REVISED EQUATION OF MOTION METHOD, SEMI-CLASSICAL LIMIT, AND MATHEMATICALLY CLOSED THEORY OF LARGE AMPLITUDE COLLECTIVE MOTION

HAL Id: jpa-00224214

https://hal.archives-ouvertes.fr/jpa-00224214

Submitted on 1 Jan 1984

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

REVISED EQUATION OF MOTION METHOD, SEMI-CLASSICAL LIMIT, AND MATHEMATICALLY

CLOSED THEORY OF LARGE AMPLITUDE COLLECTIVE MOTION

A. Klein

To cite this version:

A. Klein. REVISED EQUATION OF MOTION METHOD, SEMI-CLASSICAL LIMIT, AND MATHEMATICALLY CLOSED THEORY OF LARGE AMPLITUDE COLLECTIVE MOTION.

Journal de Physique Colloques, 1984, 45 (C6), pp.C6-111-C6-119. �10.1051/jphyscol:1984613�. �jpa-

00224214�

30URNAL DE PHYSIQUE

Colloque C6, supplément au n°6, Tome 45, juin 1984 page C6-111

REVISED EQUATION OF MOTION METHOD/ SEMI-CLASSICAL LIMIT, AND MATHEMATICALLY CLOSED THEORY OF LARGE AMPLITUDE COLLECTIVE MOTION

A. Klein

Physics Department, Technical University , Munich, 8046 Garching by Munich, F.R.G.

and

Dept. of Physics, U. of Pennsylvania, Philadelphia, PA. 19104, U.S.A.

Résumé - On décrit une nouvelle formulation de la théorie de matrice densité généralisée, qui évite les défauts de l'ancienne théorie.

Au moyen de la transformée de Wigner on arrive d'un seul coup à la théorie Hartree-Fock dépendant du temps comme limite semi-classique.

A partir de ce cadre, on éclaircit plusieurs questions concernant la théorie d'oscillation à grande amplitude, en essayant de donner une forme complète à cette théorie.

A b s t r a c t - We d e s c r i b e a new form o f t h e g e n e r a l i z e d d e n s i t y m a t r i x t h e o r y w h i c h i s f r e e of t h e d e f e c t s of t h e p r e v i o u s f o r - m u l a t i o n . The a p p l i c a t i o n o f t h e W i g n e r t r a n s f o r m p r o v i d e s an e c o n o m i c a l d e r i v a t i o n of t i m e - d e p e n d e n t H a r t r e e - F o c k a s t h e s e m i - c l a s s i c a l l i m i t . From t h i s f r a m e w o r k , we s e e k t o i l l u m i - n a t e some open q u e s t i o n s i n t h e t h e o r y o f l a r g e a m p l i t u d e c o l l e c t i v e m o t i o n , a n d t o p r o v i d e , t h e r e b y , a m a t h e m a t i c a l l y c o m p l e t e t h e o r y o f t h e p h e n o m e n o n .

INTRODUCTION

The p u r p o s e of t h i s r e p o r t i s t w o f o l d . I t i s f i r s t t o c a l l a t t e n t i o n t o t h e s i n g l e - m i n d e d e f f o r t w h i c h we h a v e made d u r i n g t h e p a s t y e a r t o d e v e l o p and i m p r o v e two d i f f e r e n t c o m p l e t e t h e o r i e s of c o l l e c t i v e m o t i o n a n d t o t r y t o u n d e r s t a n d t h e r e l a t i o n s h i p , i f a n y , b e t w e e n t h e t w o . I t i s t h e n t o f o c u s on one of t h e s e t h e o r i e s , t o e x p l a i n i t s e s s e n t i a l e l e m e n t s , t o show i n a g e n e r a l m a n n e r how i t c o n t a i n s t i m e - d e p e n d e n t H a r t r e e - F o c k t h e o r y (TDHF) i n t h e s e m i - c l a s s i c a l l i m i t , and f i n a l l y , a n d m o s t s u b s t a n t i v e l y , t o a p p l y TDHF i n t h e a d i a b a t i c l i m i t

(ATDHF). Our aim h e r e h a s b e e n t o r e s t u d y t h e p r o b l e m of l a r g e a m p l i - t u d e c o l l e c t i v e m o t i o n i n o r d e r t o a c h i e v e a new s y n t h e s i s a n d g e n e - r a l i z a t i o n of p r e v i o u s r e s u l t s . We w a n t p a r t i c u l a r l y t o e m p h a s i z e t h e l a t t e r s u b j e c t , w h e r e we i n v i t e t h e r e a d e r t o c o m p a r e t h e v e r y c l e a r - c u t and u n a m b i g u o u s a s s e r t i o n s we s h a l l m a k e ( a n d c a n s u b s t a n t i a t e ) w i t h t h e r a t h e r c l o u d y p i c t u r e w h i c h e m e r g e s from t h e s e r i e s of r e - p o r t s made a t t h e Bad Honnef m e e t i n g of J u n e , 19 8 2 / 1 / . The m a t e r i a l of w h i c h we s p e a k a b o v e h a s b e e n i s s u e d i n f i v e r e p o r t s . With t h e e x - c e p t i o n o f t h e m a t e r i a l i n t h i s s e c t i o n we s h a l l a l l u d e o n l y b r i e f l y t o t h o s e t h r e e o f t h e r e p o r t s w h i c h h a v e a l r e a d y a p p e a r e d i n p r i n t and s h a l l d e v o t e o u r m a i n e f f o r t s t o d e s c r i b i n g t h e e s s e n t i a l c o n t e n t s o f two r e p o r t s w h i c h a r e a b o u t t o b e i s s u e d i n p r e p r i n t f o r m .

