Pseudo-cluster expansion

Nuclear Physics 14 (1959/60) 648--660; ( ~ North-Holland Publishing Co., A msterdam Not

to be reproduced by photoprint or microfilm without

PSEUDO-CLUSTER

written

permission from the publisher

EXPANSION

J. FUJITAt Department o/ Physics, College o/ Science and Engineering, Nihon University, Tokyo Received 1 August 1959 Abstract: The Ursell-Mayer cluster expansion method is extended to the case of a q-number model operator for either boson or fermion systems. The many-body problem for a system with strong, hard sphere interactions can be approximately reduced to that for a weakly interacting system without hard sphere interactions. The effective Hamfltonian for the weakly interacting system is given in the form of an expansion series quite analogous to the ordinary cluster expansion. (We call it the "pseudo-cluster expansion" in this paper.) The validity of our expansion method is discussed. It can be proved that in the simplest case we are led to the approximate Bethe-Goldstone equation (or Brueckner approximation). We derive formulas expressing the total energy and the diagonal and non-diagonal matrix elements of operators by the asymptotic wave functions. Possibilities of application of our method are discussed.

1. I n t r o d u c t i o n R e c e n t s t u d i e s of t h e n u c l e a r s a t u r a t i o n p r o b l e m h a v e m a d e r e m a r k a b l e progress. T h e y are g r o s s l y classified i n t o t w o t y p e s : a) B r u e c k n e r ' s a p p r o x i m a t i o n 1, 2, 3) a n d b) t h e v a r i a t i o n a l t r e a t m e n t u s i n g t h e c l u s t e r e x p a n s i o n m e t h o d 4, 5, 6). T h e l a t t e r t r e a t m e n t h a s b e e n discussed o n l y u n d e r v e r y simplified a s s u m p t i o n s w i t h r e s p e c t t o t h e t w o - b o d y n u c l e o n correlations. T h e m o s t c o n v e n i e n t f o r m of t h e c o r r e l a t i o n f u n c t i o n is t h e so-called J a s t r o w t y p e : = MO,

(1)

= I-[/(r,~),

(2)

with

i

(~,J)

w h e r e ~ is t h e t r u e w a v e f u n c t i o n of t h e g r o u n d s t a t e s a t i s f y i n g t h e S c h r o e dinger equation H ~ = E~ (3) a n d t h e m o d e l o p e r a t o r of t h e J a s t r o w t y p e as in (2) is a s s u m e d t o h a v e t h e following properties: 0 /(r,~) = 1

for for

r~.: D r , j - + oo,

(4)

w h e r e r,~ r e p r e s e n t s t h e d i s t a n c e b e t w e e n t h e positions of i - t h a n d ]'-th p a r t i c l e s ? Present address (until June 1960): Department of Physics, Stanford University, Stanford, California. 648

PSEUDO-CLUSTER EXPANSION

649

and D the radius of a "hard sphere". Most previous treatments seem to be based on the idea that we know the model wave function ~ well and the main task is to determine the structure of the model operator M b y the Schroedinger equation (3). According to the variation method we assume the Fermi gas model or shell model for • and minimize the expectation value of the total Hamiltonian H b y changing the parameters contained in the correlation function [(r). On the other hand, the Brueckner approximation method has been formulated apparently in an entirely different way. However, we can also understand it in such a w a y that for a fixed ~5 we determine the parts of the velocity-dependent model operator M which are mostly responsible for the total energy E, b y solving the Bethe-Goldstone equation instead of solving the Schroedinger equation (3) rigorously t. The aim of this paper is to propose a new method to treat a strongly interacting system satisfying suitable conditions to be stated later. Both treatments a) and b) can be regarded as special cases and it is very easy to understand the relation between them. Roughly speaking, our main idea is to isolate in the form (1) a velocity-dependent short range correlation part: here "short range" correlation means a correlation compatible with the idea of cluster expansion. We shall derive the effective Schroedinger equation which the asymptotic wave function ¢ should satisfy: g o O = E~.

