (3.2.1a) and (3.2.8) together, we obtain, a f t e r some computation, the following pair of coupled d i f f e r e n t i a l equations: (For d e t a i l s , see Appendix F ) . ~ f T r l r ' 0 , r ' u ( r ' ) ^ o o ( r ^ ' ) - f W ( 0 = i f T [ r V r V ( r ' ) U l o 2 ( r , r ' ) (3.a.|*\u00C2\u00AB) - t J i r j r V r V ( r ' ) V a 2 1 ( n r ' ) ] ~ \u00C2\u00A3 V ( r ) = J f - r J r W r ' \u00C2\u00AB ( r ' ) V a a o ( 1 r ' ) (3.2./2b) Note that our p o t e n t i a l functions \"^2g (J.r> r ) a r e > thus f a r , p e r f e c t l y general. The presence of terms with i = 0, 1, and 2 corresponds to the contribution from, r e s p e c t i v e l y , the sc a l a r , the vector, and the tensor force. I f we now take the p o t e n t i a l functions to be l o c a l , i . e . ^g} ( (r-r')\u00E2\u0080\u0094 Vj-j q ( O S ( r - r ' ) 5 we can deduce from (3.2.12) the following f a m i l i a r deuteron equations i n B l a t t and Weisskopf(1952, p.102): A + V c(r ) U - \u00C2\u00A3 U = - J T V T ( 0 ^ (z.X.ISa) ' 'U off + [ V c ( 0 - 2 V \u00C2\u00BB > - \u00C2\u00A3 i . = - j f V T ( r ) U (3.2.131,) where we have set - j ? = \" i f r 2 ^ 2 0 = [l r* l\u00C2\u00A3\u00E2\u0080\u009Efr) = /? V T (r) 25 3 .3 The S i n g l e t Even S c a t t e r i n g S t a t e To pave the way f o r the r e a c t i o n ]\u00C2\u00BBf>->TT*e(\u00E2\u0080\u0094 the main t o p i c of Chapter 4\u00E2\u0080\u0094 we s h a l l , i n t h i s and the next s e c t i o n , w r i t e down the s c a t t e r i n g s t a t e s f o r two i n t e r a c t i n g n u c l e o n s . We d e f i n e the s i n g l e t even s c a t t e r i n g s t a t e as f o l l o w s : where we i d e n t i f y K w i t h the t o t a l momentum, and 2k, w i t h the r e l a t i v e momentum of the two nucleons when they a re a t i n f i n i t e s e p a r a t i o n . V^ (-)h) i s the s i n g l e t even s c a t t e r i n g wave f u n c t i o n obeying the Lippmann-Schwinger i n t e g r a l e q u a t i o n . (See Newton(1966, p.181 and p . 2 9 9 ) . ) But from ( 2 . 2 . 3 ) , \u00C2\u00BB d h k f ^ ' R e \" 8 \" \" ' 6 ' * = S C E - E ) , (3.3.3) t h e r e f o r e , ( 3 . 3 . 1 ) can be r e w r i t t e n as fcU,r>= (^ftffilk) s\u00C2\u00A3(S,KM\u00C2\u00B0> (3.3.**) where Note tha t f o r the s p e c i a l case t u , j O = eik'r/ti (3.3.5-*) then W , i ) = ( 2 K t f 8(hk) ( 3 . 3 . ^ ) lfe , M > = S^ fk,!<)l0> , (3.3.5-c) as expected . By n o t i n g that ( 3 . 3 . 1 ) must be i n v a r i a n t i f we change the dummy 26 i n t e g r a t i o n v a r i a b l e r_ to -r_, and u s i n g ( 2 . 2 . 1 6 ) , we have and s i m i l a r l y , , , \u00E2\u0080\u009E . , . ^(-|,k)= ^ ( l , k ) (3.3.6 fc) Demanding < K', ^ k ,/*> = S ( K'~ \u00C2\u00A3)Sft'-fc) S/y , we o b t a i n by s t r a i g h t f o r w a r d computat ion , u s i n g ( 2 . 2 . 5 a ) . Under r o t a t i o n s , we demand A p p l y i n g the r o t a t i o n operator to ( 3 . 3 . 1 ) , and s u b s t i t u t i n g i n ( 2 . 2 . 1 3 b ) , we o b t a i n Changing the v a r i a b l e s (E_(R > B.^ ) to (n , R ) , and equat ing ( 3 . 3 . 8 ) and ( 3 . 3 . 9 ) , we o b t a i n J / r / R e' \u00E2\u0080\u00A2 & / k O S.^(r,5) = J V r A B \u00C2\u00AB * Wr\u00C2\u00AB,k) S^ V, & ) which i s s a t i s f i e d i f That i s , uiffi 6 denot ing the angle between r_ and k , f(r , k ) = ^ ( r . k . e ) (3.3.(0) The angu lar dependence on 9 can be expanded out i n terms of the Legendre po l ynomia ls T^(cos0) , wh ich , by ( D . 6 ) , can be f u r t h e r expanded i n terms of the s p h e r i c a l harmonics , as f o l l o w s : 27 By ( 3 . 3 . 6 a ) , ^ (n k) must v a n i s h u n l e s s \u00C2\u00A3 i s even. The terms w i t h i-0 , 2 , correspond r e s p e c t i v e l y to the 'S0 , 'D^, p a r t i a l waves. Wi th (D.7) and ( 3 . 3 . 4 b ) , we a l s o have where <* t e ^ O - j o/r i / ^ ) ^ ( r , fc) ( 3 . 3 . | i t ) Thus, ( 3 . 3 . 4 a ) can be r e w r i t t e n as Fur thermore , one can r e w r i t e ^ ( r , fe) i n the f o l l o w i n g fo rm: (See Newton(1966, p p . 3 0 1 - 3 0 3 ) . ) ^ (r> ) = el cos SA ^ \r, k ) (3. 3. If) where V ^ f r k ) i s a r e a l f u n c t i o n , and 8g(j0 i s the phase s h i f t f o r the I t h p a r t i a l wave. 3 .4 The T r i p l e t Odd S c a t t e r i n g S t a t e Analogous to the p r e v i o u s . - s e c t i o n , we d e f i n e the - t r i p l e t odd s c a t t e r i n g s t a t e as f o l l o w s : where \"Tgg M ( r , i ) i s the t r i p l e t odd s c a t t e r i n g wave f u n c t i o n . One can a l s o d e f i n e 28 Analogous to the c o n s t r a i n t s ( 3 . 3 . 6 ) and ( 3 . 3 . 7 ) , we have S(l- k' )S v i = ( ^ ' r & ^\u00E2\u0080\u009E(r, \u00E2\u0080\u009E, (r, K ) ( 3.*.* O Since the t r i p l e t odd s c a t t e r i n g s t a t e has s p i n 1 , we demand i t to s a t i s f y the f o l l o w i n g t r a n s f o r m a t i o n r u l e under r o t a t i o n s : The l e f t hand s i d e can be r e w r i t t e n w i t h the he lp of (2 .2 .12b) as f o l l o w s : where we have changed the v a r i a b l e s ( ' r _ ( R ) \u00C2\u00A3 ( R ) to The r i g h t hand s i d e of ( 3 . 4 . 5 ) can be r e w r i t t e n as f o l l o w s : X D ' S s \u00C2\u00AB s < ^ V ) T : j + / 1 ( r,R )|o> (3.t.y) We observe tha t ( 3 . 4 . 6 ) and ( 3 . 4 . 7 ) are equal i f Le t us w r i t e 29 S u b s t i t u t i n g ( 3 . 4 . 9 ) i n t o ( 3 . 4 . 8 ) , and u s i n g the r e l a t i o n s ( D . l ) , (C.4) and ( C . 