I n one of t h e t h e o r i e s o f c o l l e c t i v e m o t i o n / 2 / , we h a v e u t i l i z e d a

g e n e r a l i z e d c o h e r e n t s t a t e a s a t r i a l s t a t e i n t h e v a r i a t i o n a l p r i n -

c i p l e of t h e t i m e - d e p e n d e n t S c h r o d i n g e r e q u a t i o n . The i m p l e m e n t a t i o n

t o a f u l l t h e o r y of c o l l e c t i v e m o t i o n r e q u i r e s t h e d e m o n s t r a t i o n t h a t

t h e t r i a l s t a t e c a n i n f a c t b e d e t e r m i n e d , a n d t h a t o n c e known, t h e

t h e c o l l e c t i v e H a m i l t o n i a n can be c o n s t r u c t e d from i t i n a f u l l y

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984613

C6-112 JOURNAL DE PHYSIQUE

quantum-mechanical f a s h i o n . One t h e r e b y a c h i e v e s a f u l l y quantum- mechanical g e n e r a l i z a t i o n o f p r e v i o u s s e m i - c l a s s i c a l f o r m u l a t i o n s of Rowe-Basserman /3/ and of Marumori e t a l . / 4 / , though t h e l a t t e r have a l s o a c h i e v e d a quantum v e r s i o n a l t e r n a t i v e t o t h e one we d e s c r i b e /5/. I n t h e same p a p e r , we have i l l u s t r a t e d how t h e weak c o u p l i n g

(anharmonic v i b r a t i o n s ) and s t r o n g c o u p l i n g l i m i t s ( l a r g e a m p l i t u d e c o l l e c t i v e m o t i o n ) may be s t u d i e d .

The second"comp1ete" t h e o r y of c o l l e c t i v e motion t o which we have de- v o t e d o u r e f f o r t s h a s been t h e s o - c a l l e d Kerman-Klein t h e o r y , r e c e q t - l y r e v i e w e d , / 6 / , where we have a d o p t e d and c a r r i e d o u t a f u r t h e r .>- f i n e m e n t and g e n e r a l i z a t i o n of t h e form of t h e t h e o r y p r e f e r r e d by Belyaev and Z e l e v i n s k y , t h e g e n e r a l i z e d d e n s i t y m a t r i x t h e o r y ( G D M )

/ 7 / . We s h a l l d e s c r i b e t h i s g e n e r a l i z a t i o n i n Sec. 11. Over t h e y e a r s ,

we have emphasized r e p e a t e d l y t h a t t h i s method c a r r i e s w i t h i t i t s own c h a r a c t e r i s t i c s e t o f v a r i a t i o n a l p r i n c i p l e s /6/. I n a r e c e n t b r i e f p u b l i c a t i o n /8/ we have been a b l e t o show t h a t t h e s e v a r i a t i o n a l p r i n c i p l e s a r e c o n t a i n e d w i t h i n t h e c l a s s o f v a r i a t i o n a l p r i n c i p l e s a s s o c i a t e d w i t h t h e time-dependent SchrBdinger e q u a t i o n . I n c o n t r a s t t o t h e Rowe-Basserman, Marumori t h e o r y which emerges from c o h e r e n t s t a t e t r i a l f u n c t i o n s , t h e p r e s e n t t h e o r y c a n b e d e r i v e d from t r i a l f u n c t i o n s which a r e s u p e r p o s i t i o n s w i t h e q u a l a m p l i t u d e s of t h e c o l l e c t i v e s t a t e s . The same p a p e r a l s o c o n t a i n s r e f i n e m e n t s o f o u r p a p e r on t h e c o h e r e n t s t a t e method /2/.

I - LARGE AMPLITUDE COLLECTIVE MOTION FOR BOSONS

T u r n i n g t o t h e t h e o r y o f l a r g e a m p l i t u d e c o l l e c t i v e m o t i o n , we have d i v i d e d o u r e f f o r t s i n t o two p a r t s . I n t h e f i r s t p a r t / 9 / we d e c i d e d t o s t u d y a s y s t e m o f n c o u p l e d bosons d e s c r i b e d by c a n o n i c a l v a r i a b l e s and an a s s o c i a t e d H a m i l t o n i a n

c h a r a c t e r i z e d by a p o t e n t i a l e n e r g y f u n c t i o n V and a m e t r i c t e n s o r