(5)

In this paper we would call the method to derive the effective Hamiltonian H o from the true Hamiltonian H the "pseudo-cluster expansion" method. The essence of our method is probably quite similar to the "nearest-neighbour expansion" of Riesenfeld-Watson s), though the apparent features are entirely different. In sec. 2 the method is discussed in detail. Our starting equations are eqs. (1), (3) and (5). The formalism of sec. 2 is of sufficiently general character, b u t the intuitive Jastrow case will be exemplified everywhere. If we can solve the effective Schroedinger equation (5) b y some means exactly, the eigenvalues E are the same as the true eigenvalues, and the true eigenfunctions kv can be obtained b y the relation (1). The advantage of treating (5) rather than (3) is the following: 1) The model Hamiltonian H 9 does not contain any hard core interaction, so that we have several ways of solving the differential equation, if we want to make the effort. In fact the forces involved in H 0 are quite weak compared with the original ones. This suggests the appearance of shell aspects in the effective t T h e e q u a t i o n (3) l e a d s to E -- (4, HkrJ)

(4, ~)

(H4, ~//) for a n y f u n c t i o n 4, so t h a t E c a n be

(4, ~)

d e t e r m i n e d b y i n v e s t i g a t i n g o n l y s o m e p r o p e r t i e s of t h e t r u e w a v e f u n c t i o n ,/1.

650

J. FUJITA

many-body problem (5). Many excellent theories using the Fourier components of the interaction can be applied to (5), but not to (3). 2) In the many-body problem generally, there should exist both long and short range correlations in the wave function. If the particles can be regarded as approximately free, although there exists some kind of long range correlation *, our method is useful for treating such systems. It is possible in principle to discuss which of the shell model or the collective model is more favourable in some given nucleus. On the other hand we meet some disadvantages in treating the effective many-body problem: 1) If the operator M were a unitary operator, the orthonormal complete set W~ would correspond to another orthonormal complete set O, by the relation Wi = MO~. But this is not the case generally (see fig. 1). Accordingly it is

Fig. 1. Definition of t h e a s y m p t o t i c w a v e function # (eq. (5)). The t r u e w a v e functions ~ri are o r t h o n o r m a l , b u t t h e a s y m p t o t i c w a v e functions # are n o t so rigorously in general.

necessary in quantitative discussions to take into account the effect of the small deviation from the orthonormality. A rigorous relation is resembling the 0rthogonality relation of the Hermite polynominals (~,, ~ ) = (MO o MOt) = 0

(i q: i).

(6)

However, it is shown later that the O's are approximately orthogonal. If we can choose the model operator M such that ~J, MJ = 0,

[/-/, M] -----0,

(7)

where the J a n d / 7 represent the total angular momentum and parity operators, the W's having different spins or parities correspond to exactly orthog0nal asymptotic wave functions O's. 2) As shown in this paper, the effective Schroedinger equation (5) is a complicated integro-differential equation in the higher cluster terms. We must solve the equation for instance by the method of successive approximations. In order to choose a suitable M the simplest method is to take into account only the second order cluster terms. It is shown in sec. 3 that to this order the * A s y s t e m of high viscidity c a n n o t be t r e a t e d b y our m e t h o d . However, t h e particles in a nucleus should b e a p p r o x i m a t e l y free, because it has clearly t h e shell model aspects. B o h r model etc. do n o t c o n t r a d i c t t h e a p p r o x i m a t e f r e e d o m of particle motion.

PSEUDO-CLUSTER EXPANSION

651

equation effectively corresponds to the well-known Bethe-Goldstone equation, so that this approximation corresponds to the Brueckner approximation. 2. P s e u d o - c l u s t e r

Expansion

We choose any operator M in eq. (1) which has the following properties e): (I) The product property * M(1, 2 . . . . . N) = Me")(1, 2 . . . . . n ) M ( N ~ ) ( n + l . . . . . Y ) ,

(8)

provided that r~ ~ ]x~--xj[ > 8 for every

= n + l . . . . . N;

stands for the "healing distance" z). By this equation the (generally q-number) operators M (~), M (3}. . . . M CN-1)can be defined, if M is given. In the formulation below it is not necessary at all to assume that the MC~}'sare functions only of the space coordinates of the particles. (II) Norm~!ization for M m : M(1)(i) = 1.

(9)

Since M = 1 if r~ > 8 for every i,/" = 1 , . . . , N, the wave function ~ deviates from the asymptotic wave function when any two particles approach each other within the distance ~. (III) The only essential condition imposed on the operator M and the corresponding # simultaneously is the convergence of our pseudo-cluster expansion: "The pseudo-cluster terms defined below are not of a larger order of magnitudes than the ordinary cluster terms." Strictly speaking, it is not clear whether the condition (III) is satisfied or not until we check the validity of (III) for the M and the exact # which can be obtained by solving (5). For convenience we assume that (~, ~) = 1.