5 ) , we f i n d t h a t w i t h t h i s form f o r m , ( 3 . 4 . 8 ) i s i d e n t i c a l l y s a t i s f i e d , and hence, the t r a n s f o r m a t i o n r u l e ( 3 . 4 . 5 ) i s a l s o s a t i s f i e d . With (D.7) and ( 3 . 4 . 2 b ) , we a l s o have where \u00E2\u0080\u00A2 \u00C2\u00AB> ( 3 . 4 . 2 a ) can now be r e w r i t t e n a s : We now cons ide r the p a r i t y of our s t a t e . By d e f i n i t i o n of the p a r i t y o p e r a t o r , We r e q u i r e the s t a t e to be of odd p a r i t y , i . e . = -!-!<, k , ( 3 . ^ . ( 3 ) S u b s t i t u t i n g ( 3 . 4 . 1 ) and ( 3 . 4 . 2 a ) s e p a r a t e l y i n t o ( 3 . 4 . 1 3 ) , we o b t a i n the c o n s t r a i n t s : %\u00C2\u00AB(*rk) = , \\u00C2\u00BBJtri>-V,ft .k) (3-*- '\u00C2\u00AB-) These, together w i t h ( 3 . 4 . 3 ) , when a p p l i e d to ( 3 . 4 . 9 ) and ( 3 . 4 . 1 0 a ) l ead to the f o l l o w i n g c o n d i t i o n s T^( r , l0 = S^(1i,l0== 0 u n l e s s both JL and i ' a re odd . IS\") i i The f u n c t i o n s can be f u r t h e r expressed i n the f o l l o w i n g fo rm: (See Newton(1966, p . 4 5 7 ) . ) 30 where a re r e a l phase s h i f t s , ^Y^P a re r e a l f u n c t i o n s , and are o r thogona l r e a l m a t r i c e s i n v o l v i n g mix ing parameters . 31 Chapter _4 Systems of Nucleons and P ions To study systems of nucleons and p i o n s , we use the f o l l o w i n g H a m i l t o n i a n , H= H 0+ V u t c+.o.i) w i t h , , , . H0 = K0+ VN N ( i f . o . i ) where K0 i s the t o t a l energy of the f r e e nuc leons and p i o n s . i . e . K. = JA [ Z \u00C2\u00A3N(k) F^ (k)Fv(kH Z ejk) B>)B,(fe)3 w i t h mN and 1% the masses of the nuc leon and the p i o n . V N N i s the i n t e r a c t i o n p o t e n t i a l f o r the n u c l e o n s , as d e f i n e d i n ( 3 . 2 . 9 ) . (See a l s o ( 3 . 1 . 1 ) and ( 3 . 1 . 9 ) . ) Hence V N N i s of the form JVT^PVFF , and the p o t e n t i a l f u n c t i o n s VNN can be t a k e n , f o r i n s t a n c e , to be the Reid(1968) p o t e n t i a l s . V^rt, the i n t e r a c t i o n p o t e n t i a l f o r the nucleons and p i o n s , can be subd iv ided as f o l l o w s : Vut ~- V f f N N + V T N + V f f f f (t-.o.t-) VTTNN i s a p i o n p r o d u c t i o n (and a n n i h i l a t i o n ) p o t e n t i a l c o n t r o l l i n g the ^rrNN^FFFB + a d j o i n t . Governing p i o n - n u c l e o n s c a t t e r i n g i s the p o t e n t i a l V\u00E2\u0080\u009E. N , chosen to have the form ]V^ NF8FB . The p o t e n t i a l f u n c t i o n s V\"-^ can be taken to be the Landau-Tabakin(1972) p o t e n t i a l s . D e s c r i b i n g the i n t e r a c t i o n s v^B B B8 \u00E2\u0080\u00A2 Here, we emphasize the f a c t that a l l these p o t e n t i a l s ( V N N , VJTNN J V-tiH and ) have a t l e a s t two a n n i h i l a t i o n o p e r a t o r s each . When o p e r a t i n g on Fn^ \oy and B^l0^ > these p o t e n t i a l s c o n t r i b u t e n o t h i n g . Hence, 32 and B i , ' 0 ) * a r e e igenkets of the H a m i l t o n i a n H. We, t h e r e f o r e , s t r e s s the f a c t tha t F^ , |o> and B u l 0 ^ r e s p e c t i v e l y represent a p h y s i c a l nuc leon and a p h y s i c a l p i o n . In t h i s c h a p t e r , we s h a l l be f o c u s s i n g on the WNN p o t e n t i a l , and the r e a c t i o n f>+ F TT + + c( (f.o-s-; TTNN i s taken to be composed of two p a r t s , V and V, r e s p e c t i v e l y , the A s i n g l e t - t r i p l e t and t r i p l e t - t r i p l e t i n t e r a c t i o n p o t e n t i a l s . V and V w i l l be d i s c u s s e d i n d e t a i l i n the next two s e c t i o n s , w i t h f i n a l forms g i ven by ( 4 . 1 . 1 ) , ( 4 . 1 . 1 1 ) , ( 4 . 2 . 1 ) , ( 4 . 2 . 3 ) . We s h a l l denote the momenta of the two incoming protons by \"k & + k and \"iK -k . , the momentum of the p i o n by + K , and the momenta of the two nucleons i n the deuteron by and 7 p K ~ i k ' _ ^ , (so tha t the deuteron as a whole has momentum i\u00C2\u00A3 ~ k )\u00E2\u0080\u00A2 F i g - 1 i l l u s t r a t e s the s i t u a t i o n : F i g . 1. Momentum diagram f o r the r e a c t i o n p+|=-?\"rr++d \u00E2\u0080\u00A2 E v e n t u a l l y , we s h a l l go i n t o the cen t re of mass frame where K=0. The c o o r d i n a t e system w i l l be chosen such tha t the i n c i d e n t p r o t o n , p o l a r i z e d i n the y - d i r e c t i o n , t r a v e l s a long the + z - a x i s , (w i th the u n p o l a r i z e d t a r g e t p ro ton t r a v e l l i n g i n the - z d i r e c t i o n ) . The p i o n then e x i t s w i t h angles ( 9, $ ) i n the s p h e r i c a l p o l a r c o o r d i n a t e system. 33 4.1 The Potential V for Singlet-Triplet Reactions We introduce the following potential for s i n g l e t - t r i p l e t reactions: where \"adj.\" denotes the adjoint of the f i r s t term i n braces, and V\u00E2\u0080\u009E are functions. Our d e f i n i t i o n of V ensures i t to be Hermitian. We s h a l l demand this V to be invariant under various transformations, and this w i l l impose constraints on the functions Vm (k, %,\u00C2\u00A3). F i r s t l y , we demand V to be invariant under s p a t i a l displacements. From (2.2.9) and (2.1.15), we have - V ( w e get which equals V i f ^ ( f c , i+w^,*, K-i\u00C2\u00BBvO= Vk,fe!