. We a s k f o r t h e consequences o f t h e a s s u m p t i o n t h a t t h e low e n e r g y p a r t of t h e s p e c t r u m o r some p o r t i o n o f i t c a n b e d e s c r i b e d by a Harniltonian H d e f i n e d o v e r a s p a c e o f p < n d i m e n s i o n s and h a v i n g t h e same form a s ( 1 . 2 1 , namely

T h i s means t h a t t o implement a program f o r d e t e r m i n i n g H , we must s p e c i f y ( i ) t h e c l a s s o f t r a n s f o r m a t i o n among which we s h a l l s e e k t h e r e l a t i o n s h i p between t h e v a r i a b l e s S and t h e v a r i a b l e s

and ( i i ) t h e s e n s e i n which t h e o r i g i n a l H a m i l t o n i a n a and t h e c o l l e c t i v e Hamil- t o n i a n H a r e t o be r e g a r d e d a s d y n a m i c a l l y e q u i v a l e n t i n t h e p-dimen- s i o n a l s u b s p a c e , s i n c e such e q u i v a l e n c e can n e v e r be e x a c t e x c e p t i n t r i v i a l l i m i t i n g c a s e s . Our answer f o r ( i ) i s t h a t we imagine t h a t among t h e c l a s s of l o c a l l y i n v e r t i b l e p o i n t t r a n s f o r m a t i o n s

S*= ##(X'. .. ^{X ^ )} , F=fg(P ... ^{fa) >} ( 1 . 5 )

t h e r e e x i s t s a t l e a s t o n e , w i t h t h e p r o p e r t y t h a t the s u b s p a c e t o which (1 - 3 ) and (1 . 4 ) r e f e r may b e d e f i n e d by t h e c o n d i t i o n

x ~ = , . . sK"to 9 fYs+"c3 , ( 1 . 6 )

s u c h t h a t (1 . 6 ) and ( 1 . 7 ) may be viewed a s e q u a t i o n s o f p r o j e c t i o n from t h e f u l l n-space t o t h e s m a l l e r p-space. The a s s o c i a t e d dynami- c a l c o n d i t i o n s ( i i ) a r e s p e c i f i e d w i t h i n t h e framework of H e i s e n b e r g m a t r i x mechanics, t h e e q u a t i o n s

t o be s a t i s f i e d w i t h i n t h e p-space. Combining t h e s e w i t h t h e c a n o n i - c i t y c o n d i t i o n s ( 1 . 4 ) , we can a r r i v e a t a s e t o f e q u a t i o n s , now c a l l e d V i l l a r s ' e q u a t i o n s /10/, c o n s i s t i n g o f k i n e m a t i c a l and two dynamical s e t s of e q u a t i o n s , namely

(11) (111)

where t h e f o r d e r i v a t i v e s .

Concerning Eqs (1-111) and t h e i r d e r i v a t i o n we have e s t a b l i s h e d t h e f o l l o w i n g r e s u l t s , which a r e , a s f a r a s we can u n d e r s t a n d t h e pub- l i s h e d l i t e r a t u r e , new o r p a r t l y new: ( i ) Eqs (11) and (111) a r e u n d e r s t o o d t o b e c o n d i t i o n s o f z e r o and f i r s t o r d e r i n t h e momentum.

We have p r o v e d t h a t t h e s e e q u a t i o n s a l s o e x h a u s t t h e dynamics t o s e - cond o r d e r i n t h e momentum. ( i i ) T o g e t h e r w i t h v f s ) , i t i s u s e f u l t o c o n s i d e r a second s c a l a r c h a r a c t e r i s t i c of m, namely t h e ( s q u a r e o f ) magnitude o f t h e p o t e n t i a l ,

V J ~ = X ~ @ ~ ~ , (1 . l 4 1 From (1-111) a n d - t h e s p e c i a l c h o i c e of g a u g e .

g A = p ( 3 ) x i ^{> ( i = ~} ...P) , (1 . l 5 1 we d e r i v e t h e f i r s t - o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n s

( i i i ) These d i f f e r e n t i a l e q u a t i o n s a r e i n v a r i a n t i n form under any p o i n t t r a n s f o r m a t i o n w i t h i n t h e p-space. ( i v ) The d i f f e r e n t i a l equa- t i o n s d e s c r i b e f o r p=l a l l s t a t i o n a r y p a t h s on t h e n-dimensional s u r - f a c e V ( f ) . For p > l , a unique s o l u t i o n i s o b t a i n e d by s p e c i f i y i n g a s boundary c o n d i t i o n one o f t h e ( p - l ) d i m e n s i o n a l " s t a t i o n a r y " s p a c e s p r e v i o u s l y found. T h i s i s a c o n s t r u c t i v e p r o c e d u r e s t a r t i n g w i t h p = l . I t c a n f u r t h e r m o r e be proved t h a t t h e p ) l d i m e n s i o n a l s o l u t i o n s do i n d e e d r e p r e s e n t s t a t i o n a r y s p a c e s . ( v ) E q u a t i o n s (1 - 1 6 ) and (1 - 1 7 ) imply and a r e i m p l i e d by t h e l o c a l harmonic a p p r o x i m a t i o n . The equa- t i o n s of t h i s a p p r o x i m a t i o n a r e t h e c o v a r i a n t form o f (1 .16Y and a s a r e p l a c e m e n t f o r (1 . l 7 ) t h e e q u a t i o n s

