A remark on the generator-coordinate method and the quasi-boson approximation

Volume27B, number 10

A REMARK

PHYSICS

ON

LETTERS

14 October 1968

THE GENERATOR-COORDINATE QUASI-BOSON APPROXIMATION

METHOD *

AND

THE

H. UI ** a n d L. C. BIEDENHARN

Department of Physics, Duke University, Durham, North Carolina, USA Received 25 July 1968

By use of Bargmann's integral transform and the coherent state formalism, it is proved that the generator-coordinate method with quadratic approximation to the kernels (= Gaussian kernels) is completely equivalent to the quasi-boson approximation to the original Hamiltonian.

R e c e n t l y , s e v e r a l a u t h o r s [1] have r a i s e d the q u e s t i o n a s to the r e l a t i o n b e t w e e n the m e t h o d of g e n e r a t o r - c o o r d i n a t e s [2] a n d the q u a s i - b o s o n a p p r o x i m a t i o n [3]. In the m e t h o d of g e n e r a t o r c o o r d i n a t e s , one c o n s i d e r s f i r s t a f a m i l y of N p a r t i c l e f u n c t i o n s qS(x, ot), w h e r e ot s t a n d s for a s e t of p a r a m e t e r s . T h e t r u e wave f u n c t i o n i s g e n e r a t e d by the g e n e r a t i n g f u n c t i o n 0 (x, ot ) by the o p e r a t i o n :

be u n d e r s t o o d a s dot ~- d ( R e o t ) . d (Imot). It is to be n o t e d that (5) c o n t a i n s a s a s p e c i a l c a s e the conventional Gaussian overlap kernel e x p { - (½). (ot _ot,)2} for r e a l ot. G e n e r a l i z a t i o n to the c a s e of n p a r a m e t e r s w i l l b e d i s c u s s e d l a t e r . Let u s i n t r o d u c e n e x t the m o d i f i e d weight f u n c t i o n F(ol) d e f i n e d by

(1)

T h e n , t h e i n n e r p r o d u c t ( ~ i I~)) can b e w r i t t e n a s

~o(x) : f o(ot,x) f(ot) dot .

A p p l y i n g the v a r i a t i o n a l p r i n c i p l e to the e x p e c t a t i o n v a l u e of the H a m i l t o n i a n , H, of the s y s t e m , 6[(~ I H I ~ ) ] = 0, l e a d s t h e n to an i n t e g r a l e q u a t i o n for d e t e r m i n i n g the w a v e - p a c k e t - l i k e weight f u n c t i o n f ( a ) :

f K(ot',ot)f(ot) da = E f I(ot',ot)f(ot) dot .

(2)

H e r e the H a m i l t o n i a n k e r n e l a n d the o v e r l a p k e r n e l a r e defined, r e s p e c t i v e l y , by

K(ot',ot) = (4)(X,ot') IH [0(X,ot)) ,

= 1 exp(_½1a 12)F(@).

(6)

1

<~il~j) = f dxCPi(x)~j(x ) =~-~ f F j ( S ' ) ×

(6)

X e x p ( - I o t ' 12 - lot 12+~'ot) Fi(~)dot'dot' U s i n g the i d e n t i t y [4] 1 f exp (flot - lot 12) F ( 5 ) dot = F(~) ,

(7)

we o b t a i n

(3)

and

= fPj(S')'Fi(~)

d~(a),

(8)

w h e r e the G a u s s i a n m e a s u r e dlz (a) i s d e f i n e d by

l(ot',ot) = <~(x, ot') [ 0(x, ot)) .

(4)

In t h i s note, we s h a l l i n v e s t i g a t e the c o n s e q u e n c e s of the o v e r l a p k e r n e l s of the following f o r m ,

I(ot',ot) = exp((~'ot - ½ 1 o t i 2 - ½ [ o t ' l 2) ,

(5)

w h e r e o7 d e n o t e s t h e c o m p l e x c o n j u g a t e of ot. S i n c e ot i s c o m p l e x , da in eqs. (1) a n d (2) s h o u l d * Supported in part by the Army Research Office (Durham) and the National Science Foundation. ** On leave of absence from Department of Physics, Tohoku University, Sendai, Japan. 608

f(~)

1

d~ (ot)= ~ exp (-lot12) dot . In terms of F(~) and d#(a), eq. (I) reads as

qo(x) = f dp(x, ot) exp (½1otl2) F(a) dl~(ot) .