*,K) ^\u00E2\u0080\u00A2'\u00E2\u0080\u00A23) (4.1.3) holds i f and only i f i.e. V can only depend on k and K v i a the combination i \"K+k'. Because of having neglected terms of order \u00E2\u0080\u0094 , our potential V i s only approximately Galilean invariant. Thirdly, we demand V to be invariant under rotations. By (2.1.23), (2.2.12) and (2.2.13b), we have ft fepO V rffy * ) = J A A'/fr A Z. V* (k, h\ * ) S^T (JL\u00C2\u00AB, S <\"f y) by appropriate change of variables. If we require then i t follows by (C.3a) that VU'.&JS) (4.1-8) By an analogous manoeuvre w i t h the v a r i a b l e k., we have tfJ-U',= ^ ( k , k ' 4 , K ) (4. M ) Summarizing a l l the c o n s t r a i n t s on V^, W e have: = - ^ ( k , % , - \u00C2\u00B1 ! < - k ' ) = - ^ - k . - 4 , - t k - k ' ) W - . M \u00C2\u00B0 \u00C2\u00AB ) v - m ( k 4 ^ ^ ) = ( - 0 V * ( k,|^lS+ k ' ) (^.i.lob) ^ ^ V i V U = 2 . \u00C2\u00BB J \u00C2\u00AB p ) l U k , ^ H + k ' ) O M - H 36 We now choose a form for v m that s a t i s f i e s these constraints. Let X ^ i ' ^ f c ^ ' O (4.1. I I ) where H8,f tJ'i 3 a r e functions that depend only on the magnitudes , and |iK + k'/ . This form for Vw s a t i s f i e s (4.1.10) i f we impose some re s t r i c t i o n s on the functions ^ju^'J3 Using (D.2), (4.1.10a) i s s a t i s f i e d i f (k,%li^H'l) = 0 unless JI, , B2 are both even and i 3 i s odd. C 4 - . U 2 ) Substiuting i n the complex conjugate of (D.3) for the three spherical harmonics i n (4.1.11), changing the indices ( w'< w 1 ( \u00C2\u00BB ) 3 ) to (-m1,-i\"1,,-\u00E2\u0080\u00A2\"i,-\"^') and using (B.5a), (4.1.10b) i s seen to be s a t i s f i e d i f H ) W V ' < ^ ^ ( K ^ , l i . ^ J ) = ^ i t i ' i,(fc . j,li!c +J }'J) which, with (4.1.12), yields the condition that the functions ^ ^ must be r e a l . Using (D.l) and (C.7), i t i s straightforward to show that (4.1.10c) i s s a t i s f i e d with no constraints on the functions V j ^ j ' / Analogous to section 3.1, we can rewrite our potential V i n terms of ni may) the operators y introduced i n section 2.3. With the help of (E.12) and (4.1.12), one can show that v = s s v/- (k, t,m^yyfjM<'\^\u00C2\u00BB'^) where 37 (4.1.l3c() and i s a Racah c o e f f i c i e n t . By ( E . l l ) , we also have .i,+a-r(2i1+i)fai'+iX2j+0\" ' 4.2 The Potential V for T r i p l e t - T r i p l e t Reactions We introduce the following potential for t r i p l e t - t r i p l e t reactions: where i&j.njt a r e the 2-Fermion transfer operators defined i n (2.5.1), and Vjin are functions. Our d e f i n i t i o n of V ensures i t to be Hermitian. A Exactly similar to the l a s t section, we require V to be invariant under various transformations. Using (2.5.2)-(2.5.7), we find that V f u l f i l l s these invariance requirements i f we impose the following constraints on ^a,k\u)- v^a^-u+k') = -) (4-3.^) Hence, to obtain an e x p l i c i t expression for the cross section, we must f i r s t evaluate the four traces i n (4.3.6)\u00E2\u0080\u0094 a straightforward, but tedious task. We s h a l l start off by calculating Tr(VWV). The orthogonal relations (2.2.7) guarantee that only a single term from W, (W given by (G.10)), survives i n VWV, i . e . V W V = \u00C2\u00A3v|!U,i> > the singlet even scattering state for the two i n i t i a l protons, i s given by (3.3.4a). Substituting i n (4.1.1) for V, i t i s straightforward to show that V W V = \u00C2\u00A3V> \u00E2\u0080\u0094 the t r i p l e t odd s c a t t e r i n g s t a t e f o r the two i n i t i a l p r o t o n s , one can show that V W V = Z dCv><'&n + ^ 5 ( ^ | o X - l i , - ^ l o > < - l l l*s S S 1 where w i t h vL g i ven by ( 4 . 2 . 3 ) , , by ( 3 . 4 . 1 0 a ) , a n d ? , by XT*. I S where j> i s the p r o b a b i l i t y of the ( p a r t i a l l y ) p o l a r i z e d i n c i d e n t p roton hav ing i t s s p i n o r i e n t e d a long the +y d i r e c t i o n . Fu r ther computat ion y i e l d s : 42 where, i n v o k i n g ( E . 6 ) , L = Z \".-.ITMXJF, I ^^^^\iK^0Y*Jk)Y . ftfc+i') ^. 3.12b) ^ U ^ k 1 ) = i g ^ ( ) i n the s p h e r i c a l p o l a r c o o r d i n a t e system. With the help of ( B . 5 ) , ( D . 3 ) , ( 4 . 1 . 1 2 ) and ( 4 . 2 . 4 ) , the f o l l o w i n g symmetry r e l a t i o n s can be proven : X . M = ( - i f X, x = (-o'-*s+m t A d i r e c t consequence of (4 .3 .20b) i s X o . = 0 ( 4 . 3 . 2 1 ) A We s h a l l approximate X\u00E2\u0080\u009E, and Xnj\u00E2\u0080\u009E by t a k i n g on ly the f i r s t few terms i n the i n f i n i t e sums ( 4 . 3 . 1 8 ) and ( 4 . 3 . 1 9 ) , i . e . , we s h a l l n e g l e c t c o n t r i b u t i o n s from the h igher order p a r t i a l waves. Table 1 shows the nature of the terms we s h a l l r e t a i n . (4-. 3.20 a) (4.3. Zo\u00C2\u00A3) I n i t i a l pp s t a t e (deuteron s t a t e - p i o n s t a t e ) T S i n g l e t s : 's. 0 1 z 1 T r i p l e t s : T 3 P , C3s,- *\ 1 1 o ' P , ( 3 s,<0, 1 1 z 7. 1 Z. Table 1 . P a r t i a l wave c o n t r i b u t i o n s i n c l u d e d i n our c a l c u l a t i o n of ^\u00C2\u00A3 D i r e c t eva luat i o n of ( 4 . 3 . 1 8 ) and ( 4 . 3 . 1 9 ) y i e l d s the r e s u l t s g i ven i n Table 2 . 44 -1 O i ^(w-o , - i i^ , )Y, 0 jferte. + fc^OY,, \ m : -1 A 0 -t &(/\u00C2\u00A3AW4*0Y\u00E2\u0080\u009E 0 0 I ^ a / i ^ ^ ^ j Y , , Table 2 Table of X \u00E2\u0080\u009E and \u00E2\u0080\u009E c o e f f i c i e n t s . ( N o t a t i o n : Yjpw = Y ^ t M ) ) S u b s t i t u t i n g the v a l u e s from Table 2 i n t o ( 4 . 3 . 