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Submitted on 1 Jan 1987
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ON SOME ASPECTS OF SEMICLASSICAL
PROPAGATION AND TIME-DEPENDENT DENSITY
FUNCTIONAL THEORY
H. Kohl, P. Schuck
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C2, suppl6ment au n o
6,
Tome
48,
j u i n 1987
ON SOME ASPECTS OF SEMICLASSICAL PROPAGATION AND TIME-DEPENDENT
DENSITY FUNCTIONAL THEORY
H. KOYL
and
P.SCHUCK
I n s t i t u t des Sciences NuclBaires,
5 3 ,
Avenue des Martyrs,
F-38026
Grenoble Cedex, France
Resume
.
Nous d i s c u t o n s d ' u n e m a n i e r e s i m p l e l e s r e l a t i o n s f o n d a m e n t a l e s q u i e x i s - t e n t e n t r e l e s champs d e s i n t 6 g a l e s d e mouvement, l a p r o p a g a t i o n semi- c l a s s i q u e e t l a t h e o r i e d e l a f o n c t i o n e l l e d e l a d e n s i t 6 dependant d u temps.A b s t r a c t
.
Fundamental c o n n e c t i o n s between t h e f i e l d s o f time-dependent i n t e g r a l s o f m o t i o n , s e m i c l a s s i c a l p r o p a g a t i o n and time-dependent d e n s i t y f u n c t i o n a l t h e o r y a r e d i s c u s s e d i n a s i m p l e p e d e s t r i a n ' s way. I n p r a c t i c e we o f t e n e n c o u n t e r t h e f o l l o w i n g s i t u a t i o n . At some a r b i t r a r y t i m e t = 0 a s t a t e ( a>
i s p r e p a r e d w h i c h i s a n e i g e n s t a t e ofsome
o p e r a t o r ( s ) A A : A / a>
= aI
a>.
T h i s p r e p a r e d s t a t e / a>
i s t h e n p r o p a g a t e d i n a quantum s y s - tem o f i n t e r e s t i n t o l a t>
= ( t ) l a>,
6
( t ) b e i n g t h e t i m e - e v o l u t i o n o p e r a t o r . S i n c e t h e quantum L i o u v i l l e s p a c e h a s proved t o be a f l e x i b l e background f o r semi- c l a s s i c a l c o n s i d e r a t i o n s and s i n c e quantum p r o p a g a t i o n i s l i n e a r , we may i n t r o d u c e,.
,.
* *-I t h e f o l l o w i n g c o n c e p t . D e f i n e t h e o p e r a t o r I ( t ) : = U ( t ) A U ( t ) , t h e n o b v i o u s l y A t h e e q u a t i o n h o l d s : Y A ( t ) l a t>
= a ( a > . O p e r a t o r s o f t h a t t y p e have a c o u p l e o f i n - t t e r e s t i n g p r o p e r t i e s . Among t h e most i m p o r t a n t o n e s i s t h e f a c t , t h a tI*
( t ) i s a n Ai s o s p e c t r a l d e f o r m a t i o n o f t h e o p e r a t o r
a.
I f t h e l a t t e r i s bounded from below, ? A ( t ) can b e used t o d e f i n e a minimum p r i n c i p l e on s e t s of time-dependent (TD) s t a - t e s [ 1 , 3 , 4 ].
I n t h e Weyl-Wigner r e p r e s e n t a t i o n t h e o p e r a t o r*I
( t ) obeys t h e f o l l o - - A w i n g e q u a t i o n o f m o t i o na l ~
2 45 ----+
s i n(T
)I
H = 0 , I ( t = ~ ) = A,A
=vH
.$
-vH
.
v1
a t A p q q P ( 1 )w h i c h i s t h e quantum a n a l o g u e of t h e c l a s s i c a l e q u a t i o n a t I + { I , H } ~ . ~ . = 0 . The ge- n e r a t o r s o f t h e k i n e m a t i c symmetries o f H a r e always among t h e s o l u t i o n s of e q . ( 1 ) .
However, t h e r e a r e (TD) s y s t e m s w i t h a d d i t i o n a l and n o n t r i v i a l c l o s e d s o l u t i o n s of e q . ( l ) . ( S e e e . g .
11-51
and f o r some i m p o r t a n t r e s u l t s f o r t h e c l a s s i c a l c o u n t e r p a r t o f e q . ( 1 ) compare [7,8].
An i l l u s t r a t i o n of t h e a p p l i c a t i o n o f t h e G r e e n ' s func- t i o n method t o e q u a t i o n s o f t y p e ( 1 ) i s g i v e n i n r e f . [ 9 ] ) . It s h o u l d be s t r e s s e dC2-318 JOURNAL
DE
PHYSIQUEt h a t t h e r e a r e many open and e x c i t i n g q u e s t i o n s , c o n c e r n i n g t h e p r o p e r t i e s of t h e s o l u t i o n s o f eq. ( 1 ) . For example t h e r o l e o f t h e "%-dependent terms" i n t h e Sin- o p e r a t o r has not been r e a l l y understood s o f a r (compare 191 f o r a n a n a l y t i c a l s t u d y of a model s y s t e m ) . Of c o u r s e , t h e q u e s t i o n which Hamiltonian systems admit c l o s e d s o l u t i o n s of eq. (1) t o u c h e s a wide arid l i t t l e e x p l o r e d f i e l d a s w e l l .