(I0)

The original Ha_miltonian H is written as H =-- H ( ' ) + H (~ = ~.. H~{I~-} - ~ H (a). ~9.

-

(ll)

~ A , + ~., V , .

2 M ,,

(~)

The expectation value E can be expressed as

E = [-~fllogl,(m?,_o,

(12)

t The expression on the right hand side of eq. (8) is consistent with the symmetry property o. the left hand side, because we restrict ourselves to some specific region of configuration spacer

652

]. FUJ ITA

where

I~ ~N~= (qL M+e~In'~'+n"')Mqb).

(13)

Now, quite analogously to the ordinary cluster expansion method, we define the pseudo-cluster integrals xa"~ successively: I~ (1) ~

(14a)

(~, M,(I'+eZH'm M ~ ( ~ ¢ ) = l+O(fl), M~y I q~) = (q~, e pln'''+n,''') q~) (1 -+-x#(2)),

I #(2) ~- ,_,(¢~...M(*l+e_

(14b)

17e(3) ~ ((l~, 71Ar(3)+'~B(H*/*)+HJ*(*)+H**~*~)+P(H*~I~+H~C*~+H**I~)21//'(8)(/~ : (~b,e#(n,'*'+n,'*'+n*'I')qb) (I-[-3CsxD(2)-}-xp(3)),

(14c)

I~(4) = (~,)i/f(4)+~(Hc$(S)+HIk(2I+H~|{Z)+Hj~{S)+Hj~(2)+H~t (z))+~(H,(1)+H~(1)-bH~(1)+HI (I))~]/r(4)tr/~

= (q~, e'8(H"I}+Hd[1}+HI*'I)+Ht'l)) ~D) (l -~-4Csxfl(2)-J-4C3xfl (3)-~-xS(4V)

(14d)

= (¢),eZ(~,"'+u/"+n,"'+H,")q~) (l_~4C2xfl(2).3vgC328(3)_~½4C22C2xfl(2)'.~f_xs(4)),

.ire(N) ~- (¢, M + e zgc'%'~gm M(7I)) = (qL e p"'I' q~) (1 +NCsXp (2' +NCaXD(a) +NCaXp (4) + . . . -J-NCNXfl (4))

= @, ePW'~) I+NC2x~(2)+NC3X~(3)+ ~sC22x~ ~ 1

2! 1

(NC22--NC4 4C22C2)xfl(2)s~--NC4X~(4)-~-

(~, e$n(l'~b) exp N I ~ N C z x z ( ~ ) +

""

1

.}

NCax~TM

1 , NC4 } 2 !N (NC22--NC4 4c2 2C2)xfl(2) -~- - Y - xfl(4)~- . . . .

(lge)

The equations (14b), (14c), (14d) . . . . (14e) give the definitions of x~ (2~, x~(3~, x~(4;, x~(4~. . . . x~ (N) successively. The x~(N)'s do not depend on the specific numbers of particles, provided that q~ is symmetrized or antisymmetrized with respect to all the numbers. For the systematic derivation of the last step in (lge) we can refer to the paper by Tolhoek et al. ~). If we choose plane waves in place of q~, the series of x~ (n) in the exponent of (14e) just corresponds to the ordinary cluster expansion series. The only difference between the ordinary and our pseudo-cluster integrals is that each cluster integral implicitly involves suitable higher cluster terms in ~.

PSEUDO-CLUSTER

653

EXPANSION

By solving the equations (14) we obtain

Ip(~)

xp (s> =

(~, eP(U~,l,+H/~,)~)

-

-

1

Ip( s )

Xp (s)

= (~, ep(a/,,+n/1,+n ,,) ~) --aC2 xp ~s)

Xp(4) =

(15)

Ip(4)

(~, eP~H¢"+n/"+H~'"+nt") ~) --4C2xP(2)-- 4Caxp(3)--½4C~sC~xp(~)~"

Now we are at the stage of discussing the convergence of the series in the exponent of (14e). The necessary condition for the convergence of the ordinary cluster expansion has already been discussed by m a n y authors 4, 5, e). As an illustration let us choose the Jastrow type operator M (eq. (2)). Then, the explicit forms for Xo(~), Xo(s). . . . are given by x0(2) = (~, h(r,~)O),

(16a)

Xo(a) ---- 3(~, h(r,,)h(r~k)qs)+ (qb, h(r,j)h(r,,)h(rk,)q~),

(16b)

