Bound states of a many-particle system

Nuclear Physics 7 (1958) 4 2 1 - - 4 2 4 ; ( ~ ) N o r t h - H o l l a n d Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

B O U N D S T A T E S OF A M A N Y - P A R T I C L E S Y S T E M F. C O E S T E R

Department o/ Physics, State University o/ Iowa, Iowa City, Iowa Received 10 April 1958 A b s t r a c t : R i g o r o u s f o r m a l solutions of t h e b o u n d s t a t e S c h r 6 d i n g e r e q u a t i o n are c o n s t r u c t e d in t e r m s of a n a r b i t r a r y c o m p l e t e set of single particle w a v e f u n c t i o n s . F r o m t h e s e s o l u t i o n s o n e sees w i t h o u t effort t h a t t h e R a y l e i g h S c h r 6 d i n g e r p e r t u r b a t i o n e x p a n s i o n of t h e e n e r g y does n o t c o n t a i n m a t r i x e l e m e n t s r e p r e s e n t e d b y p r o d u c t s of u n l i n k e d d i a g r a m s . T h e c o m p o n e n t s of t h e s t a t e v e c t o r a r e related in a simple m a n n e r to f u n c t i o n s r e p r e s e n t e d b y linked d i a g r a m s only.

The validity of the Brueckner approximation to the bound state energy of a many particle system depends on the absence of "unlinked dusters" in the perturbation expansion of this energy. Brueckner l) has shown that such terms are absent from a few orders of the perturbation series. General proofs for all orders of the perturbation series have been given b y several authors 2-4). All these proofs are based on a detailed inspection of perturbation terms of arbitrary order. The purpose of this note is to cast the basic equations into such a form that the absence of unlinked terms from the energy becomes evident without detailed inspection of all n'th order perturbation matrix elements. We are concerned with the formal perturbation expansion of the bound state ~Q of a many particle system in terms of the eigenstates of a suitable independent particle Hamiltonian H 0. The expansion in powers of the perturbation W ~ - H - - H o need not converge. However, if the interaction potential is bounded, the perturbation expansion of the state Y2,, (Ho+~W)~2,t = E~Q~

(1)

converges for sufficiently small ~, 0 < ~ <_ 1. There is an unambiguous correspondence between ~2~ and the zero order eigenstate ¢ defined by t ¢ = lim ~ , y=0

Ho¢=

E0¢.

The formal properties of the terms in the perturbation series are the same for all values of 7- After a suitable selective summation of the leading terms in the series 9.) the limit ~7 -+ 1 may exist even if it did not exist for the original series. The correspondence ¢ ~ , ~ persists. ¢ is the "model state" of Eden and Francis 6) and the "chosen configuration" of Bethe 7). For an t F o r proof of t h e s e s t a t e m e n t s see for i n s t a n c e F. Riesz a n d B. Sz. N a g y , ref. ~). 421

422

F. COESTER

infinite repulsive core W is not bounded and the above considerations are not directly applicable. However, we m a y replace the infinite repulsive core by a finite one and go to the limit after the potential has been eliminated in favor of the reaction matrix 2.8). The discussion of useful approximation procedures and the convergence for ~ = 1 and infinite repulsive cores is beyond the scope of this note. We merely intend to establish formal properties of the bound state solution of the Schr6dinger equation. We shall therefore put ~ = 1 in the following for the sake of simplicity and without regard for questions of convergence. We assume that the eigenvalues E and E 0 are non-degenerate. However, our formalism can be easily adapted to the important case of a ( 2 I + 1 ) fold degeneracy of a state with spin I. D is represented b y the amplitudes for the excitation of k particles from the configuration ~. It is normalized b y the condition

o)

=

1.

Let the indices p and v label respectively the single particle states which are occupied or e m p t y in the configuration ~. ~ can be represented in the form 0 = F~ (3) with N

F =

1+

(4) k=l

1

F¢ =

a p+ l " ' " a ,+~ b ,+l . . , Pl'"P~¢

+ ~-~ b~,

F¢(pl.

• • pC, ~ 1 - . .

~¢)

(~)

vl'"vk

where ap+ and b, + are anticommuting Jordan-Wigner operators which create respectively a particle in the state p and a hole in the state v of the configuration ~b.The function ~ satisfies b y definition the equations ap ~ -= 0 and b~~ = 0 for all p and v if ap and b, are respectively particle and hole annihilation operators. N is the total number of particles. One might proceed from here to derive the formal perturbation solution for the functions F~ and examine the structure of these functions to all order. However, the discussions of H u b b a r d 4) and Hugenholtz 3) indicate that simpler results can be expected if one defines an operator S b y F ---- e s

(O)

with N

s =

X

(7)

k=l

1 S~ =

Y,

Pl"lPI¢

Z

Vl*''Vk

a+

Pl

" " *

a+

+

+

Pkbvl " " " b"k ~-_)2

&(pl

. . . p~,

vl . . . v~).