1 7 ) , we have vi = C(y\u00C2\u00B0+ *>s*e) - x? s^ ecoscf) (x\u00E2\u0080\u009E + x.cose + A^ e ) ] (4.3. where Yo = r(fc,k')L(^0o+2/rb0, + b \u00E2\u0080\u009E V (^ M-/2t l 3 - / i 5 b 1 + + 4 b 3 3 - i ? b 3 1 f + f t ^ ) ] (f.3;\u00C2\u00A3ib) ^ = r ( U T ( - ^ be i+ 3 b \u00E2\u0080\u009E ) + (3^b 1 3 4 3/IS b,^- I b 3 3 + I b^.)] ( 4 . 3 . \u00C2\u00AB c ) = T(kM')(Wl C.| - 4 C 0 3 - *C 1 S I - i JT C i S ) (if, 3. 22.4) X , = TU,k')(*fi5\"c I H. + ? J s c 3 ( t . ) ( < f . 3 . \u00C2\u00AB e ) ^ = TtkX) ( l / I C \u00C2\u00AB 4 3jio C ^ ) (q-. 3. i* f ) w i t h the n o t a t i o n : b0= ^o, , L A 1 b 2 = wii , I A r 1 ( 4 . 3 . 2.3 a\") b , v r e + r zc O S*e (4-. 3.2?) The t o t a l scattering cross section CT i s obtained by integrating' 46 (4 .3 .22a ) over JlSL- [(>\u00E2\u0080\u009E + ^xjp '/cose) + i*,Pl(cose) + ^ A z P 3 ' ( \u00C2\u00AB s e ) ] (4.3.31) where and a r e , r e s p e c t i v e l y , the Legendre po lynomia ls and the a s s o c i a t e d Legendre f u n c t i o n s . These are or thogona l f u n c t i o n s , obey ing : l ^ ) f M O - X ^ ' ^ X = ^ = U , S k m ( 3 3 ,c) Taking advantage of these o r t h o g o n a l i t y p r o p e r t i e s , an exper imenter can r e a d i l y o b t a i n the c o e f f i c i e n t s X0 , Yx , \ , X, and \ x from the e x p e r i -mental d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n by an a p p r o p r i a t e i n t e g r a t i o n . For i n s t a n c e , ^ - T f \u00C2\u00ABKe* s e) (4.3.33a) X * = \"If { _( ' j l ^ l P B ( \" ^ \u00C2\u00AB\u00E2\u0084\u00A2*\/ I \u00C2\u00AB** ' (/r. 3. 33b) e t c . 47 4.4 A Perturbation Study of the Deuteron State As an example i l l u s t r a t i n g the various types of calculations one can perform with our penta-linear potentials, we give, i n th i s section, a simple discussion of the deuteron state using f i r s t order perturbation theory. We require the deuteron state lot^C^)^ (with momentum K and spin m) to s a t i s f y the following eigenvalue equation: H I ^ ) > H ( H 0 + V ,^)I^ (<)>= eC0|*UG> (4.4.1) with Ho given by (4.0.2). Since t h i s section i s mainly of pedagogical value, we w i l l make a s i m p l i f i c a t i o n by taking V^t to be composed of only one penta-linear potential: v\u00C2\u00AB = v where v = j A A ^ A ZI ( < 0 C z, (i i//|o o) s^ a,^ -k')B (^iK+k')]Tmeo^ Js) A (We could have included V and V i n V^- , but i t turns out that these potentials do not contribute i n f i r s t order perturbation theory. However, a t r i p l e t - t r i p l e t potential of the form \" T \u00C2\u00B0 B T E + adj.\", which does contribute i n f i r s t order, has been neglected for sim p l i c i t y . ) Analogous to section 4.1, we can check the constraints imposed on from the various invariance requirements on V. I t turns out that the functions v m have to s a t i s f y the same constraints on , namely (4.1.10). Hence, we can l e t ^ have the same form as v\u00C2\u00BB, , i . e . (4-. 4. 3 ) 4 8 where Vjtj \u00C2\u00A3'ji3 are r e a l functions. To lowest order, we take the deuteron state to be composed of two nucleons only, | = | E>M0<)> + higher order terms (4.4.4a) where . , , e^ |D,(k)>=^^l^^|lm)J^kU,(k)Y i m/k)TK,s0Ci^|o> (4.4.4k) analogous to (3.2.3) and ( 4 . 3 . 4 ) . 1 \"D\u00E2\u0080\u009E(\u00C2\u00A3S> s a t i s f i e s the equation Hoi TUfc)> = g^CWlDJ^ (4.4.*) For convenience, the following notations w i l l be used interchangably: The i d e n t i t y operator i i n the Fock space of Fermions and Bosons can be expressed i n terms of the eigenkets of H 0 : i - l\u00C2\u00B0X\u00C2\u00B0l+ F* (fc)|o>IO><0|B/4)]).^) + P 3 U \ 4 C Z ^ a ^ ^ l ^ l B ^ ^ ^ L k ) ^ \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2] + {-\u00E2\u0080\u00A2\u00E2\u0080\u00A2} ( 4 . 4 . 7 ) where {\u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2} denotes terms with two or more Boson creation operators, and terms with more than two Fermion creation operators. D^tK,^), J-* CJS, k), \u00E2\u0080\u00A2t 4 \"to4 Dm are the creation operators f o r , respectively, the singlet even, the singlet odd, the t r i p l e t even and the t r i p l e t odd scattering state. Analogous to (3.3.4) and (3.4.2a), we l e t Dsok k \u00C2\u00BB = &ZTJM r\u00C2\u00B0(u) s;!a, * \u00C2\u00BB (4.4. % b) 49 These s c a t t e r i n g s t a t e s are e igenkets of H0 , w i t h e igenva lues \u00C2\u00A3S (i<,k) , \u00C2\u00A3 fK/ \u00C2\u00A3), e^CS,^, and e*\u00C2\u00B0(K,k). From ( 4 . 4.1), we have i*UKY>= [eW - H.]\"' V |<*Ul<\u00C2\u00BB = [>(/<)- H.]~'l V|\u00C2\u00A3L(K)> f 4.4.?) S u b s t i t u t i n g i n ( 4 . 4 . 7 ) , and remembering the o r t h o g o n a l i t y r e l a t i o n s ( 2 . 2 . 7 ) , we have *I^\"(jc',k)> + Z [ e ( \u00C2\u00BB - I 0 K ; B > < D > ; b ) | V I ^ ) > } + W\ z [\u00C2\u00A3( K) - e, (ic)- \u00C2\u00A3T ft)]'1 D* )^B*tt)i*X\u00C2\u00BBlB\u00E2\u0080\u009E($)Dm.(*\u00E2\u0080\u00A2;v I W ) + JVicW > - ^ . k V E ^ t ) ] \" ^ ^ ; k ) B ^ ^ ) | o X o | B A ^ I ) ^ ( K ; k ) v | c D ^ ) > } +{\u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2} (4.4. lo) I n v a r i a n c e of the m a t r i x elements under s p a t i a l d isp lacements and r o t a t i o n s g i ves the f o l l o w i n g requ i rements : = ^-K'^, l(K) (4.4.11a) = S((H<')f^(K.fc) (r.4.Hb) <^lD^(K ' ,k )V|= S d C - ^ S ^ f e O H . f . l l c ) <\u00C2\u00B0!BA^ )D^(!C')i7|\u00C2\u00ABqn(!c)>= S(K-|<-\u00C2\u00A3) W (4,4. H o t ) = l \u00C2\u00A3 ( K \u00C2\u00BB + JA {Z C^(K) + ?