• o(*, = 3{(~, h ( r , ~ ) h ( . ~ ) ¢ ) - - ( ~ ,

h(..)~)~}

+ 4{(q~, h (r,,)h (r,,)h (r,,)q~) + 8 (q~. h (r,,)h (r~k)h(r,,)q~) + 3 (~b. h(r,~)h(r~k)h(rkz)h(r,)q~)

(16c)

+ 6 (~, h(r,,)h(rj~)h(~,)h(r,,)h (r,~)¢) + (q5, h(r,~)h(r,,)h(rk,)h(r,,)h(r,,)h(rj~)q~)), where /2(r.) = l + h ( r . ) .

(16d)

In order to estimate them we put

]f h(r)4~r~dr[ ----co.

(17)

Therefore, if the ~'s represent plane waves, x0(l) = O ( ~° ) , Xo(S)= O ( ° ; 1 , x0(i) = O ( ~l ° ; )

+ O (°~) ,

(18)

where ~9 means the total volume of the system. It is to be noticed that the first curly bracket on the right hand of (16c) has an order of magnitude with an additional factor 1IN. This fact comes from the well-known product property of the plane wave functions. In Jastrow's case it is clear that the necessary condition for the convergence of the series can be expressed as N °_ /2 << 1.

(19)

654

j. FUJITA

The condition (19) means that the volume where h(r) differs appreciably from zero should be much smaller than the average volume occupied b y one particle, so that the shape of h(r) should be of sufficiently short range. It should be stressed that the above order of magnitude relations (18) can be valid even if the # ' s are not exactly plane waves. For instance, they are also valid in the case in which there exist over-all correlations, but the particles are approximately free. Although # contains the residual correlations, the relations continue to be valid, in so far as q~ has a product property similar to eq. (8) *. Of course it is not the case if ~ is like a liquid of high viscidity. The condition I I I is satisfied if the relation (18) is still valid for the true effective wave function in place of plane waves, though the equations (16)'s are not adopted generally. Let us proceed with our discussion b y assuming that the relations (18) are satisfied b y the true asymptotic wave functions ~. A plausible argument for the required product property of • will be given later. Under these assumptions, eqs. (12) and (14e) lead to the pseude-cluster expansion for the energy expectation value tt

E = E(°)+E(~)+E(2)+ . . . .

(20a)

E(O) = (q~, H(~)#),

(gOb)

with E(1)

NC2((~), M ( 2 ) + { ~ - /

(I)_I_ 7J (1)_1_ 7~T(2)~M(2)("~

_ (~, ( H i ( i ) + H (1))~) (~, ~" "rf f( ~ ) +" ~" t(j ~ ) ~ l J,

(~oc)

and

NC3[{(~, ~(8)+(H,(I)+H (1)=FH~U)_{_H(2)-~(2)~=u(2)~r(~),-h~

E (2)

_ (~, (Hi(X).+H(X)+Hk(i))~) (~, ivr(a)+M'(8),-~. " " ~jk " " t j k ~ l ) --

3C2{(~ ' a/r(~)+ tH (~)~-H (1)_L_/-~ (2)~M(2)(~b~ "'~ ij ", i ~ J t iJ I lJ ! -- (~), (H~(1)+H~(1))~))(~,

-

(NC.?-~C4

M.(~)+Mo(2)

~)}]

4C.. ~C~){ (~p, ...~R(~>+...~(2),~. =J _ I}

(2od)

× { ( ~ ' ""0~/r(2)+tT4~"i(1)_a_ r4...j (1)_a_ 14(2h..~t~/"" ~j~/r(2)d)~=~-- (~, (H~(1) + H~(1)) ~ ) (~ . . .~/f(2)+ . o" ""~t,f(2)d~Yt ~ =~J

.=~NC4~{(~) ~]'(4)-F[T-T( 1 ) ~ _ .