2 (no sum

(1 . l 8 ) where t h e semi-colon n o t a t i o n i n d i c a t e s c o v a r i a n t d i f f e r e n t i a t i o n w i t h r e s p e c t t o t h e m e t r i c X , , The p r o o f o f t h e e q u i v a l e n c e o f (1 - 1 7 )

and (1 . l 8) i n c l u d e s t h e i d e n t i f i c a t i o n

JOURNAL DE PHYSIQUE

I n summary o f t h i s p a r t o f t h e d i s c u s s i o n we have shown t h a t t h e s t a n - d a r d f o r m u l a t i o n o f t h e l a r g e a m p l i t u d e c o l l e c t i v e motion problem i m - p l i e s u n i q u e l y and unambiguously t h e i n t u i t i v e l y e x p e c t e d outcome, a t

l e a s t f o r t h e boson problem. For d e t a i l e d p r o o f s we r e f e r t o t h e pub- l i s h e d p a p e r / g / . Below i n S e c I V , we s h a l l s e e t h a t a l l t h i s c a r r i e s o v e r f o r t h e fermion problem. T h i s l a t t e r problem i s s t u d i e d i n t h e l a s t o f o u r f i v e r e p o r t s , which i s c u r r e n t l y a v a i l a b l e o n l y i n p r e - p r i n t form ( U . of P A . r e p o r t 0230-T, r e v i s e d , 1 9 8 4 ) .

I1 - EQUATION OF MOTION METHOD. GDM THEORY.

R e t u r n i n g t o a d i s c u s s i o n o f t h e f u n d a m e n t a l s o f c o l l e c t i v e motion f o r t h e many fermion s y s t e m , we l i m i t o u r p r e s e n t d i s c u s s i o n t o t h e c a s e where t h e low-lying s t a t e s o f i n t e r e s t may b e d e s c r i b e d phenomenolo- g i c a l l y by a " q u a s i - l o c a l " H a m i l t o n i a n of t h e form ( 1 . 3 ) - a l l o w i n g h i g h e r d e r i v a t i v e terms when we a r e i n t e r e s t e d i n t h e problem o f an- harmonic v i b r a t i o n s . T h i s c o l l e c t i v e H a m i l t o n i a n must now b e deduced from a s t a n d a r d many-fermion H a m i l t o n i a n of t h e form

+ p ^Z6cd C'%'% % ( 2 . 1 ) ( 2 . 2 ) where v , v f a r e t h e u s u a l f e r m i o n d e s t r u c t i o n and c r e a t i o n o p e r a t o r s and v a r e t h e . a n t i s y m m e t r i z e d matriq e l e m e n t s of t h e i n t e r a c t i o n . W e l i m i t f u r t h e r d i s c u s s i o n t o c a s e s whlch can b e d e s c r i b e d c o m p l e t e l y i n terms of a s u i t a b l e s e t o f m a t r i x e l e m e n t s of t h e o p e r a t o r ( 2 . 2 ) . Thus we r e q u i r e a p r o j e c t i o n from t h e many-body H i l b e r t s p a c e o n t o a p-dimensional s p a c e 18) where i n t h i s s p a c e we r e q u i r e , i n a n a l o g y w i t h (1.8,9),,

; ~ = ~ & ~ J u ] = [ & d , U * ( 2 . 3 ) The l e v e l of a c c u r a c y a t which we p r e s e n t l y aim i s t h e f o r m u l a t i o n o f a t h e o r y c o n f i n e d t o t h e e l e m e n t s o f t h e g e n e r a l i z e d d e n s i t y m a t r i x

(GDM)

p ( a 3 1 65')s (i!'l~l~%/5). ( 2 . 4 ) We a r g u e t h a t a n a c c e p t a b l e t h e o r y can b e b a s e d on a "new" g e n e r a l i z e d Hartree-Fock f a c t o r i z a t i o n o f t h e t w o - p a r t i c l e d e n s i t y m a t r i x

f t

<_x'lY, a Jge l 3) zp@441 cdx? - ⁼ t ^{?(&X ICJ")~ @ ~ * l c 3 ; )

- ( b u d ) - (c&d) + ( b r a , f - d ) 3 , ( 2 . 5 ) whose d e s i r a b l e f e a t u r e s w i l l be d i s c u s s e d below. T h i s f a c t o r i z a t i o n

c o n v e r t s ( 2 . 3 ) i n t o a c l o s e d s e t o f e q u a t i o n s i n v o l v i n g e x c l u s i v e l y t h e e l e m e n t s o f t h e GDM ( 2 . 1 ) , which we w r i t e a s an o p e r a t o r w i t h r e s - p e c t t o t h e c o l l e c t i v e s p a c e l x ) ,