(K) -\u00C2\u00A3 S e (K ,k) ] \" ' f v (K , k )D^(<; k)l\u00C2\u00B0> 4 fe(K)+ \u00C2\u00A3(K) x D!(K')B^|C-J<')|O>+ / ^ { R W + ^ ^ - ^ K ^ - ^ O ^ O T ' Z , K r / , ( k ^ ; A ) X D*e(K', k)B* +{\u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2} (4 .4. 13) The equat ion i s so f a r e x a c t \u00E2\u0080\u0094 g i ven our cho ice of . Let us now w r i t e l \u00C2\u00A3 \u00E2\u0080\u009E ( ( 0 > = \\(^?+ \\u00C2\u00A3>l(\u00C2\u00A3)> (4 .4 . (4) and M ^ H\u00C2\u00A3-\u00C2\u00A3)\m, l }(K)= < \u00C2\u00B0 ] ^ ^ ' ) V | \u00C2\u00A3 ) m ( K ) > = : <0|Dm,^)v|cBl(g)> (+. 4 . /s-) To f i r s t order i n our p e r t u r b a t i o n theory , we have ( 4 . 4 . /6) where Cfa-ttXkW <\u00C2\u00B0IB x^)D; e^;A)(/D^(i<\u00C2\u00BB ( 4 . 4 . 1 7 ) 51 Th is m a t r i x element can be eva luated e x p l i c i t l y u s i n g ( 4 . 4 . 2 ) , ( 4 . 4 . 4 b ) , ( 4 . 4 . 8 a ) , and ( 2 . 2 . 5 b ) , y i e l d i n g : tflfM'.k)* J A A Z C ^ \" ^ C44l8fc) S u b s t i t u t i n g i n ( 4 . 4 . 3 ) and ( 3 . 3 . 1 2 a ) , and u s i n g ( E . 1 2 ) , one can show that Given , one can use ( 4 . 4 . 1 6 ) to c a l c u l a t e the p r o b a b i l i t y of f i n d i n g a p i o n i n the deute ron . We now want to c a l c u l a t e the e igenva lue \u00C2\u00A3 of ( 4 . 4 . 1 ) . S u b s t i t u t i n g ( 4 . 4 . 1 6 ) i n t o ( 4 . 4 . 1 5 ) , we get \u00C2\u00A3(t K) \u00C2\u00AB J A A lSd(K) - Bse(K\ k) - K-K'))]\" Z, C/' ft ' x> Invoking ( 4 . 4 . 1 7 ) and ( B . 6 a ) , we have e w (K )=J-IVA few- \u00C2\u00A3 s%;k)-\u00C2\u00A3.(i(<-i<'o]\"'| h (>,/<; (4.4.20) I f we now, f o r s i m p l i c i t y , r e p l a c e \u00C2\u00A3 SS(^k) by ZSe(*',k) , and approximate by some a p p r o p r i a t e average v a l u e Jfc-J^'l^ , we f i n a l l y o b t a i n , by (D.5) and ( B . 6 a ) : (4.4. 21) 52 Chapter 5^ Summary and Conc lus ion We s h a l l f i r s t g i v e a b r i e f summary of what we have done: In the b e g i n n i n g , there were the Fermion c r e a t i o n operator and' the Boson c r e a t i o n operator B^u. By c o u p l i n g two Fermions t o g e t h e r , we were l e d to the s i n g l e t and t r i p l e t opera to rs , S\u00C2\u00B00 , T\u00E2\u0080\u009E0 , ~Cja-By f u r t h e r combining these w i t h a p p r o p r i a t e C lebsch-Gordan c o e f f i c i e n t s and s p h e r i c a l harmonics , we obta ined s e v e r a l u s e f u l opera to rs i n s e c t i o n s 2 . 3 - 2 . 5 . The Fermions and Bosons were i d e n t i f i e d w i t h p h y s i c a l nucleons and p h y s i c a l p i o n s . We then s t u d i e d systems of n u c l e o n s . A t r i p l e t even p o t e n t i a l \4 e was p r e s e n t e d , and used i n d e r i v i n g a p a i r of coupled d i f f e r e n t i a l equat ions d e s c r i b i n g the deuteron s t a t e . These e q u a t i o n s , under a p p r o p r i a t e r e s t r i c t i o n s , reduce to the f a m i l i a r deuteron equat ions i n B l a t t and Weisskopf (1952, p . 1 0 2 ) . The s i n g l e t even and t r i p l e t odd s c a t t e r i n g s t a t e s were in t roduced i n s e c t i o n s 3 . 3 and 3 . 4 , f o r use i n the f o l l o w i n g c h a p t e r . F i n a l l y , i n chapter 4 , we s t u d i e d systems of i n t e r a c t i n g nucleons A and p i o n s . The p e n t a - l i n e a r p o t e n t i a l s V and V, d i s c u s s e d i n s e c t i o n s .4.1 and 4 . 2 , were employed i n s e c t i o n 4 . 3 to o b t a i n the d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n jj\u00C2\u00A3 f o r the r e a c t i o n pp-\u00C2\u00BB7T+d\u00E2\u0080\u0094 w i t h r e s u l t s i n agreement w i t h Mandl and Regge(1955). As an example i l l u s t r a t i n g other types of c a l c u l a t i o n s one cou ld perform w i t h p e n t a - l i n e a r p o t e n t i a l s , we gave a f i r s t order p e r t u r b a t i o n study of the deuteron state i n s e c t i o n 4 . 4 . Having f i n i s h e d our summary, we s h a l l t r y to answer the f o l l o w i n g three q u e s t i o n s : \"How should one proceed from h e r e ? \" , \"What are the uses of t h i s t h e o r y ? \" , and \"What are the advantages and d isadvantages of our theory compared to o t h e r s ? \" 53 As po in ted out before i n Chapter 1 , our theory , by u s i n g a p e n t a -l i n e a r form (FFFFB+ a d j . ) f o r V and V, does not have as much p r e d i c t i n g power as the c o n v e n t i o n a l approach w i t h the Chew(1954, 1956a, 1956b) H a m i l t o n i a n , where the n u c l e o n - p i o n i n t e r a c t i o n p o t e n t i a l i s of a t r i -l i n e a r form ( F *FB + + a d j . ) . However, i n the c o n v e n t i o n a l c a s e , r\"^Jo> i s not an e igenket of the H a m i l t o n i a n , (whereas By.