(1)J_ H

(1)_L_H

(1)

_~ fq(2)_L M ( 2 ) _ L f4(2)._~ M(2).J_ f-/(2)_L M(2)~ ~/~(4) 05~ ~'tJ /''ik /''i/ [''Jk I''jl /''k/ ]'~"tJkt--} _

(~, (Hin) +H(1) +H~(1)+H (1))~) (~, ~m)+~(~) o ~

? I n s t e a d of (5 in eq. (8), ]~l 2 m a y h a v e a n o t h e r h e a l i n g d i s t a n c e ~' (~' :> ~). I t is e s s e n t i a l for t h e v a l i d i t y of (18) t h a t

9

"

"tt W e c a n easily verify t h a t e a c h t e r m on t h e r i g h t h a n d side of eqs. (20) j u s t c o r r e s p o n d s to e a c h t e r m of t h e eqs. (31) a n d (32) of ref. ~) if we choose for • t h e Slater d e t e r m i n a n t of t h e oneparticle wave functions.

655

PSEUDO-CLUSTER EXPANSION

-- ({5, (Ht(1) + H~(1) + H~,(1)){5) ({5, M~(a)+M~j~{5)

(20d)

--aC~{({5,M,~(~)+ (H, (1) +H~ (1) + H . (~) )M~(~) {5)--({5, (H,(1)+H~(1)){5)t{5 M(~)+M(~){5~ X , ~1 it 1) _ ; C 2 ( ({5, ~r(=)+~r(s)o~

M (2)+/H (1)± ~ (1)±~(2)~ a/r(2)o~ - - ({5, (Hi(I) -~-H1(1)){5) ({5, "il/r(~)+ aAr(2)os~ 1~ 2_ "it ""ti ~/jjt

....

3. The Effective H a m i l t o n i a n //0 If we minimize E for fixed M b y varying {5, we obtain the pseudo-cluster expansion of the effective Schroedinger equation (5) for the asymptotic wave function {5 (Ho--E){5 = 0 and ({5, {5) = 1, (5') where H o = Ho (°)+ Ho(1)+Ho(*) + . . . . (2 la) Ho (°~ = ~ H~ (1),

(21b)

k

H0(1) ~__ w s M ( ~ ) + t r 4 (O) --

(1)_1_ g / ( 1 ) _ L g / ( i ) ~ f ( g )

(H~(t)+HJ (1)) ({5, ""M(21+ M121°5~*t "" O =J

A/f(i)+~A~(2)/(fi

--A-({5, .,.M(~)+O.,~lzrl2)d)~,t = ] ({5, ( H i (1) + H t ( 1 ) ) { 5 ) )

(21c)

= Z {M~I+[H,(1)+HJ (1), ag12)n± ~rl2)+nl~)~r(2)/ (O)

[m M~)+M~){5))(Hi(1)+-HJ {1)- ({5, (H,{1)+H~(l'){5))+ . . . . + Z ~~M(l)+M(I) ,, , -- ~', (t J)

Similarly, we can easily derive the higher cluster terms from eq. (20), b u t we shall not write them down explicitly here. Eq. (5') must be solved under the boundary conditions 0

{5 =

[x,] =

finite

for

oo

= D.

If we can obtain the solution satisfying the above conditions, ~ automatically satisfies the boundary conditions =0

for

[x~[= oo

and

r.=D.

It can be noticed that H ~ ) appears on the right hand side of (21) always accompanied b y ~rcl) Thus, we can choose M in such a manner that the effective Hamiltonian H e .has no hard core interaction. Let us consider the Jastrow case again. Eq. (21) becomes

656

J. FUJ ITA

H o = ~ H,tl'+ ~,/(ro)VJ(r,~) k

0

(~2)

~2 ¢,,~{i (r,s)3,/(r,,) + / (r,,)V, / (r, j) • V~} U

+ ~ {h(r,j)- (q~, h(r,jl~b)} {H,U)+H/I'-- (q~, (H,(I'+H/1))q~)}+ . . . .

Fig. 2. Schematic picture of the reducible 4-th cluster term (eq. (14d)). ~ stands for the healing distance. The validity of cluster expansion is due to the weakness of the coupling between the pairs (1, 2) and (3,4).

/"

"",

-

~.I*C,)A/(,)

t

0

0

~\ ~

-

~ /

°~

-

l*(r)V(r)/(r)---~/*(r)Al(r) /*(~)V(r)/(r)

Fig. 3. Effective potential in Jastrow's case (eq. (22)). See Iwamoto-Yamada (ref. 5), p. 352).

