On a probabilistic interpretation of expansion coefficients in the non-relativistic quantum theory of resonant states

Volume 33B. number 8

ON IN

PtfYSICS

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A PROBABILISTIC INTERPRETATION THE NON-RELATIVISTIC QUANTUM

21 D e c e m b e r 1970

OF EXPANSION COEFFICIENTS THEORY OF RESONANT STATES

T. B E R G G R E N Department of Mathemalical P h y s i c s , Lund lnslilute of Teclmology, S-220 07 Lurid 7, Sweden

Received 4 November 1970

It is suggested that "complex probabilities" appearing in eigenfunetion expansions involving resonant states ean be i n t e r p r e t e d in t e r m s of two real prolmbilities. This is shown to be consistent with identi fieation ambiguities due to the finite lifetime of sueh states.

D u r i n g t h e l a s t f e w y e a r s t h e i n t e r e s t in r e s o n a n t s t a t e s (often c a l l e d G a m o w s t a t e s ) h a s b e e n s t e a d i l y r i s i n g , l a r g e l y b e c a u s e of the r o l e t h e s e s t a t e s m a y p l a y in v a r i o u s d i r e c t r e a c t i o n s . Their mathematical properties, especially with r e s p e c t to n o r m a l i z a t i o n , o r t h o g o n a l i t y , a n d c o m p l e t e n e s s , h a v e b e e n s t u d i e d by s e v e r a l a u t h o r s [ 1 - 4 ] , a n d one m a y n o w s a y t h a t t h e s e p r o b l e m s h a v e b e e n s o l v e d in a f a i r l y s a t i s f a c t o r y way. H o w e v e r , so f a r no g e n e r a l a g r e e m e n t h a s b e e n r e a c h e d on h o w to c a l c u l a t e e x p e c t a t i o n values, transition rates, and cross sections i n v o l v i n g s u c h s t a t e s . T h i s l a c k of a g r e e m e n t c o n c e r n i n g q u a n t i t i e s of c e n t r a l i m p o r t a n c e f o r t h e p h y s i c a l i n t e r p r e t a t i o n of t h e t h e o r y i s due to s o m e p e c u l a r i t i e s of the G a m o w s t a t e s w h i c h l e a d to c o m p l e x v a l u e s of s o m e q u a n t i t i e s a n d f u n c t i o n s w h i c h a r e a n a l o g u o u s to p r o b a b i l i t i e s a n d p r o b a b i l i t y d e n s i t i e s in the u s u a l a p p l i c a t i o n s of q u a n t u m m e c h a n i c s . It t h u s s e e m s i m p o s s i b l e to i n t e r p r e t s u c h q u a n t i t i e s in a c c o r d a n c e w i t h t h e p r o b a b i l i s t i c i n t e r p r e t a t i o n u s e d so s u c c e s s f u l l y s i n c e t h e b e g i n n i n g of q u a n t u m m e c h a n i c s in the m i d t w e n t i e s . It i s the p u r p o s e of t h i s n o t e to p r e s e n t a n interpretation scheme which coincides with the u s u a l one f o r o r d i n a r y b o u n d s t a t e s w h i l e f o r G a m o w s t a t e s it y i e l d s r e a l q u a n t i t i e s w h i c h c a n b e u n d e r s t o o d in o r d i n a r y s t a t i s t i c a l t e r m s a n d which are fully compatible with the experimental or observational situation associated with the uns t a b l e c h a r a c t e r (finite life t i m e ) of t h e G a m o w s t a t e s . T h e k e y to t h e i n t e r p r e t a t i o n l i e s in t h e r e c o g n i t i o n of the f a c t t h a t w h i l e b o u n d s t a t e s c a n b e i d e n t i f i e d w i t h o u t a m b i g u i t y by m e a n s of a n a c c u r a t e d e t e r m i n a t i o n of i t s e n e r g y ( s u p p l e m e n t e d by o t h e r o b s e r v a b l e s if t h e e n e r g y i s d e g e n e r a t e ) , t h e i d e n t i f i c a t i o n of a r e s o n a n t s t a t e