|\u00C2\u00B0> i s ) . Hence, w i t h the Chew H a m i l t o n i a n , l<>> cannot be i d e n t i f i e d w i t h a p h y s i c a l p a r t i c l e , and one has to i n t r o d u c e the n o t i o n of \" d r e s s e d \" p a r t i c l e s \u00E2\u0080\u0094 l e a d i n g to very compl icated t h e o r i e s f o r processes i n v o l v i n g complex n u c l e i . In our theory , w i t h two or more a n n i h i l a t i o n opera to rs i n a l l our p o t e n t i a l s , both h^ 1\u00C2\u00B0/ and B |o> are e igenkets of our unconvent iona l H a m i l t o n i a n . Due to the l o s s of p r e d i c t i n g power, our theory cannot make p r e d i c t i o n s f o r a r e a c t i o n as fundamental as pp - \u00C2\u00BBn + d . However, we can work backwards; i . e . , we can use the exper imenta l data f o r pp-7>Tt+d to o b t a i n e x p l i c i t \u00C2\u00A3 . A T f u n c t i o n a l forms f o r our p o t e n t i a l f u n c t i o n s and vSi . . (Th is i s analogous to u s i n g NN->NN data to o b t a i n e x p l i c i t forms of the n u c l e o n -nuc leon i n t e r a c t i o n p o t e n t i a l f u n c t i o n s i n low energy n u c l e a r p h y s i c s ) . A p o s s i b l e procedure i s o u t l i n e d below: Q One makes \" i n s p i r e d guesses\" f o r the f u n c t i o n a l forms f o r ( f e , ^ , ^ ) A T ) I \ and ~^itz\n3\k'lfi * / from prev ious work. That i s , one w r i t e s vl\h{K h O = * > \u00C2\u00AB r , \u00C2\u00AB \u00E2\u0080\u009E O (S.O.U) A where f and f are s p e c i a l f u n c t i o n a l forms w i t h (X1^ . . . , 0 ^ parameters to be determined from d a t a - f i t t i n g . > ^ , % s (see ( 3 . 3 . 1 4 ) and ( 3 . 4 . 1 6 ) ) have been found by p rev ious exper imenta l works . S u b s t i t u t i n g these i n t o ( 4 . 3 . 1 0 c ) and ( 4 . 3 . 1 2 c ) , one can o b t a i n by n u m e r i c a l i n t e g r a t i o n : 54 A where ^ and ^ are aga in f u n c t i o n a l forms depending on the parameters , tf,,\u00C2\u00B0(r. \u00E2\u0080\u00A2 Wi th \u00C2\u00AB , determined , | j f and v ^ z i J f j > and hence, V A and V are a l l known e x p l i c i t l y . A Having V and V determined from the p p - \u00C2\u00BB n d d a t a , our theory can then proceed to make p r e d i c t i o n s f o r p i o n p r o d u c t i o n i n v o l v i n g an i n c i d e n t proton c o l l i d i n g w i t h a heav ie r t a r g e t n u c l e i , i . e . r e a c t i o n s of the type P + A -> i r + + ( A + 0 , A > 2 . (5r.o.3) Assuming the pr imary r e a c t i o n mechanism to be the NN-?TNN mechanism, A our V and V can be used to p r e d i c t the d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n of ( 5 . 0 . 3 ) . A l s o of i n t e r e s t are i n t e r m e d i a t e s t a t e p i o n a b s o r p t i o n and p r o d u c t i o n e f f e c t s i n e l a s t i c s c a t t e r i n g of p ions o f f n u c l e i . That i s , p rocesses of the t y p e , TT++A A + 7T++A (S.O.if) A g a i n , our p o t e n t i a l s can be employed i n s tudy ing t h i s k i n d of r e a c t i o n . With these f u t u r e goa ls i n mind , we b r i n g t h i s t h e s i s to an end. Bibliography B l a t t , J.M. and Weisskopf, V.F. 1952. Theoretical Nuclear Physics (Wiley, New York). Brink, D.M. and Satchler, G.R. 1968. Angular Momentum, 2nd ed. (Clarendon, Oxford) . Chew, G.F. 1954. Phys. Rev., 94, 1748. Chew, G.F. and Low, F.E. 1956a. Phys. Rev., 101, 1570. Chew, G.F. and Low, F.E. 1956b. Phys. Rev., 101, 1579. deShalit, A. and Feshbach, H. 1974. Theoretical Nuclear Physics v . l (Wiley, New York). Goldberger, M.L. and Watson, K.M. 1964. C o l l i s i o n Theory (Wiley, New York). Landau, R.H. and Tabakin,F. 1972. Phys. Rev. D, _5, 2746. Mandl, F. and Regge, T. 1955. Phys. Rev., 99_, 1478. Messiah, A. 1958. Quantum Mechanics v . l (Wiley, New York). Newton, R.G. 1966. Scattering Theory of Waves and P a r t i c l e s (McGraw-H i l l , New York). Reid, R.V. J r . 1968. Annals of Physics, 50, 411. Rose, M.E. 1957. Elementary Theory of Angular, Momentum (Wiley, New York). 56 Appendix A The Mot ion R e v e r s a l Operator *J The mot ion r e v e r s a l operato r , a p p l i e d to a system, r e v e r s e s i t s momentum, but leaves i t s s p a t i a l c o o r d i n a t e s unchanged. By c l a s s i c a l ana logy , the angular momentum(=position v e c t o r x momentum) must a l s o be reversed under jT . S i m i l a r l y , we demand the s p i n to be reversed under motion r e v e r s a l , which i s e q u i v a l e n t to hav ing the observer ( i . e . c o o r d i n a t e system) r o t a t e by 180\u00C2\u00B0 a long some a r b i t r a r y a x i s . We choose a r o t a t i o n w i t h E u l e r angles (0,71,0\") . The nucleons w i t h s p i n \ , thus t ransforms by Dmm,(o~n0) , as shown below: The p i o n s , on the other hand, hav ing no s p i n , t ransform, t r i v i a l l y by D\u00C2\u00B00(o\"iro) \u00E2\u0080\u00A2 From Rose(1957 , p p . 6 2 - 7 3 ) , we have D* (OTTO) = ) - ( A - 3 ) S u b s t i t u t i n g (A.3) i n t o ( A . l ) and (A.2) y i e l d s ( 2 . 1 . 2 9 ) and ( 2 . 1 . 3 0 ) by s imple computat ion . Appendix 13 Some P r o p e r t i e s of the C lebsch-Gordan C o e f f i c i e n t s (j , tix w, \"^ j In t h i s appendix , we l i s t some of the p r o p e r t i e s of the C l e b s c h -Gordan c o e f f i c i e n t s that are needed elsewhere i n t h i s t h e s i s . From Rose(1957, p p . 3 2 - 4 7 ) , we have (h iin'^z\iim^)-\u00C2\u00B0 u n l e s s + = w 3 ( B - l ) (ji h m*\hm3)-0 u n l e s s h',+ j z | * i 3 > Ik+islsj,, and | j 3 + 0',|< j 2 .(B - O C J i 1\ \" i l \" i l j , B 3 ) = ' 0 u n l e s s K | < j , . I \u00C2\u00BB*| < g 2 ; and l \" M < j , ( B - 3 ) ( 3 , 0 M , o | m3) = g \u00E2\u0080\u00A2 ^ S m i \u00E2\u0080\u009E 3 and (o ^ o m z | j , * 3) = S . ^ S ^ ( B - 4-) From Rose(1957, p p . 3 8 - 3 9 ) , we have the f o l l o w i n g symmetry r e l a t i o n s : (j. U.n,.w*Us - ( L 0 4 , + 3* lB(l\u00C2\u00AB 3 \u00C2\u00BB - ^ l a \" , - W 3 ) CB . S-ti) \ Z j, + t / 1 From B l a t t and Weisskopf (1952, p . 