The first three terms of the effective Hamiltonian H o consist of the ordinary kinetic energy T, the screened potential energy /(r)V/(r) and the apparent (repulsive) potential resulting from the curvature of the correlation function ] (r). Therefore, it is clear in this case that H o does not contain any hard core inter-

PSEUDO-CLUSTER

EXPANSION

057

action and the effective potentials in H 0 are greatly reduced compared with the original interaction V. So far we have not adopted any special form for M. Provided that it satisfies the conditions (I) to (In), M can be chosen arbitrarily. However, it is practically impossible to handle the complicated higher cluster terms. The practically best way of choosing M ~2~is the minimization of the approximate expression for the energy, all the higher cluster terms H0 ~2~,Ho (8~, . . . being cut off. If the M ~ so obtained satisfies the condition (19) we adopt it and then solve the eq. (5'). Now let us vary the M(2)'s in E a ~ + E (2~ of eq. (20) for a fixed ~:

2 E_UH(~)~M(~)~

(23)

(o)

Eq. (23) is quite similar to the ordinary wave equation governing the scattering of a pair of particles, since M (2~ --> 1 for r,j -+ ~ . Let us assume that • is a Slater determinant of plane waves, which describes a fermion assembly:

= ]~,H¢,/~,)

(24)

where x~p stands for the space, spin and isospin coordinates of ip-th particle. We get from (23) and (24) (~) i,~(~)¢,,.(,)4~,.(x~) x Z Z ~ , H,(,+H;1)-- ~ E + H ,~ II' 4~,(~) = o. (25) I:~,1

(0) P

Taking into account the exclusion principle,

Itt) P

k:#t,t

(28) where Q~v~P is the projection operator ~) which omits the Fourier component corresponding to the occupied levels except iv and /'p. If we put

W,p(x,, x,) ~ i~){¢~,(x,)¢p(xj)--¢p(x,)¢a(xj) } =/~p(~,~){¢~(~',)¢p(~)-¢l,(~,)¢~(~)},

(27)

the solution of the Bethe-Goldstone equation ¢ (H,U) + H ~ ( 1 ) - e a p ) ~ a a ( x , x t ) = -Qaa[-t~)v2a#(X,X~) * The equation O{H~'*~ 4- HjCt)--eap+ H~i"'}Vap = 0

is equivalent to if the 4's are eigenfunctions of H cI*.

(28)

658

J. FUJITA

with = (ap)

roughly satisfies the equation (26), since on the average ~ ~ (2/N)E. Therefore, it is practically the best way to choose M (2) such that the / ~ defined by the relation (27) satisfy the Bethe-Goldstone equation. 4.

Pseudo-Cluster

Expansion

of

Matrix

Elements

Let us assume that the solutions of the effective Schroedinger equation (5') could be obtained and let a set of solutions be #0, #z . . . . . Now, we will consider a matrix element of the operator ~ between the states ~vm and Tn: "~gmn = (k~m'( . J ( t ) + v g ( 2 ) ) ~ )

(27)

where X = ~ £ ( z ) + W '~) = ~ ~£k(~)+~,s X ~ ) stands for the sum of one-body and two-body operators. We m a y write

~/ (#m, M+M#m) (~,~, M+M#.) = ¢ (¢~'M+(JI(I'+'ZI'~))M¢') X ~ (@,,, M+M@,,)

(@'' M + ( d / ' l , + " E ' 2 ) ) M ¢ " )

(28)

(¢n, i+i¢,,)

By the same prescription as in section 2, we can obtain the pseudo-cluster expansion series of each quantity under the square root in the right hand side of (28)-

___ L~-~ log (#.., M + eP(~m~+~(~')Me~m)l J P=,y =0

(~,,,, M+M#,,,)

+ [~-~log(#.~,M+e#~"'+x""M#,~)],=o [~---~log(#.~,M+Me'~=#m,],=o = (q'm,

(¢m,

(q,,,,,

--2 ~ ( ~ , #.){(#m, M(~)+f~ (1)_L~£ (1)J_ £(a)~M(2) # m/ ~ iS ~ t i S i i.t I tS (o)

--(#m M(~)+M(2)# ~/,~ tJ tS m/("t'm "'~ tS

J

tj /

tJ

~

( ~ "(i)+,~tc/(1))~m) }

nl

(is)

-- (~m, ;'~"ttt a/r(S)+M(~)d~ i S - - n / ,~ ~C m

,

(.~/i(1)_~_~¢1(1))~m)

--

(~m,

U , ,(2)+i .