is always more or less uncertain, firstly because of i t s f i n i t e l i f e t i m e a n d s e c o n d l y b e c a u s e it a p p e a r s in the m i d s t of a b a c k g r o u n d of c o n t i n u u m s t a t e s w h i c h m a y o r m a y not r e p r e s e n t t h e d e c a y p r o d u c t s of the r e s o n a n t s t a t e . F r o m the f o r m a l p o i n t of v i e w , t h e i n t e r p r e t a t i o n p r o b l e m s t e m s f r o m the f a c t t h a t the G a m o w w a v e f u n c t i o n i s n o t s q u a r e i n t e g r a b l e . By d e f i n i tion, a Gamow function is a formal eigenfunction of the H a m i l t o n i a n h a v i n g only o u t g o i n g w a v e s at i n f i n i t y [5]. I t s t i m e r e v e r s e i s a l s o a f o r m a l e i g e n s t a t e b u t w i t h only i n c o m i n g w a v e s a s y m p t o t i c a l l y . By m e a n s of c o n v e r g e n c e f a c t o r s [ 1 , 2 , 4 ] o r a n a l y t i c c o n t i n u a t i o n [3] u n i t a r y i n n e r p r o d u c t s c a n b e d e f i n e d p r o v i d e d one of the f a c t o r s i s a n o u t g o i n g , the o t h e r f a c t o r a n i n c o m i n g , G a m o w s t a t e . T h u s t h e two t y p e s of s t a t e f o r m t o g e t h e r a biorthogonal set, and completeness relations c a n b e d e r i v e d [2] w h i c h m a k e e i g e n f u n c t i o n e x p a n s i o n s p o s s i b l e . A c o m p l e t e s e t of s t a t e s t h u s m a y c o n t a i n the b o u n d s t a t e s of the H a m i l t o n i a n , a s e l e c t e d n u m b e r of o u t g o i n g G a m o w s t a t e s , a n d a c o m p l e m e n t a r y s e t of c o n t i n u u m states. This complete set forms a basis, the r e c i p r o c a l b a s i s of w h i c h c o n t a i n s t h e b o u n d s t a t e s , the i n c o m i n g G a m o w s t a t e s c o r r e s p o n d i n g to t h e o u t g o i n g o n e s in t h e f o r m e r s e t , a n d a c o m p l e m e n t a r y s e t of c o n t i n u u m s t a t e s w h i c h a r e e s s e n t i a l l y t h e c o m p l e x c o n j u g a t e s of t h o s e in t h e f o r m e r s e t . T h e r e f o r e , an e i g e n f u n c t i o n e x p a n s i o n of t h e i n n e r p r o d u c t (tpll ~2} h a s t h e f o r m

+ / dak <'11 where



I~n} a n d 1¢0k} a r e e i g e n s t a t e s

(1)

of t h e 547

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H a m i l t o n i a n b e l o n g i n g to t h e d i s c r e t e (bound a n d resonant states) and the continuous spectrum, r e s p e c t i v e l y . T h e s t a t e s !~n} a n d ] ~ k ) a r e c o r r e s p o n d i n g s t a t e s in the r e c i p r o c a l b a s i s . If [tp1} = 't~2}= [~) i s a n o r m a l i z e d s t a t e d e s c r i b i n g a s y s t e m u n d e r o b s e r v a t i o n in a s t a t e p r e p a r e d in a d e f i n i t e way, we h a v e :

= n

(qSn[~,: + continuum contribution.

(2) In t h i s s u m , the t e r m s c o n t a i n i n g b o u n d states are real and positive quantities having the w e l l k n o w n i n t e r p r e t a t i o n a s p r o b a b i l i t i e s of f i n d i n g t h e s y s t e m in t h e c o r r e s p o n d i n g b o u n d e i g e n s t a t e s of the H a m i l t o n i a n . T h e b o u n d s t a t e s a r e t h e i r own t i m e r e v e r s e s , a n d t h i s p r o p e r t y l e a d s to t h e r e a l i t y of t h e c o r r e s p o n d i n g t e r m s in the e x p a n s i o n [2]. T h e t e r m s c o n t a i n i n g t h e r e s o n a n t s t a t e s , h o w e v e r , n e e d not b e r e a l , a n d t h e r e f o r e we c a n n o t i n t e r p r e t t h e s e t e r m s a s probabilities. This also holds for the continuum contribution. H o w e v e r , a s we h a v e a l r e a d y i n d i c a t e d , t h e i d e n t i f i c a t i o n of a r e s o n a n t s t a t e i s i n e v i t a b l y to s o m e e x t e n t m n b i g u o u s . In o r d e r to a p p l y statistical methods or use statistical concepts o n e m u s t b e a b l e to i d e n t i f y t h e o u t c o m e of a n experiment which is repeated a sufficient numb e r of t i m e s so t h a t f r e q u e n c y r a t i o s c a n b e taken as "experimental" values for probabilities. L e t u s i m a g i n e a n e x p e r i m e n t (e.g. i n e l a s t i c s c a t t e r i n g by an a t o m i c n u c l e u s ) t h e o u t c o m e of w h i c h we w i s h to c l a s s i f y in t e r m s of p o p u l a t i o n of v a r i o u s b o u n d a n d r e s o n a n t s t a t e s t o g e t h e r w i t h a b a c k g r o u n d . T h e i d e n t i f i c a t i o n of t h e s t a t e w h i c h i s p o p u l a t e d in t h e p r o c e s s m u s t b e a c c o m p l i s h e d by m e a s u r i n g t h e e n e r g y of t h e s y s t e m (in t h e e x a m p l e m e n t i o n e d a b o v e by m e a s u r i n g t h e e n e r g y l o s s of the s c a t t e r e d p r o j e c t i l e o r r a d i a t i o n ) . W h e t h e r we c o n s i d e r a n i n d i v i d u a l e v e n t o r a l a r g e s a m p l e of e v e n t s s u c h a s a n e x p e r i m e n t a l s p e c t r u m , we c a n s o m e what schematically classify the events with resp e c t to a p a r t i c u l a r r e s o n a n t s t a t e ] 4~n}in (at least) three categories:

(i) practically certain identification as ICn); (it) practically certain identification as not 1qbn); (iii) uncertain whether [qhn) or not. Note that by using this classification the problem is now formulated in terms of certain or uncertain identification of the state, i.e. on the observational level, not in terms of population of the states of the system, which is on the level of the internal structure and dynamics of the 548

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i s o l a t e d s y s t e m . A s s u m i n g t h a t we c a n d e c i d e w h i c h of t h e c a t e g o r i e s (i), (ii) o r (iii) a g i v e n m e a s u r e m e n t o r o b s e r v a t i o n b e l o n g s to, we a r e now in a p o s i t i o n to a p p l y s t a t i s t i c s in t h e u s u a l way a n d t h u s a s s i g n p r o b a b i l i t i e s an, b n a n d c n f o r the o u t c o m e s (i), (ii) a n d (iii), r e s p e c t i v e l y , of t h e identification. Clearly a n + bn + cn = 1

(3)

since the categories (i), (it) and (iii) comprise all possibilities and are mutually exclusive. The probability that out of N identifications K are in (i) and L are in (it) is then given by the expression N: a nil bnL c N - K - L / which follows from elementary statistical principles. T h e n e w p h y s i c a l e l e m e n t in t h e a n a l y s i s i s t h e p o s s i b i l i t y t h a t cn, t h e p r o b a b i l i t y of a n u n c e r t a i n i d e n t i f i c a t i o n of t h e s t a t e I ~bn), i s g r e a t e r than zero however carefully the identification method is designed, because such methods must u l t i m a t e l y r e s t on a n e n e r g y d e t e r m i n a t i o n . In t h e c a s e of a G a m o w s t a t e w i t h f i n i t e l i f e t i m e one cannot improve the energy resolution beyond t h a t g i v e n b y H e i s e n b e r g ' s p r i n c i p l e of i n d e t e r m i n a c y a n d t h e l i f e t i m e of t h e s t a t e , s i n c e in t h e time interval needed for a more accurate energy d e t e r m i n a t i o n t h e s y s t e m w o u l d too o f t e n h a v e disintegrated before the measurement was comp l e t e . One w o u l d t h e n not b e s u r e w h e t h e r t h e r e s u l t o b t a i n e d i s p e r t i n e n t to t h e s y s t e m in t h e G a m o w s t a t e o r to t h e d e c a y p r o d u c t s . On t h e o t h e r h a n d , if t h e t i m e i n t e r v a l i s m u c h s h o r t e r t h a n t h e l i f e t i m e of t h e s t a t e , t h e e n e r g y r e s o l u t i o n m a y n o t e v e n b e s u f f i c i e n t to d i s c r i m i n a t e against other resonances with nearly the same energy and the same quantum numbers otherwise. In accordance with these general remarks we expect that the number Cn, which represents the degree of ambiguity in the identification of the state I ~bn), increases with decreasing lifetime. Thus we need for each resonant state two real numbers, e.g. an and Cn, to characterize the observational possibilities. Now t h e f a c t t h a t e q . (2) p r o v i d e s a c o m p l e x n u m b e r Pn @lqhn)(qhnlq.,} f o r e a c h r e s o n a n t s t a t e t u r n s out to b e a d v a n t a g e o u s r a t h e r t h a n p u z z l i n g . It o n l y r e m a i n s to f i n d a n a t u r a l r e l a t i o n b e t w e e n t h e c o m p l e x n u m b e r Pn = r n + i q n a n d the t h r e e p r o b a b i l i t i e s an, bn, a n d c n (of w h i c h one m a y b e e x p r e s s e d i n t e r m s of t h e o t h e r two b y v i r t u e of eq. (3)). T h e p r e s c r i p t i o n we p r o p o s e i s t h e f o l lowing: i

