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ON WIGNER’S FUNCTION AND OTHER FUNCTIONS ASSOCIATED WITH OPERATORS
N. Balazs
To cite this version:
N. Balazs. ON WIGNER’S FUNCTION AND OTHER FUNCTIONS ASSOCIATED WITH OPER- ATORS. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-11-C6-14. �10.1051/jphyscol:1984602�.
�jpa-00224203�
JOURNAL DE PHYSIQUE
Colloque C 6 , s u p p l e m e n t au rt°6, Tome 4 5 , juin 1984 page C 6 - I 1
ON WIGNER'S FUNCTION AND OTHER FUNCTIONS ASSOCIATED WITH OPERATORS
N.L. B a l a z s
Department of Physics, State University of New York at Stony Brook, Stony Brock, New York 11794, U.S.A.
Abstract - We discuss a general method of associating functions with dynamical operators, and offer some guiding principles to search for a suitable
association.
Sganarelle: "Votre religion, a ce que je v o i s , est done ].'arithmetique ? II faut avouer qu'il se met d'etranges folies dans la tete des hommes, et q u e , pour avoir bien etudie, on est bien moins sage le plus souvent."...
Moliere, Don Juan Acte III, scene 1.
This note contains some ideas about the pattern of semiclassical approximation schemes • I do not offer new results, but, a different point of view which may bring new results. The basic ideas are not novel; their application in this context may well b e . The literature references can be found in a review article written
together with D r . B. Jennings, and which will appear shortly in Physics Reports.
I do apologize to the many people whose work. I have not adequately discussed.
x x x -
The expectation values of observable quantities in quantum mechanics can be computed by the rule
9. , = Trace (fJp) obs
where Q is the operator describing the observable and p is the density operator describing the state of the system. For a given observable and state, fiODS
is a function of Planck's constant. The problem of semi-classical approximations is to identify the class of states and operators for which this Planck constant dependence is simple and can be computed systematically through an approximation scheme which starts from the knowledge of the classical behavior of the system.
There are a variety of methods grappling with this problem. All of the ones I know of can be subjugated to the following scheme.
One associates a function f(p,q) with p , a function £2(p,q)with il, a functional F[f,.Q] with the product pQ, and the integral /dp dq F[f,f>] with Tr(pfi). At this stage p,q are simply integration parameters. In the classical limit, however, p,q are the canonical momenta and coordinates, £i(p,q) the classical dynamical quantity described by the operator fi, and f(p,q) the classical phase space distribution function specifying the state of the system (or of an ensemble of systems);
in this limit the functional K[f»S2] is simply the product, fs>. If, however, h is not zero, f,Q depends on h ; the functional F i s , in general, no longer a product, and dependson h; p,q have no longer the simple physical meaning of momenta and coordinates. The different semi-classical calculations distribute differently the Planck constant dependence in the expression dp dq [f.fi] among
Résumé - Nous discutons d'une méthode générale permettant d'associer des fonctions aux opérateurs dynamiques et nous présentons quelques points de repère en vue de recherches d'une association appropriée.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984602
C6-12 JOURNAL
DE
PHYSIQUEt h e f u n c t i o n s f , R , t h e f r ~ n c t i o n a l . F, and t h e p a r a m e t r i s a t i o n p , q . S i n c e t h e h d e p e n d e n c e o f t h e i n t e g r a l i s u n i q u e l y s p e c i f i e d by ~ r ( j f l ) t h i s r e s h u f f l i n g c a n o n l y s e r v e t h e p u r p o s e t o a i d t h e i n t u i t i v e u n d e r s t a n d i n g and t h e c o m p u t a t i o n a l e a s e o f t h e i n d i v i d u a l e n t i t i e s f , R , F, p, q .
To u n d e r s t a n d t h e d i f f e r e n t s c h e m e s i t i s u s e f u l t o p o s e two q u e s t i o n s : (A) How t o a s s o c i a t e f u n c t i o n w i t h d y n a m i c a l o p e r a t o r s ?