7 9 1 ) , we have the o r thogona l r e l a t i o n s Z ^ O . O z ^ ^ l ^ X i j ' z ^ ^ N ' \u00C2\u00AB ' ) = S . . / 8Mffl, ( B . \u00C2\u00A3 a ) \u00C2\u00A3 ( 0 , 3 * *. \" 1 I 3 \u00C2\u00AB ) ( j , Oi I 0 \u00C2\u00BB ) = S W ( n ; S ^ , ( B . 6 b) 58 i Appendix C Some Properties of the Rotational Matrices Dm m'(\u00C2\u00AB In this appendix, we l i s t some of the properties of the matrices DrW^pY) > which are needed elsewhere i n t h i s thesis.. From Rose(1957, p.54), we have The orthonormal p r o p e r t i e s , from Rose(1957, p . 7 3 ) , a r e : From Rose(1957, p.58), we have where the common arguments (<*|3y) of the D-matrices have been omitted for brevity, and the summation over yu, and ^ i s such that = = constant. Correspondingly, we have Relabelling O\|,,>/0 by (i', \u00C2\u00BB*<',/*') i n (C.5), multiplying the equation through by 2, (j'j3i*)'m3|jm)I^ 3m > and remembering (B.l^we have (j, j , i>, A i ) g . s (j- j , \u00C2\u00AB ' \u00C2\u00BB31 j \u00C2\u00BB-> c i v D ; ; \u00E2\u0080\u009E S Substituting (C.5) into the l e f t hand side, we obtain v>i rv>* TV's C C. 7 ) 59 where the summation i s such that m remains f i x e d . S i m i l a r l y , we have w i t h /*\u00E2\u0080\u00A2 f i x e d . 60 Appendix D_ Some P r o p e r t i e s of the S p h e r i c a l Harmonics Yim In t h i s appendix , we l i s t some of the p r o p e r t i e s of the s p h e r i c a l harmonics f r e q u e n t l y r e f e r r e d to i n t h i s t h e s i s . From Rose(1957, p . 6 0 ) , the s p h e r i c a l harmonic Yf\u00E2\u0080\u009E(r) t ransforms under r o t a t i o n by where i t i s understood tha t Y?\u00E2\u0084\u00A2^ depends on ly on the o r i e n t a t i o n but not the magnitude of the v e c t o r jr; and where, i n C a r t e s i a n c o o r d i n a t e s , \u00C2\u00A3.z) V ^ / ^ P , ( \" s e ) ( J > ^ where !L-(r,0,<^) i n s p h e r i c a l p o l a r c o o r d i n a t e s , and ^ i s a Legendre p o l y n o m i a l . ^ P , ( c 0 5 . ) % \u00C2\u00A3 Y>)Y,Jr') ( D - 0 where o( i s the angle between r_ and r'. where ^ i s a s p h e r i c a l B e s s e l f u n c t i o n op With the d i f f e r e n t i a l e ra to rs L T and L de f ined below, the s p h e r i c a l harmonics t u r n out to be e i g e n f u n c t i o n s of these o p e r a t o r s : 62 Appendix E Wigner 3 - j , 6 - j , 9 - j Symbols and Racah Coefficients This appendix l i s t s some of the properties of the Wigner 3 - j , 6 - j and 9 - j symbols and those of the Racah c o e f f i c i e n t s . For a more complete l i s t , see Brink and Satchler ( 1 9 6 8 , pp.136-145) or deShalit and Feshbach (1974, pp.925-932) . The Wigner 3 - j symbol i s related to the Clebsch-Gordan co e f f i c i e n t by From (B . 6 ) , we have the corresponding orthogonality relations: v /i,uu\(i< i * i i \ ~ J \u00E2\u0080\u0094 s .\u00C2\u00AB s , (E Za) From the symmetry relations (B . 5 ) , i t can be shown that the 3 - j symbol i s invariant under c y c l i c permutations of i t s columns, but i s multiplied by the extra factor ( - 1 ) + J i + 5,3 under non-cyclic permutations. The 6 - j and 9 - j symbols arise from the coupling of, respectively, three and four angular momenta. They can be expressed i n terms of the 3 - j symbols, as follows: K I + m ? mc ' Other useful relations.are: 63 The orthogonality relations for the 6-j and 9-j symbols are: ZL ( a ^ O O ^ - o f t V H P ' ' A = \y ( E . 7 ) The 9-j symbols can also be expressed i n terms of the 6-j symbols: \> M T J From the symmetries of the 3-j symbols, i t follows that: (1) The 6-j symbol i s invariant under interchange of columns, and also under interchange of the upper and lower arguments i n any two columns. (2) The 9-j symbol i s multiplied by the factor + + Jii+i 3q- +3 l 3 +U + T under an odd permutation of rows or columns, but i s invariant under an even permutation or a transposition. The Racah co e f f i c i e n t W(0ida^3 H > is i$) , frequently used i n place of the 6-j symbol, i s defined as follows: w(ii> u,,-j 5 ^ = ( - o i t l 2 + i , t i M i i ; t ) (E'lo) The orthogonality r e l a t i o n (E.7) becomes 05\" (E .5) can be r e w r i t t e n i n terms of the Racah and C lebsch-Gordan c o e f f i c i e n t s : 65 Appendix F_ D e r i v a t i o n of the Deuteron Equat ions In t h i s appendix , we show how the deuteron equat ions ( 3 . 2 . 1 2 ) are der i ved from the e igenva lue equat ion ( 3 . 2 . 1 1 ) . S u b s t i t u t i n g ( 3 . 2 . 1 a ) (w i th K= 0) i n t o ( 3 . 2 . 1 1 ) y i e l d s : tXJ + D L > ( X l where, w i t h ( 3 . 2 . 8 ) , ( 3 . 1 . 1 ) and ( 3 . 1 . 2 ) , we have + ^ S %^\u00C2\u00AB^|.\u00C2\u00AB^r)r;^x JB)}io> ( F.ib) + ^ Z (*i ^ 11m) Y2M,(r)Twet (r, E)\|o> (f. Ic) [1 ] = \u00C2\u00A3 ( i k f I ^ R Y 0 0 ( x ) T ; > , S) + ^ Z/* I 11 *0 Y^dOT^ (r, )}|o> (F. W ) Let us focus on term C1] f i r s t . From ( 2 . 1 . 8 ) and ( 2 . 1 . 1 ) , we have { F \u00E2\u0080\u009E f l O , F^ , (r)} = fef e ^ - ' r A S m \u00E2\u0080\u009E , S / , ( F . ^ Us ing t h i s and ( 2 . 2 . 1 ) , we have Not ing tha t we can r e w r i t e ( 2 . 1 . 1 ) a s : (F-4) we can combine (F .3) and (F .4) to get 66 We now turn to term^E.