There i s a n o t h e r c l a s s of expansio: of t h e s o l u t i o n s of eq. ( I ) , which s b d d be mentioned i n t h i s c o n t e x t
1.41
.
Be A:=EL
+ V ( q j t = 0 ) and H =A
'
+
V ( q , t ) . TheCU 2m 2m
a n s a t z I =
i
-$
I l e a d s t o a r e c u r r e n c e r e l a t i o n f o r t h e functions I p , which Rt u r n s o u t to%$! e z s y t o s o l v e i n c l o s e d form. For t h e a s s o c i a t e d o p e r a t o r I one ob- t a i n s t h e f o l l o w i n g e x p a n s i o n
where t h e d o t i n d i c a t e s 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 time.
The i n t e r e s t i n g p o i n t i s t h a t t h e s e e x p a n s i o n s a r e u n i v e r s a l f o r e a c h equi- v a l e n c e c l a s s of s y s t e m s c o n s i d e r e d , i n t h e s e n s e t h a t t h e p o t e n t i a l and i t s d e r i v a - t i v e s e n t e r i n a f u n c t i o n a l form. I n t e r a c t i n g many body systems may be t r e a t e d i n t h e same f a s h i o n [4] b u t t h e d e t a i l e d convergence p r o p e r t i e s of t h i s m-'-expansion. a r e n o t i n v e s t i g a t e d s o f a r .
I n p r i n c i p l e d e n s i t y f u n c t i o n a l t h e o r y (DFT) i s one of t h e b a s i c r e p r e s e n - t a t i o n s of quantum mechanics, s i n c e i t t r e a t s w i t h a q u i t e fundamental q u e s t i o n : g i - ven t h e s t a t e
11)
( t )>
of t h e system a t t i m e t and some o b s e r v a b l e 8 , b e p ( q , t ) t h e one p a r t i c l e d e n s i t y c o r r e s p o n d i n g t o t h e s t a t e I $ ( t )>.
I s t h e r e a map m g s u c h t h a t t h e f o l l o w i n g diagram of maps commutes ?The e x i s t e n c e of t h e map me would imply t h a t t h e e x p e c t a t i o n v a l u e 8 of t h e ob-
L
The q u e s t i o n o f e x i s t e n c e o f t h e map m, h a s b e e n s t u d i e d r e c e n t l y [1,2,4] u s i n g t h e c o n c e p t s i n t r o d u c e d a b o v e . It h a s b e e n shown a s w e l l t h a t t h e same means may b e employed t o t o u c h t h e p r a c t i c a l a s p e c t s of t h e p r o b l e m and d e t e r m i n e f u n c t i o n a l s e x p l i c i t e l y ~ i n c l u d i n g s h o r t t i m e and a d i a b a t i c f u n c t i o n a l s . The p o s s i b i l i t y t o de-
C
f i n e minimum p r i n c i p l e s w i t h t h e h e l p o f t h e o p e r a t o r s o f t y p e I ( t ) e n t e r s a s a n
A
e s s e n t i a l i n g r e d i e n t i n a l l t h e s t e p s . Work a l o n g t h e s e l i n e s c o n t i n u e s .
ACKNOWLEDGEMENTS
.
One o f u s (H,k)
i s i n d e b t e d t o R.M. D r e i z l e r ( F r a n k f u r t ) a n d W M . Durand ( G r e n o b l e ) f o r numerous d i s c u s s i o n s . H i s s t a y a t t h e ISN was made p o s s i b l e by a NATO g r a n t .REFERENCES
H. KOHL, T h e s i s , U n i v e r s i t y o f F r a n k f u r t ( 1 9 8 6 ) , u n p u b l i s h e d H. KOHL and R.M. DREIZLER, P h y s . L e t t . 98 A ( 1 9 8 3 ) 95 H. KOHL and R.M. DREIZLER, P h y s . Rev. L e t t . 56 ( 1 986) 1993 H. KOHL, u n p u b l i s h e d
H. KOHL and R.M. DREIZLER, Chem. P h y s . L e t t . 128 (1986) 189
P . RING a n d P. SCHUCK, The n u c l e r many body p r o b l e m , S p r i n g e r (1980)
H.R. LEWIS andP.G.L.LEACH, J. Math. Phys. 23 ( 1 9 8 2 ) 2371
H.R. LEWIS, J. Math. P h y s . 25 (1984) 1139
H. KOHL, P. SCHUCK a n d S. STRINGARI, Nucl. P h y s . A 459 (1986) 265