(B) How t o c h o o s e t h i s a s s o c a t i o n s o t h a t
a ) t h e s e f u n c t i o n s s h o u l d g i v e t h e c l a s s i c a l d y n a m i c a l q u a n t i t i e s when h = 0 ;
b ) s h o u l d b e e a s i l y c o m p u t a b l e i f h # 0 b u t " s m a l l " ( a n d t e l l me what
" s m a l l " means) ;
c ) t h e p r o p e r t i e s o f t h e f u n c t i o n s c o u l d b e u n d e r s t o o d q u a l i t a t i v e l y i f h f i n i t e ;
d ) t h e f u n c t i o n a l F [ f , Q ] s h o u l d n o t h e t o o h o r r i b l e when h f i n i t e b u t s m 3 l l .
L e t u s l o o k a t q u e s t i o n ( A ) . We n o t i c e i n o u r scheme t h e f o l l o w i n g s a l i e n t p o i n t s . T h e r e e x i s t new m n t h e m a t i c a l o b j e c t s , o p e r a t o r s , whir11 a r e n o t c numl)ers; we d e s i r e t o d e s c r i b e them b y c n u m b e r s ; we a s s o c i a t e a b i l i n e a r f u n c t i o n a l w i t h t h e p r o d u c t o f t h e s e new o b j e c t s . P h r a s e d i n t h i s manner we n o t i c e i m m e d i a t e l y a c l o s e a n a l o g y w i t h v e c t o r c a l c u l u s . V e c t o r s a r e new g e o m e t r i c a l o b j e c t s w h i c h a r e n o t c n u m b e r s ; t h e i r s c a l a r p r o d u c t ( a c number) i s b i l i n e a r . A t t h e same t i m e we
can
d e s c r i b e them b y c numbers b i n t r o d u c i n g c o m p o n e n t s w i t h r e s p e c t t o a c o m p l e t e s e t o f +~s v e c t o r s . I f
2
i s a v e c t o r , a n d e ( a ) i s t h e a - t h b a s i s v e c t o r ( a = 1, 2 , 3 ....),
t h e n
n'
= L :(a> A ( a ) ,w h e r e t h e a - t h compone2t A ( a ) i~ g i v e n ( f o r o r t h o g o n a l b a s e s ) a s A ( a ) = g ( a ) . A , -f
t h e s c a l a r p r o d u c t o f e ( a ) a n d A. The l a b e l a s i m p l y e n u m e r a t e s t h e b a s e s .
We i m i t a t e now t h i s scheme by t h i n k i n g o f t h e o p e r a t o r s a s p o i n t s i n a l i n e a r v e c t o r s p a c e . ( A f t e r a l l , we c a n a d d o p e r a t o r s , a n d m u l t i p l y them by s c a l a r s ) .
.. ..
Thus we a t t e m p t t o w r i t e a d y n a m i c a l o p e r a t o r R ( p , q ) a s
A A A e A , .
R ( p , q ) = I d a w(a) B ( p , q , a ) w h e r e
. . . A
.. ..
A-
B ( p , q , a ) i s a b a s i s s e t o f d y n a m i c a l o p e r a t o r s c o n s t r u c t e d from p , q , i ? ~ - ~ [ ~ , q ] = 1 and a i s a c o n t i n u ? u s l a b e l e n u m e f a t i n g t h i s s e t ; w ( a ) i s a f u n c t i o ~ o f a a s s o c i a t e d w i t h t h e o p e r a t o r (2 i n t h e b a s i s B. F o r many b a s e s w ( a ) i s T r a c c (R B ) . [ T h e r e a r e p r a c t i c a l c o m p l i c a t i o n s b o t h i n t h e v e c t o r i a l p r o b l e m and t h e o p e r a t o r p r o b l e m a b o u t t h e d e t a i l s ( e . g . t h e u s e o f a b a s i s and a r e c i p r o c a l b a s i s , e t c . ) ; b u t t h e s e a r e p u r e l y t e c h n i c a l q u e s t i o n s , and c a n b e r e s o l v e d . ]
kee
tlze di66ehent h e m i - d t m ~ i c n ! c~pptoximation ochernu ah$ oh
t k i n6 0 m , and they
n h p L ydi66eh &tom each o t h m
byth& choice
06 t h e n c t 8.A A
S i n c e a l l b a s e s B a r e c o n s t r u c t e d f r o m p and q we no l o n g e r i n d i c a t e t h e ppq d e p e n d e n c e ; i n a d d i t i o n , i t - i s n e c e s s a r y t o u s e two s e t s oE e n u m e r a t i o n v a r i a b l e s , p , q ; o n e is r e l a t e d t o t h e p d e p e n d e n c e , t h e o t h e r t o t h e q d e p e n d e n c e . We g i v e now a few e x a m p l e s o f d i f f e r e n t b a s e s , w h i c h l e a d t o s c h e m e s a s s o c i a t e d w i t h t h e names.