L;d6 = $ ^[%)!,,6

Here

i s a q e n e r a l i z e d Hartree-Fock H a m i l t o n i a n and

does n o t depend on t h e s i n g l e p a r t i c l e v a r i a b l e s . The e v a l u a t i o n o f

t h e l a s t form of ( 2 . 8 ) u t i l i z e s c o n s i s t e n t l y , t h e f a c t o r i z a t i o n ( 2 . 5 1 To c o m p l e t e the t h e o r y a s s o c i a t e d w i t h Eqs (2.6-2.8) , we n e e d a n o r - m a l i z a t i o n c o n d i t i o n . The a p p r o p r i a t e c o n d i t i o n

P65 = $ & f i r + : f i h p ~ . ( 2 . 9 ) can b e d e r i v e d from a s u i t a b l e a v e r a s i n a of ( 2 . 5 ) . To summarize, o u r t h e o r y c o n s i s t s of ( 2 . 6 ) and ( 2 . 9 ) w i t h a t h e d e f i n i t i o n s ( 2 . 4 ) , ( 2 . 7 ) and ( 2 . 8 ) , t h e l a t t e r a l s o p l a y i n g t h e r o l e o f a s e l f - c o n s i s t e n c y c o n d i t i o n . The e s s e n t i a l e l e m e n t i s t h e f a c t o r i z a t i o n ( 2 . 5 ) (which w i l l of c o u r s e b e g e n e r a l i z e d when w e w i s h t o e x t e n d t h e p h y s i c a l con- t e n t of t h e t h e o r y ) . Concerning t h i s r e l a t i o n we w i s h t o emphasize three p o i n t s . F i r s t , t h e v a l i d i t y o f o u r t h e o r y d o e s n o t r e q u i r e t h e a c c u r a c y o f ( 2 . 5 ) on a p o i n t by p o i n t b a s i s . We r e q u i r e o n l y t h a t two

a v e r a g e s b e a c c u r a t e , t h e f i r s t of t h e s e t h e a v e r a g e w i t h t h e elements o f V&d t h a t l e a d s t o t h e e q u a t i o n o f m o t i o n ( 2 . 6 ) a n d t h e s e c o n d t h e a v e r a g e needed f o r t h e s u b s i d i a r y c o n d i t i o n ( 2 . 9 ) . Second, t h e decom- p o s i t i o n ( 2 . 5 ) h a s been p r o p o s e d p r e v i o u s l y i n t h e l i t e r a t u r e , b u t t h e n e i t h e r r e j e c t e d /l l/ f o r r e a s o n s unknown t o t h e w r i t e r , o r else /l 2/ t h e s u b s i d i a r y c o n d i t i o n ( 2 . 9 ) was o v e r l o o k e d and t h u s t h e a t t e m p t s a t s t u d y i n g ( 2 - 6 ) p r o v e d i n d e c i s i v e . T h i r d , t h e f a c t o r i z a t i o n h a s a number of p h y s i c a l l y a t t r a c t i v e p r o p e r t i e s . We emphasize t h e most i m p o r t a n t : ( i ) I t i s c o n s i s t e n t w i t h the symmetry p r o p e r t i e s o f

t h e two-body d e n s i t y m a t r i x . ( i i ) By s u i t a b l e c h o i c e o f t h e s p a c e t h e r e s u l t i n g quantum t h e o r y m a i n t a i n s o r " r e s t o r e s " t h e symmetries v i o l a t e d by t h e u s u a l Hartree-Fock t h e o r y . ( i i i ) The r e s u l t i n g equa- t i o n s of motion a r e c o n s i s t e n t w i t h t h e c o n s e r v a t i o n laws i m p l i e d by t h e o r i g i n a l H a m i l t o n i a n H . ( T h i s p o i n t i s d i s t i n c t from ( i i ) ) .

( i v ) The r e s u l t i n g e q u a t i o n s o f motion a r e ( i n a s u i t a b l e s e n s e ) con- s i s t e n t w i t h b o t h e n e r g y and e n e r g y s q u a r e d w e i g h t e d sum r u l e s . W e h a v e shown i n o u t l i n e t h a t t h e f u l l s t r u c t u r e o f t h e t h e o r y b a s e d on ( 2 . 6 ) and ( 2 . 9 ) l e a d s t o a p l a u s i b l e form f o r t h e t h e o r y o f dam- p i n g of s i n g l e - p a r t i c l e e x c i t a t i o n s and o f g i a n t r e s o n a n c e s , b u t t h i s would b e t o o l e n g t h y a d i g r e s s i o n t o c o n s i d e r h e r e . F o r t h e r e m a i n d e r o f t h i s p a p e r w e s h a l l c o n s i d e r o n l y t h e s e m i - c l a s s i c a l l i m i t o f t h e p r e v i o u s t h e o r y .