(9)

We i n t e r p r e t t h i s f o r m u l a a s the d e f i n i t i o n of a m a p p i n g F ( 5 ) - . ~0(x), the i n t e g r a l , k e r n e l of which is g i v e n by 0(x, ot)" exp (½lot Iz). If we t a k e the r i g h t - h a n d s i d e of eq. (8) as the d e f i n i t i o n of the i n n e r p r o d u c t ( F I I F i ) , the f u n c t i o n s p a c e { F ( 5 ) } can be m a d e J i n t o a H i l b e r t s p a c e H B d e -

Volume 27B, number 10

PHYSICS

f i n e d by t h i s H e r m i t i a n s c a l a r p r o d u c t . H e n c e , eq. (9) t o g e t h e r with (8) d e f i n e s a u n i t a r y m a p p i n g by w h i c h H B i s m a p p e d onto t h e u s u a l H i l b e r t s p a c e L 2 of t h e s q u a r e - i n t e g r a b l e w a v e f u n c t i o n s (x) in c o n f i g u r a t i o n s p a c e . T h e f o l l o w i n g two p o i n t s a r e e s s e n t i a l to o u r a r g u m e n t s . Firstly, s u c h a kind of m a p p i n g a s m e n t i o n e d a b o v e h a s a l r e a d y b e e n s t u d i e d in d e t a i l by B a r g m a n n [5], who s h o w e d that t h e i n t e g r a l k e r n e l A ( s ,x) of the m a p p i n g can b e uniquely determined as *

A(d,x) =- q~(s ,x) exp (½Is

Further,

14 October 1968 f r o m ( l l a ) and ( l l b ) , we h a v e (~3]a) = exp (~s - ½ I s

exp (- ~ 2 + 2~x - ½x2) = ~n

¢ ( x , a ) = exp ( - ½ I s 12) exp ( - ½ s 2 + ~ f 2 a x - ½ x 2) (10)

~¼

u n d e r t h e s a m e r e s t r i c t i o n ** on F ( ~ ) . Secondly, t h e g e n e r a t i n g f u n c t i o n o b t a i n e d above is precisely the same as the normalized w a v e f u n c t i o n of t h e " c o h e r e n t s t a t e " in q u a n t u m o p t i c s [6], a s w i l l be p r o v e d in t h e f o l l o w i n g * * * T h e c o h e r e n t s t a t e I s ) i s u s u a l l y d e f i n e d a s an e i g e n s t a t e of b o s e d e s t r u c t i o n o p e r a t o r a w i t h a complex eigenvalue a: and


(11)

where a + is the bose creation operator: [a, a +] = 1. Using the complete set { }n)}, where In> = = (n')-½(a+)n]0), we have

.

(13b)

(in~n:) • Hn(X) exp

(- ½x2),

one s e e s i m m e d i a t e l y that t h e g e n e r a t i n g f u n c tion ~b(x,a) d e t e r m i n e d by eq. (10) is i d e n t i c a l to t h e n o r m a l i z e d c o h e r e n t s t a t e w a v e f u n c t i o n in this coordinate representation*: ¢(x,a)

u n d e r t h e r e s t r i c t i o n t h a t t h e f u n c t i o n F ( s ) i s an e n t i r e a n a l y t i c f u n c t i o n of 8 . H e n c e , we h a v e

12-½t~12)

A d o p t i n ~ the c o o r d i n a t e repr~esentation of [n) = {n~2Kn '. }-½Hn(x) exp (-½x z) and u s i n g the g e n e r a t i n g f u n c t i o n of t h e H e r m i t i a n p o l y n o m i a l s

12) =

_ 11 e x p ( _ ½ s 2 + ~ f 2 s x _ ½ x 2)

a l s ) = s Is>

LETTERS

= la).

(14)

S i n c e t h e g e n e r a t i n g w a v e f u n c t i o n is p r o v e d to be d e t e r m i n e d uniquely a s the c o h e r e n t s t a t e , it i s p o s s i b l e to i n t r o d u c e t h r o u g h eq. (11) t h e b o s e c r e a t i o n and d e s t r u c t i o n o p e r a t o r s in o u r p r o b l e m . Now, let us w r i t e e q s . (1) and (2) of t h e g e n e r a t i n g c o o r d i n a t e m e t h o d in t e r m s of t h o s e b o s e o p e r a t o r s . F o r t h i s p u r p o s e , it is c o n v e n i e n t to n o t e t h e f o l l o w i n g u s e f u l i d e n t i t i e s f o r t h e m a t r i x e l e m e n t s of t h e c o h e r e n t s t a t e : f o r t h e m a t r i x e l e m e n t of an e n t i r e f u n c t i o n F(a +) of a + b e t w e e n a c o h e r e n t s t a t e and t h e v a c u u m s t a t e , we h a v e

(a ]F(a +) IO>= exp

(- ½1a I) F ( ~ )

(15)

and f o r the m a t r i x e l e m e n t of an o r d e r e d o p e r a t o r H(a+,a), we h a v e


(16)

= exp ( ~ ' ~ - ½Is[ 2 -2b±ls' 12) H ( s ' , s ) both of w h i c h a r e d i r e c t c o n s e q u e n c e s of eqs.

Is>-- 2

n

and

(°la+)n

[0)

(n'.)~

(Ii) and (13). Eq. (1) can now be written as*$

(12a) =

~n



=

T h e n o r m a l i z a t i o n constant through

n

n.

•

(12b)

<0] a> is determined

= exp(-½1sl

2) .