T h e r e
.. ..
p
-*
f w ( p . q ) 2. T r a c e p A ( p , q ) ;- ,.
R -+ Qw(p , q ) 2. T r a c e ilA(p , q ) ;
a a
( w i t h q f = q j i = q ; pf = pn = p. a f t e r d i f f e r e n t i a t i o n ) . Kirkwood :
*
,.
P + f k ( p , q ) C T r a c e P A k '
A A
R + i ? k ( ~ , q ) 2. T r a c e RA k '
a a
F ( f . 2 ) c [ e x p
(ih E, -
e x p ( i t-
-) f ( q sP ) c ( q c $ ~ s l )a q f apR a q R a p f f f
( w i t h q f = qG = q ; p f = pR = p a f t e r d i f f e r e n t i a t i o n ) . G l a u b e r
.. .. ' v 2 / r w +
u mw) 2
B = A % I d u d,, eiu(P-P) + i v ( ; - q ) e - Z ( N
C u b i c p h a s e :
-
A i u ( p - p )+
i v (q-q) - i h v c o n s t 3 B = A 2. I d u d v e3
[ F o r f u r t h e r d e t a i l s a n d e x a m p l e s c o n s u l t P h y s i c s R e p o r t s . ]
T h e u s u a l t e c h n i q u e , t h e n , c o n s i s t s o f s e l e c t i n g a p a r t i c u l a r b a s i s , F i n d i n g f , R . F t o a g i v e n o r d e r i n 11, r e a s s e m b l i n g t h e i n t e g r a m d a n d i n t e g r a t i n g . I t i s n o t h a r d t o show t h a t a l l t h e s e s c h e m e s a r c e q u i v a l e n t . How s l l o u l d we t h e n c h o o s e anlong t h e m ? T h e r e i s n o a n s w e r i n g e n e r a l . However, o n e !nay h o p e t h a t a g u i d i n g
p r i n c i p l e c a n b e f o u n d t o c h o o s e a p a r t i c u l a r b a s i s f o r a p a r t i c u l a r p r o b l e m . Some n a t u r a l s u g g e s t i o n s come t o mind i m m e d i a t e l y .
h t , i f possible, a l l s y n l n l e t r i e s 111 t l ~ c b a s i s . Make t h e h . i s i s r,nuge i ~ i v . i r i a ~ i t .
M i n i m i z e t h e l'lanck c o n s t a n t d e p e n d e n c e o f t h e f u n c t i o n s a s s o L i a t e d w i t h a b a s i s ( w h i c h , h o w e v e r , w i l l make t h e f u n c t i o n a l F h o r r i b l e ) . C h a n g e t h e p a r a m e t r i s a t i o n o f t h e b a s i s .
T r y t o a s s o c i a t e p o s i t i v e f u n c t i o n w i t h p o s i t i v e o p e r a t o r s .