I11 - SEMI-CLASSICAL LIMIT O F GDM THEORY

The s e m i - c l a s s i c a l l i m i t o f t h e GDM t h e o r y i s e a s i l y r e a c h e d from t h e form i n which we h a v e p r e s e n t e d t h e t h e o r y , namely by c a r r y i n g o u t a Wigner t r a n s f o r m w i t h r e s p e c t t o t h e c o l l e c t i v e - v a r i a b l e s . We w r i t e

f o r an o p e r a t o r A which may b e a l i n e a r form on t h e d e n s i t y o p e r a t o r o r any o f t h e s e v e r a l H a m i l t o n i a n s ,

f ₎ ) ( 3 . 1 ) (3.2 A p p l i c a t i o n s a r e b a s e d on t h e well-known c o n v o l u t i o n theorem which r e s u l t s i n an e x p a n s i o n f o r p r o d u c t s t h a t w e do n o t r e p e a t h e r e . I f a s e m i - c l a s s i c a l a p p r o x i m a t i o n - c a n b e j u s t i f i e d , t h e n t h e l e a d i n g t e r m s s u f f i c e . Thus i f C=AB,

and i f D=AB-BA,

D(& f ) ~ i [ A , B&,.

C6-116 JOURNAL DE PHYSIQUE

where P . B . means P o i s s o n B r a c k e t w i t h r e s p e c t t o t h e " c a n o n i c a l " s e t X , P. The j u s t i f i c a t i o n f o r t h e v a l i d i t y of t h e a p p r o x i m a t i o n s ( 3 . 3 ) - and ( 3 . 4 ) can b e made p r e c i s e l y under t h e c o n d i t i o n where t h e a p p r o x i - mate d e c o u p l i n g o f a c o l l e c t i v e subspace can Be j u s t i f i e d , b u t we c a n n o t e n t e r i n t o a d i s c u s s i o n of t h i s p o i n t h e r e .

I t remains o n l y t o a p p l y ( 3 . 3 ) and ( 3 . 4 ) . From ( 2 . 9 1 , u s i n g o n l y ( 3 . 3 ) we have

p a d (8J p 3 ^=fa, f) fed ($)_P), ( 3 . 5 ) which i d e n t i f i e s p ( x , P ) a s t h e d e n s i t y m a t r i x o f 2p p a r a m e t e r f a m i l y o f S l a t e r d e t e r m i n & t ~ . - T O o b t a i n t h e s e m i - c l a s s i c a l l i m i t o f ( 2 . 6 ) we need b o t h ( 3 . 3 ) and (3.41. The r e s u l t i s

[ X , p S d l R B .

w h e r e w ( ~ , ~ ) i s now t r u l y a HF H a m i l t o n i a n ,

and

( X P)

% 4 - f ~ ; ) ! ⁼ L6 ^{+ Ua6cd} pdd ^-2-

L P P K"'[T) ⁺ V ( g )

%c=.

>

a r e , r e s p e c t i v e l y , t h e " c l a s s i c a l " c o l l e c t i v ~ Hamiltonian and t h e Hartree-Fock e n e r g y a s s o c i a t e d w i t h t h e d e n s i t y m a t r i x p ( K g ) . I n

( 3 . 6 ) ( t o g e t h e r w i t h ( 3 . 5 ) ) , we have n o t o n l y TDHF t h e o r y , b u t a l s o a c l a s s i c a l L i o u v i l l e e q u a t i o n which i d e n t i f i e s X, P a s t h e c a n o n i c a l p a i r s we have a l r e a d y i m p l i e d them t o be by o u r c h o i c e of n o t a t i o n . W e have t h u s r e a d i e d t h e t o o l s t o s t u d y l a r g e a m p l i t u d e c o l l e c t i v e motion by c o n s i d e r i n g t h e a d i a b a t i c l i m i t o f t h e above formalism.

I V - VILLARS' EQUATIONS FOR LARGE AMPLITUDE COLLECTIVE MOTION

To s t u d y t h e a d i a b a t i c l i m i t we expand t h e d e n s i t y m a t r i x i n powers

d e n s i t y m a t r i x of a S l a t e r d e t e r m i n a n t , - t h a t i n t h e r ) e p r e s e n t a t i o n i n which p)is d i a g o n a l , t o which we h e n c e f o r t h r e s t r i c t o u r s e l v e s , p(fi) h a s o n l y non-vanishing ( p h ) and ( h p ) m a t r i x e l e m e n t s , and t h a t t h e

( p p * ) and (hh') m a t r i x e l e m e n t s o f pcaii)are d e t e r m i n e d by t h e known e l e m e n t s of and i t i s t h u s c o n s i s t e n t w i t h t h e kinema. i c s t o second o r d e r i n P; t o s e t t h e ( p h ) and ( h p ) e l e m e n t s of z e r o . I t .;urns o u t t o be c o n s i s t e n t a l s o w i t h t h e dynamics and t h e r e f o r e p a d c a n b e d i s m i s s e d from f u r t h e r c o n s i d e r a t i o n . T h i s d i s c u s s i o n t h u s

d i s p o s e s of t h e s p e c i a l e l e m e n t s o f t h e many-fermion problem which d i s t i n g u i s h e s i t from t h e boson c a s e .