Eq. (6) generates a finite set of equations for the F~ in terms of the S~:

BOUND STATES OF A MANY-PARTICLE SYSTEM F 1 =

S 1

F2 :

S 2 + ~ SI ~2

423

1 F3 =

83~-S182~-

(s)

31 S13

These equations cart always be solved for the S,. The definition (6) of S is therefore always possible. Multiplying eq. (1) (7 = 1) by e -s we have {[H o, S]+e-SWeS--E--eo} ¢ = 0

(9)

since e-Silo es = H 0 + [ H o, S]. The expansion 1 e-SWeS = W + [ W , S ] + ~. [[W, S], S ] + . . .

(10)

breaks off with the forth order term if H contains only two-body forces. F r o m (9) follows

E - - E o : (¢, e - S W e S ¢ ) = (¢, {W+[W, ( S I + S 2 ) ] + { [ [ W , Sl] S1] } ¢ ) . (11) Let A be the projection into ¢. Multiplying (9) by 1--A we have

([Ho, S ] + ( 1 - - A ) e - S W e S ) ¢ = 0.

(12)

In order to solve (12) for S it is convenient to define an operator G N

G=

~G~

with

G, =

~,

~, a+Pl

p,...p~ v,...~

" " "

a+ Pk b+ t'1 .

1

.

b+,~T-f2Gk(pl . . . . (k.)

.

.

pk,. Vl

vk) (13)

by

Gk(pi...p,, vl...vk)=

(¢, b~ . . . b~,ap . . . % l e - S W e S ¢ ) .

(14)

In other words G is obtained from e-SWe s by shifting all annihilation operators to the right and retaining only those terms which contain after ordering no annihilation operators and at least one pair of creation operators. With these definitions there follows from (12) [Ho, S ] + G = 0

(15)

since [Ho, S] and G contain only creation operators. Eq. (15) can be solved for S by ~) S ---- --i lim fo ~---~0 d - - ° °

dz e ** e i " ° r Ge -in0".

(16)

On account of the definition of G the energy denominator in (16) is guaranteed not to vanish. Eqs. (16) and (14) together with the definition (6)

4~4

F. COESTER

take the place of the original Schr6dinger equation. No approximations and no infinite expansions have been used so far. (14) and (16) can be solved b y systematic expansion in powers of W. The result is equivalent to the Rayleigh-SchrSdinger perturbation series. The matrix elements in this expansion m a y be represented b y diagrams quite similar to Feynmann diagrams t. It is now easy to prove the following theorem 3,4): The diagrams representing the matrix elements o/ S contain no unlinked parts. It is clear from (14), (10) and (16) that G contains no unlinked diagrams if S does not contain any. In an iteration solution the n'th order term G (~ is according to (14) equal to an expression containing only S ~-1~ and lower order terms. Since the theorem is true for n = 1 it is thereby proven b y mathematical induction for all orders. The matrix elements of F do have unlinked parts but there are no diagrams or unlinked parts of diagrams without external lines. Since S contains no unlinked parts the energy shift (11) also contains no unlinked parts. The choice of the single particle wave functions and energies has been quite arbitrary. Our results are independent of a "self-consistent" choice for these wave functions. t These diagrams are the same as those used by J. H u b b a r d (reI. 4)).

References 1) 2) 3) 4) 5) 6) 7) 8) 9)

K. A. Brueckner, Phys. Rev. 100 (1955) 36 J. Goldstone, Proc. Roy. Soc. A 239 (1957) 267 A. Hugenholtz, Physica 23 (1957) 481, 533 J. Hubbard, Proc. Roy. Soc. A 2240 (1957) 539 F. 1Riesz and B. Sz. Nagy, Functional Analysis (New York, 1955) p. 373 If I~. J. Eden and N. C. Francis, Phys. Rev. 97 (1955) 1366 H. A. Bethe, Phys. iRev. 103 (1956) 1353 H. A. Bethe and J. Goldstone, Proc. Roy. Soc. A 238 (1957) 551 J. M. J a u c h and F. Rohrlich, Theory of Photons and Electrons (Addison Wesley, Cambridge, Mass., 1955) p. 129