^. Using (2.4.1) and (2.2.6), we have \i\ <8^r: r; g\")Tm^ fc R \u00C2\u00BB Z (li -s|X SX-CT' T*l Cr'; &\"). Rs(r +r> s(r-r')>> Substituting this into term \X\ , integrating over Jfr , and writing r' as t , we get With (F.5) and (F.6) substituted back into ( E . l ) , we apply Z,TT0(E',&\") to both sides of equation (F.la). We again focus on the f i r s t term. Using the anti-commutator relations (2.1.5) and (2.1.8), and (B.6a), i t i s straightforward to show that J Z ^(i;E')X A - fej { [ e ^ ( ^ + e^ ( t- r ' , A + e ^ ( - r - r ^ + e^(r+r')/fc]|o> Thus 67 S ince ^)-(4r^el^^\ and ^ = ^ [ f } e l x ^ , d=i,^ ( F . I ) the i n t e g r a t i o n s over J1^ and then over <^ r y i e l d : where the proper ty (D.2) has been invoked . In s p h e r i c a l p o l a r c o o r d i n a t e s , the L a p l a c i a n V* can be w r i t t e n a s : V* = + 2 2- - J - TZ (F- II) where the d i f f e r e n t i a l operator L i s de f ined i n (D .8b) . Us ing (D .9b ) , we o b t a i n Turning back to term , we have Z Teo(rU\")En] = [Z , A ^ ( r V ) ( a k f { . K -\u00C2\u00AB \ U0(-OT Y 0 0 + ^ >2L ^M^ s|i w)(n<- W s|r^)(-0' , S y2m(r)l|o> (F-13) a f t e r u s i n g ( F . 6 ) , ( 2 . 2 . 6 ) and ( 3 . 1 . 6 b ) . S u b s t i t u t i n g i n ( 3 . 1 . 9 ) f o r ^ (r',^) , u s i n g (D.5) and ( B . 4 ) , we o b t a i n + z z a 2 \u00C2\u00AB , r v i r * x * i ^ ) 1\u00C2\u00AB)(11 K-^ |Jrf)C-ons^ trr >\u00C2\u00BBY^ -r')V> (F.I4) Turning to term Tin:] , we have, w i t h (2.2.6) and ( D . 2 ) : Combining ( F . 1 2 ) , (F .14) and ( F . 1 5 ) , and i n t e r c h a n g i n g L and l ' , we get Apply J^Y,*!) t o ( F - 1 6 ) , where - IT Using the f a c t that Y 0 0 = Yo0 , and the orthonormal p roper t y ( D . 5 ) , we o b t a i n + K*' ( H w - \" | oo)(-i? Vla0 (r> r') + jr2+0W(2\u00C2\u00A3 II \u00E2\u0080\u00A2, 21 ) Z (21 m j \u00C2\u00AB . j | i\u00C2\u00BB0 ( F - Itfci) where W(\u00C2\u00ABbcc/;e-f) i s a Racah c o e f f i c i e n t . Thus, substituting (F.19) into (F.18), and cancelling out common factors, we have: -\u00C2\u00A3S'*Z ~ C^)-J^\r'Jr'u(r,)^r' r < ) - f W+0W(aHI; *0/ r V r W ' ) v ^ ( r , r ' ) = ( F . 2 o ) From Rose(1957, pp.225-226), the Racah coefficients can be evaluated, yield i n g : W ( 2 0 l l ; a l ) = ^ f , W^lU;2l>^/i ^ W ( 2 2 H ; 2 l ) = ^ Substituting these values into (F.20), we obtain the second deuteron equation (3.2.12b). 70 Appendix (3 The Dens i t y Operator W In t h i s appendix , we i n t r o d u c e a d e n s i t y operator W f o r a two-p a r t i c l e s t a t e , w i t h one of the p a r t i c l e s p a r t i a l l y p o l a r i z e d i n the y d i r e c t i o n . F i r s t , cons ide r a s p i n \ system. We use X, ^ , and 2 to denote the three u n i t v e c t o r s a long the x , y , and z d i r e c t i o n s . We a l s o w r i t e l \u00C2\u00AB > = |*t> , and ||8> a |n> where |$T}> and |\u00C2\u00A3l)> a r e , r e s p e c t i v e l y , the s p i n up and s p i n down s t a t e s w i t h respec t to the z - a x i s . For an a r b i t r a r y d i r e c t i o n n , s p e c i f i e d by the angles \u00C2\u00A9 and <\u00C2\u00A3> i n a c o n v e n t i o n a l s p h e r i c a l p o l a r c o o r d i n a t e system, we can w r i t e the s p i n up and s p i n down s t a t e s w i t h respect to the ri d i r e c t i o n , as |= -e~^^||\u00C2\u00AB> + cos||p> ($.a.l0 For the s p e c i a l case ft=ij , we have 6 ~\ , and $ = \ . Thus, I and I S O - H ^ + I P V ] ^ - 3 ) For a system c o n s i s t i n g of two s p i n h. p a r t i c l e s , we can form two-p a r t i c l e s t a t e s from ( G . l ) and (G.3) as f o l l o w s : iqtzi> = feO/j> + ^ > > A [ f H v > + ^ s > + l M > ] | ? i * T>=^[M^> + |(30(>>^[Mt,l>+jU|t ,o>- /t|s>] ($. = jfe[i|\u00C2\u00ABp>+|^>] = ^ [ ^ | t > o > + i | S > + | t J - i V ] 7 1 where it, r>?k*>, i^ \u00C2\u00B0>=^Ci^ > + lr>] , K-i>=|(3/9> (Q.^b) a r e , r e s p e c t i v e l y , the s i n g l e t and t r i p l e t s t a t e s r e s u l t i n g from the c o u p l i n g of two s p i n % p a r t i c l e s . Suppose the f i r s t p a r t i c l e i s p a r t i a l l y p o l a r i z e d a long the y - a x i s , such that the p r o b a b i l i t y of i t hav ing s p i n up (a long the y - a x i s ) i s p . We can i n t r o d u c e a d e n s i t y operator W as f o l l o w s : S u b s t i t u t i n g (G.4) i n t o (G.6), and l e t t i n g f = \ > , we have V(5)\u00C2\u00AB-!FI*,'><*.I| - jj|I*.\u00E2\u0080\u00A2><*><>I + s a ^ ' X ^ I + ^K-f> < t ) H | + ^ | 5 > < s | ^ . 7 ) I f we l e t (T3 = |9t2t><3TST| + | S t ^ X 9 T l i | - l 9 i^><5iStl- l 9 i H X ^ 4 i \ and \u00C2\u00A3 = Q-r'-r)]? = ^r-o5 = \u00C2\u00AB 3 we can r e w r i t e (G.6) as = i d t P - o ; ) ($.\u00C2\u00ABu) where 1 i s the i d e n t i t y o p e r a t o r . P i s commonly r e f e r r e d to as the p o l a r i z a t i o n v e c t o r , and i s the s p i n v e c t o r w i t h P a u l i s p i n m a t r i c e s 72 (for 2-particle states) as i t s components. We now turn to a r e a l i s t i c two-nucleon system, where we must take into account, besides spin, the momenta of the p a r t i c l e s and isotopic spin. In p a r t i c u l a r , l e t the two nucleons be two protons. Hence, the singlet and t r i p l e t states are respectively, |K,Js,/<,> i n (3.3.13), and [ K, fe, i n (3.4.11), with /< = I . Thus, we write the density operator W for a two-proton state with one of the protons p a r t i a l l y polarized i n the y di r e c t i o n , as follows: w= ^l ia.uXU.Ul - Uiu,u> "Thesis/Dissertation"@en . "10.14288/1.0094388"@en . "eng"@en . "Physics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "A new hamiltonian for systems of nucleons and pions"@en . "Text"@en . "http://hdl.handle.net/2429/20997"@en .