The remainder o f o u r d i s c u s s i o n i s aimed a t showing t h a t t h e fermion v e r s i o n of V i l l a r s l Eqs ( I - I I I ) , (1.10-12) can b e i d e n t i f i e d i s o - m o r p h i c a l l y w i t h t h e boson e q u a t i o n s , s o t h a t a l l t h e s o l u t i o n t h e o r y developed f o r t h e l a t t e r can be c a r r i e d o v e r w i t h o u t change. The framework of t h e argument f o l l o w s . W e f i r s t i d e n t i f y t h e e l e m e n t s o f t h e fermion problem which c o r r e s p o n d t o t h e s t a r t i n g v a r i a b l e s 2 ,z

of t h e boson c a s e . From T h o u l e s s ' theorem t h e ( p h ) and ( h p ) e l e m e n t s

of P(&, E ) c o n s t i t u t e a complete s e t f o r t h e s p e c i f i c a t i o n of t h e

s p a c e of S l a t e r d e t e r m i n a n t s . L e t b e t h e m a t r i x which e q u a l s f l i n

t h i s s u b s p a c e and i s z e r o o t h e r w i s e . The t r a n s f o r m a t i o n

C

2 ( 4 . 2 )

i c l e n t i f i e s and /#*as complex c a n o n i c a l c o o r d i n a t e s , and t h e l i n e a r t r a n s f o r m a t i o n

( 4 . 4 ) c o m p l e t e s t h e i d e n t i f i c a t i o n . What f o l l o w s n e x t i s t h e e s s e n t i a l step i n o u r p r o c e s s . The f a c t t h a t t h e 2 , a r e a complete s e t o f

c a n o n i c a l v a r i a b l e s and t h e 5 , P_ a p a r t i a l s e t can be e x p r e s s e d by t h e c l a s s i c a l Laqranqe B r a c k e t c o n d i t i o n s

Using (4.1-4) , t h i s can 6 e t r a n s f o r m e d i n t o t h e r e l a t i o n e q u i v a l e n t t o (I),

(1 ( 4 . 6 ) T h i s s u g g e s t s t h e i d e n t i f i c a t i o n

( 4 . 7 ) ( 4 . 8 )

. ^-

W e v e r i f y t h i s i d e n t i f i c a t i o n by d e r i v i n g t h e e q u a t i o n s of motion which f o l l o w from t h e a p p l i c a t i o n o f t h e e x p a n s i o n ( 4 . 1 ) t o (3.6-3.9) The immediate outcome i s n o t o b v i o u s l y o f t h e forms (11) and (111).

For one t h i n g a s e c o n d - o r d e r c o n d i t i o n must be shown t o be d e p e n d e n t on t h e o t h e r c o n d i t i o n s and s e c o n d , p a r t of t h e c o n d i t i o n c o r r e s p o n - d i n g t o (111) must be shown t o be i d e n t i c a l l y s a t i s f i e d . When t h e smoke c l e a r s , we a r e l e f t w i t h t h e e q u a t i o n s

(11) ( 4 . 9 )

; ~ 7 ~ ; p*) ^{+ ;} j [ @ F!]

⁽¹¹¹⁾ ( 4 . 1 0 )

~ e r e p i n ( 3 . 7 ) h a s been expanded

( 4 . 1 1 ) The r e c o c r n i t i o n t h a t (11) and (111) a r e o f the r e a u i r e d form u t i l i z e s

( 4 . 7 , 8 ) &d t h e f u r t h e r i d e n t i f i c a t i o n s

and t h e f o l i o w i n g f o r m u i a s ' f o r t h e e l e m e n t s o f X ,

~f~~'?'=2~&~/-~'~)J A A h " $ 4 ; 4 r / s ~ 4 ) : ,@l

( 4 . 1 4 )

C6-118 JOURNAL DE PHYSIQUE

V - ELEMENTARY APPLICATIONS

For t h e boson problem, we have founded t h e t h e o r y of l a r g e a m p l i t u d e c o l l e c t i v e motion on Eqs (1.16) and ( 1 . 1 7 ) . The s o l u t i o n o f s p e c i f i c boson problems i s a l s o u s e f u l l y c a r r i e d o u t w i t h t h e h e l p o f t h o s e e q u a t i o n s . For t h e fermion problem, however, i t a p p e a r s t o be u s e f u l

t o make a d i s t i n c t i o n between t h e o r y and p r a c t i c e . Though we have l a b o r e d t o e s t a b l i s h t h a t t h e t h e o r e t i c a l s t r u c t u r e o f t h e fermion problem i s i s o m o r p h i c t o t h a t o f t h e boson problem, t h u s g u a r a n t e e i n g

t h e p h y s i c a l n a t u r e of t h e s o l u t i o n , i n p r a c t i c e t h e s o l u t i o n of t h e fermion problem a p p e a r s t o b e e f f e c t e d more n a t u r a l l y by methods which a r e g e n e r a l i z a t i o n s of time-honored, s t a n d a r d p r o c e d u r e s . A s an

example, we c o n s i d e r a s o l u b l e monopole model d e s c r i b e d by t h e Hamil- t o n i a n ( f i r s t - q u a n t i z e d form)

where

i s a p a r t i c l e l a b e l . For t h i s problem i t i s n a t u r a l t o choose t h e r e l a t i o n c o r r e s p o n d i n g t o ( 1 . 7 ) , where f o r a s i n g l e c o l l e c t i v e c o o r d i n a t e , Q , we w r i t e ( s e e end o f d i s c u s s i o n f o r a c r i t i q u e of t h i s

s t a t e m e n t ) z

Q = Z n p .