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a,,+

,

b,~+ % =

[t -Pnl: ((1 rn)2 +qln)l/2 -

(5)

T h e q u a n t i t y ]Pnl i s t h u s the p r o b a b i l i t y of a p o s s i b l e i d e n t i f i c a t i o n of the s t a t e a s 16 n) w h i l e ]1 - Phi i s t h e p r o b a b i l i t y of a p o s s i b l e i d e n t i f i c a t i o n of the s t a t e a s not I(Pn)- S i n c e t h e s e two a l t e r n a t i v e o u t c o m e s a r e not m u t u a l l y e x c l u s i v e (if the i d e n t i f i c a t i o n i s u n c e r t a i n it m a y b e i n c l u d e d in b o t h p o s s i b i l i t i e s ) t h e s u m of t h e s e p r o b a b i l i t i e s m a y e x c e e d u n i t y ; in f a c t we g e t f r o m eq. (3) by a d d i n g (4) and (5)

Ip~i + I1 - p~l - ~ ÷ ~,~. Eqs.

(4) and (5) yield after some

(6) trivial algebra

1

r n : a n + Cn(a n + ~c n) ,

(7)

q2 = (2 + C n ) ( a n + ½Cn)(a n + ½Cn)C n .

(8)

T o f i r s t o r d e r in Cn, t h e r e a l p a r t r n of Pn i s g i v e n by r n ~ a n + anCn, i.e. t h e p r o b a b i l i t y we would get if we d i s t r i b u t e d the u n c e r t a i n i d e n t i f i c a t i o n s a m o n g t h e c e r t a i n c a t e g o r i e s (i) a n d (ii) in proportion to their respective probabilities. Therefore r n may be interpreted as an "average" probability for populating the resonant state l~bn) when the system is prepared in the state ]~h). But r n is by itself insufficient to describe all aspects of the situation, as we have seen above. The imaginary part qn evidently represents the influence on the observable properties of the state from the processes which constitute the decay of the state and which may intervene in the measuring process. Some support for such an interpretation is found in the fact that if we expand (g/I ~P) in terms of incoming resonant states, which describe capture instead of disintegration, the states l~Sn) and (~n] change places in eq. (2) and the imaginary parts of the terms change their signs, which apparently corresponds to a reversal of the directions of the elementary processes mentioned above. (Note that eq. (8) is independent of the sign of qn') An important condition for the interpretation given by eqs. (4) and (5) is that the resulting probabilities an, bn, and c n are all non-negative numbers smaller than unity. This provides a natural

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criterion for selection of the resonant states one wants to include in the complete set of states. A different criterion is given by a comparison of the life time of the state with the time it takes for the relative distance between the fragments to change by an amount equal to the range of the interaction between them. We expect the resonant state to be of physical importance only if the fragments are held together for a significantly long time on this time scale. Without having any concrete example it is difficult to determine which of these criteria is the more restrictive one, but intuitively one believes that they both act in the s a m e direction. The extension of the interpretation given by eqs. (4) and (5) to cases in which several resonant states are considered separately is immediate and need not be discussed here. An interesting question is if the algebraic sums of the contributions to the expansion (2) from two different resonant states can be directly interpreted in an analoguous manner. Preliminary studies indicate that the answer to this question is in the affirmative, again due to identification ambiguities. The complex terms in eq. (2) therefore represent an additive oroperty of the resonant states, similar to the probability of a bound state. We intend to pursue this problem in a forthcoming note, in which also the definition and interpretation of expectation values of observables in systems that may be considered to be in a resonant state is discussed in the light of the interpretation proposed here. I wish to thank Dr. Bengt E. Y. Svensson for several discussions and critical reading of the manuscript.

References [1] Y a . B . Z e l ' d o v i c h , Zn. Eksp. i Teor. Fiz. 39 (1960) 776; Soy. Phys. J E T P 12 (1961) 542; N.Hokkyo, P r o g . Theor. Phys. 33 (1965) 1116. [2] T . B e r g g r e n , Nucl. Phys. A109 (1968) 265. [3] W . J . R o m o , Nucl. Phys. A l l 6 (1968) 617. [4] B. Gyarmati and T. V e r t s e (preprint, Institute of Nuclear R e s e a r c h of the Hung. Acad. of Schiences, Debrecen, Hungary, 1970). [5] G.Gamow, Z. Phys. 51 (1928) 204; A . J . F . S i e g e r t , Phys. Rev. 56 (1939) 750.

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