(13a)

* Note that eq. (9) is precisely the same as the inv e r s e mapping of Bargmann, eq. (2-11) of ref. 5. ** The same restriction on F((~) is already used in eq. (7), although not stated explicitly. *** To the best of our knowledge, the fact that the integral kernel of the Bara'mann transform A(x,S) is identical to exp (~1S 12)'~ Or> has not yet been mentioned previously, although the relation between the coherent state and Bargmann's work has been discussed by Glauber [6].

l/c~(d,x) exp(-½]s 12) _ 1 fds 7r

F(K,) d s =

Is>

(17)

= F(a+)] 0) , where use is made of the over-completeness r e lation [6,9]

1--

/dsls>
:~ The coherent state has been known as the "mininimum uncertainty state" since the late twenties. Indeed, the derivation of the coordinate r e p r e s e n tation of the coherent state given here is due to SchrSdinger [7]. $$ Note that the second form of eq. (17) is the so-called "diagonal coherent state representation" [8]. 609

Volume 27B, number 10

PHYSICS LETTERS

uniquely a s the product of the c o h e r e a t s t a t e s ,

F u r t h e r , if we adopt the u s u a l f o r m of the quadratic a p p r o x i m a t i o n to the H a m i l t o n i a n k e r nel,

g(ot',~) = I(ot',ot).H(~',ot) ,

q~(x, ot) = U t~i )" Such a p r o c e d u r e would be well $

adapted to i n t r o d u c i n g boson r e a l i z a t i o n s of app r o x i m a t e s y m m e t r y groups, for example in defining effective quadrupole o p e r a t o r s through SL(3, R).

(18)

w h e r e I ( a ' , a ) i s given by eq. (5) and H ( ~ ' , ~ ) is a p o l y n o m i a l of a and ~ ', the i n t e g r a l equation (2) can be w r i t t e n in an analogous way to (17) as

(or' I H( a+, a)F(a+)[ O) = E(ot' t F(a+) Io> •

(19)

References

Since (19) holds for any state ]~'>, we have an equation

g(a +, a)F(a+)]O) = E F(a+) I O) ,

(20)

which i s the second quantized f o r m of the S c h r ~ clinger equation of a s y s t e m with the H a m i l t o n i a n H(a +, a). Since we a r e t r e a t i n g a f e r m i o n s y s t e m , eq. (20) can be i n t e r p r e t e d as an a p p r o x i m a t e Schr~dinger equation of the s y s t e m , in which the bose a p p r o x i m a t i o n * has been employed to the exact Schr~dinger equation of the s y s t e m . It is then concluded that the g e n e r a t o r coordinate method with the q u a d r a t i c a p p r o x i m a t i o n , eqs. (5) and (19), to the k e r n e l s is completely equiva l e n t to the bose a p p r o x i m a t i o n to the o r i g i n a l Hamiltonian. A g e n e r a l i z a t i o n to the c a s e of the n - p a r a m e t e r s with the o v e r l a p k e r n e l , l(a',ot)=exp½{.~(-ta i]2

[ot~12+25~oti)},

$

can be p e r f o r m e d s t r a i g h t f o r w a r d l y . In p a r t i c u l a r , the g e n e r a t i n g function ~b(x, ~) can be d e t e r m i n e d • Here, by the bose approximation are meant not only the conventional ones, in ref. 3, but also more general bose expansion techniques such as in ref. 1O. •

610

14 October 1968

****

1. B.Jancovici and D.H.Schiff, Nucl. Phys. 58 (1964) 678; D. M. Brink and A. Weigung, Phys. Letters 26B (1968} 497. 2. J.J.Griffin and J.A.Wheeler, Phys. Ref. 108 (1957) 311; J.J.Griffin, Phys. Rev. 108 {1957) 328; R. E. Peeirls and J. Yoccoz, Proc. Phys. Soc. {London) A7O (1957) 381. 3. A.M. Lane, Nuclear theory (Benjamin, 1964}; G. E. Brown, Unified theory of nuclear models (North-Holland, Amsterdam, 1967). 4. S.S. Schweber, J. Math. Phys. 3 (1962) 831. 5. V.Bargmann, Comm. pure and Appl. Math. 14 (1961) 187; Proc. Natl. Acad. Sci. 48 {1962} 199; J. M. Jauch, Foundation of quantum mechanics (Addison-Wesley, 1967} Ch. 12. 6. R.J.Glauber, Phys. Rev. 131 (1963} 2766; C.L.Mehta and E.C.G.Sudarshan, Phys. Rev. 138 (1965) B274. 7. E. Schr~dinger, Z. Physik 14 (1926) 664; P. Carruthers and M. M. Nieto, Rev. Mod. Phys. 40 (1968} 411. 8. J.R. Klauder, J. McKenna and D.G. Currie, J. Math. Phys. 6 (1965} 734. 9. J.R.Klauder, Ann. Phys. (N.Y.) 11 (1960) 123. 10. S.T.Beliaev and V.G.Zelevinsky, Nucl. Phys. 39 (1962) 582; T. Morumori, M. Yamamura and A. Tokunaga, Prog. Their. Phys. (Kyoto) 31 (1964) 1009; B. Sorens~n, Nucl. Phys. A97 (1967) 1.