~,,,)(~,., (.£, (1) +dl s(1) )~,,)}

(2)

(29)

PSEUDO-CLUSTER EXPANSION

+[(4~

~(1)4~)+~{(4~, (o)

2 × [(4~, 4 , ) + ~ Z { ( 4 ~ , (o)

659

~/r(z)+¢~£ (1)2_~ u~2_~(2)~M(~) 4

(4~, a/r(~)+M(2)o~ ~/o~

--

(..~t(1)-{-,~ti(1))(~m)} j

i($l+i(')4o ,~ ,~

Let us consider the special case (30)

(4~, i +i4,~) %/(4,,,, M+M4,.) (4n, M+M4,,)

V (4,~, M+M4,) (4,~, M+M4,~), (4,~, M+M4,~) (4,, M+M4~)

with (4~, 4~) = (4~, 4~) = 1. We have

(4,~, M+M4~) = (4,~, M+M4,~) =

M+MeP~4,~)

log(4,~

'

a=0

2

(31) •

thus

As a first approximation, the 4~'s can therefore be regarded as orthogonal. Therefore, eq. (29) is an expansion series with respect to a quantity of the order O(No~/g2): the true matrix element ( k u , ~,(1~ 5v) of a one-body operator ~,(1~ can be approximated b y (4~, M¢(1~4 , ) with an error of the order 0 (No/g2). On the other hand, the matrix element of a two-body operator .~,(2) should be nearly ~ ( 4 , ~ , M (~)+~(~)a/r(~)~~J v ~ "'~o ~-/,~ which m a y considerably differ from (4,~,

5. D i s c u s s i o n

The most important necessary condition for the validity of our method is (19). As seen from refs. 3) and 5), No~/g2 m a y have a magnitude of about { in nuclear matter. Therefore, our method seems to be useful to understand the many apparently contraditory aspects of nuclear models. The most remarkable result is eq. (29) to calculate the matrix elements. The nearest-neighbour expansion a) seems to be essentially similar to our method and formally more rigorous, b u t our method might be sometimes more useful because of its fairly simple and intuitive character. A plausible argument for the convergence of our pseudo-cluster expansion is

660

j. FUJITA

that the effective Hamiltonian H o involves only weak interactions and ensures the approximate freedom of particle motions in the asymptotic wave function Though our method is also applicable to an imperfect Bose gas, we shall only mention nuclear problems here: 1) The weakness of the effective interactions in the effective Hamiltonian H o suggests both the appearance of the shell model aspects and the saturation property. The saturation condition should be investigated by the eq. (21) instead of the original eq. (3). 2) The foundation of the configuration mixing method 7) is given by eqs. (21) and (29). It can easily be shown that our prescription involves short range correlation effects (higher cluster terms O (No/Q)) as well as the configuration mixing effect due to the residual interactions in the asymptotic wave function #. 3) It is possible in principle to extend the Bethe-Goldstone equation to the case including the three-body correlations, to investigate the relation between the shell and collective models and the possibility of explaining the spin-orbit force in a nucleus by the three-body correlations or to examine the alpha-model, the superfluid model, the deuteron model, etc., from a unifying view point. The author would like to express his sincere gratitude for stimulating discussion by the members of the Tokyo group for nuclear studies, especially Drs. F. Iwamoto, T. Tamura and Y. Ichikawa. He also thanks Profs. M. Nogami, N. Fukuda and T. Yamanouchi for valuable discussions and suggestions. References 1) K. A. Brueckner, Phys. Rev. 96 (1954) 508; 97 (1955) 1353; 97 (1955) 1344; 100 (1955) 36; H. A. Bethe, Phys. Rev. 103 (1956) 1353; Bethe and Goldstone, Proc. Roy. Soc. A 238 (1957) 551 2) Gomes, Walecka and Weisskopf, Am. of Phys. 3 (1958) 241 3) Riesenfeld and Watson, Phys. Rev. 104 (1956) 492 4) R. Jastrow, Phys. Rev. 98 (1955) 1479; Drell and Huang, Phys. Rev. 91 (1953) 1527; J. Da,browski, Proc. Phys. Soc. 71 (1958) 658; V. J. Emery, Nuclear Physics 6 (1958) 585; J. B. Aviles, preprint 5) Iwamoto and Yamada, Prog. Theor. Phys. 17 (1957) 543; 18 (1957) 345 6) Hartogh and Tolhoek, Physica 24 (1958) 721, 875, 896 7) R. J. Blin-Stoyle, Proc. Phys. Soc. A 66 (1953) 1158; Arima and Horie, Prog. Theor. Phys. 11 (1954) 509; 12 (1954) 623