C6-14 J O U R N A L DE PHYSIQUE
I p r o p o s e h e r e ~ r ~ o t h e r p o s s i b i l i t y inotivatt,tl by Ca I - t a n ' s idc.8 o f u s i n g a r e p & r e n.obi.le i n t 1 i f f c t r c n t i ; a l g c c m e t r y . 111 t h e scllemcs grol)osell ( 1 1 1 ~ f i x e r t h e b a s i s i n a d v a n c e a n d c a l c u l a t e s t h e c u f l f i c i c n t s 01-der b y o r d e r a c c o r d i n g some a p p r o x i i l l a t i o n scheme. I n s t e a d of t h i s , do n o t f i x t h e b a s i s , b u t c h a n g e b a s i s anti c o e f f i c i e n t s t o g e t h e r making t h e r e b y t h e a p p r o x i m a t i o n scheme more a J a p t a b l e t o t h e i d i o s y n c r a s i e s o f t h e s m a l l p a r a m e t e r a t h a n d .
For e x a m p l e , i f we s t u d y w i g n e r 9 s f u n c t i o n f o r a s t a t i o n a r y s t a t e we f i n d t h a t t h e b e h a v i o r i s v a s t l y different n e a r t h e H ( p . q ) = E s u r f a c e and f a r away from i t . T h i s s r l g g e s t s t h a t wc s l ~ o c l d a l t e r t ! ~ c ~ ( p . 1 ) h n r i q f o r p , q p o i ~ l r s n e n r t h c e n c r g v s u r f a c e . O r , i f w e s t u d y t i m e d e p e n d e n t p r l ~ b l e m s we s h o u l d n o t o n l y g a k e t h e b a s i s t i m e d e p e n d e n t ():ui~i): s i n t p l y ) i n t o t h e H e i s e n b e r i : p i c l u r e , l ( p , q ) + .-jut ,",(p,q)eitl~
b u t c h a n g e t h e b ; ~ s i s a s we1 I., s o a s t o remove t h e quantuni m e c h a n i c a l c o m p l i c a t i o n s a r l s i n g f r o m t l l e p a r t i c u l a r
I?
u s e d , o r remove Llle c l a s s i c a l t i m e d e p e n d e n c e a l t o g e t h e r . The a s t u t e r e a d e r c a n f i n d many s i m i l a r p o s s i b i l i t e s .To s u m m a r i z e :
I t i s u s e f u l t o t h i n k o f 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 s c h e m e s a s t h e s t u d v of
f u n c t i o n s which a r e e x p a n s i o n c o e f f i c i e n t s o f o p e r a t o r s wiLh r e s p e c t t o a s t a r t d a r ~ l s e t of o p e r a t o r s . T h e s e f u n c t i o n s a r e d i s p l a y e d i n :I m a n i f o l d p a r n m c t - r i s e d by t h e p a r a m e t e r s e n u m e r a t i n g t11i.s s t a n d a r d s e t . T h e s e m ; l n i f o l d s a r e t i l e mock-pllase s p ; l c e s . T h e r e is n o t h i n g s a c r e d abotlt a p a r t i c u l a r s t a n d a r d s e t o f o p e r a t o r s , and w e s h o u l d a d a p t o u r s t a n d a r d s e t t o t h e e x i g e n c i e s of a p a r t i c u l a r p r o b l e m . A t t h e same t i m e we a l s o shoul(1 r e t a i n ( a s much a s p o s s i b l e ) c e r t a i n c h a r i t c t e r i s t i c f e a t u r e s o f t i l e mock-phase s p a c e , which e n a b l e s u s t o g r a s p i n t u i t i v e l y t h e p r o p e r t i e s of t h e
f u n c t i o n s a s s o c i a t e d w i t h o p e r a t o r s w i t h o u t a c t u a l l y p e r L o r m i n g a l l t h e c a l c u l a t i o n s . The f a s c i n a t i o n o f t h e f i e l d o f s e m i c l a s s i c a l methotis l i e s p e r h a p s i n t h i s
e x p l o i t a t i o n o f m a t h e m a t i c a l f l e x i l ~ i l i t y i n t r y i n g t o r e c o r ~ c i l e i n c o m p a t i b l e s i m p l i f i c a t i o n s a l l d e s i r e d i n t u i t i v e l y .