We t h e n have

With t h e s e i d e n t i f i c a t i o n s , ~ q ' ( 4 . g ) , c o n d i t i o n (11) i s e a s i l y s e e n t o t a k e t h e form

1

We t h e r e f o r e a d j o i n t o ( 5 . 6 ) t h e s e l f - c o n s i s t e n c y c o n d i t i o n ( c f ( 5 . 1 ) )

> ^{( 5} - ⁷⁾

-

where eh(Q) a r e t h e e i g e n v a l u e s o f c o r r e s p o n d i n g t o t h e o c c u p i e d o r b i t a l s . E v a l u a t i n g t h e f i r s t t e r m of ( 5 . 7 ) by t h e v i r i a l theorem and remembering ( 5 . 6 ) , we a r e l e d t o a f i r s t o r d e r d i f f e r e n t i a l equa- t i o n

2 3

y'(0?)+(~9,/4)=&4+, + T n w h ~

w i t h t h e s o l u t i o n

SO

f a r u n d e t e r m i n e d ( s e e b e l o w ) .

Verv b r i e f l y , t h e e v a l u a t i o n o f c o n d i t i o n ( 1 1 1 ) y i e l d s a f t e r some c o m p u t a t i o n - t h e e q u a t i o n

and combined i i t h c o n d i t i o n ( I ) , p r o v i d e s a f o r m u l a

which i s a s t a n d a r d sum r u l e e v a l u a t i o n . F i n a l l y , w i t h t h e c h a n g e of v a r i a b l e

w e o b t a i n c o r r e s p o n d i n g t o ( 3 . 8 ) ,

+ f m ~ , Z X 2 + + 3 n 4 4 : r c ( 5 . 1 3 )

where t h e v a l u e

( 5 . 1 4 )

2

i s i n t e r m s of t h e e q u i l i b r i u m v a l u e o f x . For s m a l l v i b r a t i o n s a b o u t x , we o b t a i n t h e f r e q u e n c y

2

U*= +L&&' + 6 & 2 ~ X , . ( 5 . l 5 )

-

The c a l c u l a t i o n d e s c r i b e d above i s , i n f a c t , n o t h i n g more t h a n a v a r i a n t o f a s e l f - c o n s i s t e n t c r a n k i n g c a l c u l a t i o n . To p r o v e t h a t i n t h i s c a s e i t i s t h e c o r r e c t s o l u t i o n , w e c a n show t h a t ( 5 . 4 ) a c t u a l l y s a t i s f i e s ( 1 . 1 6 ) a n d ( 1 . 1 7 ) . I n more g e n e r a l c a s e s , ( 5 . 3 ) c a n s e r v e a s a s a s t a r t i n g c a l c u l a t i o n f o r i t e r a t i o n .

T h i s work was s u p p o r t e d i n p a r t by t h e U.S. D e p a r t m e n t o f E n e r g y u n d e r C o n t r a c t No. 401 32-5-20441.

RE FE REN CE S

1 . GOEKE, K . a n d REINHARD, P. G. - e d s . , Time-Dependent H a r t r e e - F o c k

a n d Beyond, L e c t u r e N o t e s i n P h y s i c s , v 0 1 171 ( S p r i n g e r - V e r l a g , B e r l i n , 19821, S e c . V I .

2. KLEIN, A . , MARUMORI, T . , a n d UNE, T . , P h y s . Rev. E( ^{19 84)} , 240.

3. ROWE, D.J. and BASSERMAN, R. C a n . J . P h y s . = ( 1 9 7 6 ) , 1941.

4. MARUMORI, T . , P r o g . T h e o r e t . P h y s . = ( 1 9 7 7 ) , 112.

5 . MARUMORI, T., SAKATA, F . , UNE, T . , a n d HASHIMOTO, Y . , r e £ / l / , p. 3 0 8 .

6. KLEIN, A. i n P r o q . i n P a r t a n d N u c l . P h y s . , e d . by D. W i l k i n s o n (Pergamon, O x f o r d , 1 9 8 3 ) v o l . 1 0 , p . 34.

7. ZELEVINSKY, V . G . , P r o g . T h e o r e t . P h y s .Suppl.Nos. 75 a n d 76 (1 983) , 252 8 . KLEIN, A. and TANABE, K . , P h y s . L e t t . 1 3 5 ~ (1 9841, 255.

9. KLEIN, A . , N u c l . P h y s . A 4 1 0 ( 1 9 8 3 ) , 7 4 . 1 0 . VILLARS, F . , N u c l . P h y s . A285 ( 1 9 7 7 ) , 269.

1 1 . KAMLAH, A. and MEYER, I., Z e i t . f . P h y s . 211 ( 1 9 6 8 ) , 293.

1 2 . FRIEDRICH, A. a n d GERLING, W . , Z . N a t u r f o r s c h . 319 ( 1 9761, 872.