the generator-coordinate theory as a flexible formulation of the many-body schrödinger equation

I [ Nuclear Physics A147 (1970) 545--562; (~) North-Holland Publishing Co., Amsterdam 1.D.2 I

Not to be reproduced by pho~oprint or microfi/m without written permission from the publisher

T H E G E N E R A T O R - C O O R D I N A T E T H E O R Y AS A F L E X I B L E

F O R M U L A T I O N O F T H E M A N Y - B O D Y S C H R O D I N G E R E Q U A T I O N

CttUN WA WONG Department of Theoretical Physics, Oxford University

and Department of Physics, University of California, Los Angeles, California *

Received 23 Jarmary 1970

Abstract: By using generator coordinates as state labels we formulate the generator-coordinate tkeory of collective motion as non-orthogonal representations of the many-body SchrOdinger equation in a subspace of the I-Iilbert space of many-body state vectors. The weight function of the usual generator-coordinate theory is generalized to become the components of the state vector in one of the two bi-orthogonal representations labelled by generator coordinates. The well-known case of the Gaussian-overlap approximation is studied in order to show how the new formalism also permits a solution of the problem using the original real generator coordinates. The concept of generator coordinates is clarified by studying (i) the connection between the redundancy of generator coordinates and the linear dependence of base vectors labelled by these generator coordinates, and (ii) the construction of states having the required properties under translation and rotation. Finally, the consideration of rotational properties leads to a double-projection method for constructing internal states of good angular momentum and for removing spurious states of c.m. motion.

1. Introduction

The genera to r -coord ina te (GC) theory o f Griffin, Hi l l and Wheeler ( G H W ) ~- 4)

is well k n o w n as a theory o f collective wave functions. Several genera t ing funct ions

have been suggested and s tudied in the l i te ra ture s - 8). But even for the s imple case

o f the Gauss i an -ove r l ap a p p r o x i m a t i o n the weight funct ion canno t a lways be found

wi thin the context of the or iginal var ia t iona l p rob l e m 4, 8). We would like to repor t

in this pape r a new fo rmula t ion of the G C theory which clarifies many aspects o f the

use o f G C wave functions, and which is in a sense equivalent to the m a n y - b o d y

Schr6dinger equat ion .

The new fo rmula t ion o f the G C theory given in sect. 2 s tar ts f rom the observa t ion

tha t a set of states l e ) , label led by the genera tor coord ina tes ~ can be defined such tha t

(x l c~), where x = {r~} is the m a n y - b o d y coordinates , is the m a n y - b o d y wave funct ion

q)(x, e) used as a genera t ing funct ion in the G H W theory. The states l e ) fo rm a sub-

space S o f the Hi lbe r t space o f m a n y - b o d y state vectors. I f these I c~) are used as a

non -o r thogona l represen ta t ion o f the subspace S, then the m a n y - b o d y SchrSdinger

equa t ion has the form of the G H W var ia t iona l equat ion, except t ha t the weight func-

t ions o f the theory should be rep laced by componen t s o f the eigenvectors a long the

Present address.

545

546 C H U N W A W O N G

base vectors I~) bi-orthogonal to J@. The correct way of using these GC states as bases of the subspace S is discussed. The result is called the new GC theory.

The new GC theory contains unusual features connected with the lack of orthogonality of the basis lc~}. It is shown in sect. 3 how these unusual features permit a solution and clarification of a well-known difficulty ~, 3, 4, 8) in the G H W theory (subsect. 3.1) in the Gaussian-overlap approximation. Subsect. 3.2 repeats a solution of the problem by the method of complex generator coordinates a). Another solution is obtained in subsect. 3.3 in terms of the original real coordinates e provided that the state components (g[f} are suitably interpreted. In subsect. 3.4 the nature of the simplification in the Gaussian-overlap approximation is pointed out.

The generator coordinates can be either collective or sing!e-particle in nature. They are also "redundant" in the sense that the generating function or transformation already contains a reference to the many-body coordinates. In subsect. 4. ! it is shown that this redundancy is not always of the classical type, but generally manifests itself in the GC theory as the linear dependence of certain base vectors in the basis ]e}. Consequently the removal of redundancy takes the form of the removal of all linearly- dependent base vectors. A closely related problem - the isolation of internal states from states of c.m. motion in the total state of the system - is considered in subsect. 4.2. In the translational case we obtain the Peierls-Thouless wave function 4), In the rotational case the separation of internal states of good angular momentum suggests a new double-projection method which differs from the method of Peieris and Thou- less 4).

2. Generalization of the generator-coordinate theory to a many-body SchrSdinger equation in a subspace of Hilbert space

The many-body SehrOdinger equation

(H- -EI ) IE , . . . ) = O , (2.1)

where H is the many-body Hamiltonian and ]E . . . . } is the energy eigenstate with all the necessary state labels, has an approximate solution under the restriction that the state vector be limited to a subspace S of the Hiibert space of many-body state vectors. We write the truncated problem as

( ~ s - E s J s ) l E s . . . . } = 0, subspace S, (2.2) where

~'~s = Ps HPs, (2.3)

Ps being the projection operator onto the subspace S, and ]Es, . . .} is restricted to the subspace. Eq. (2.2) represents a very special approximation of the original problem in that improvement cannot be made without first enlarging the subspace 9). For this reason we may call the solutions of eq. (2.2) the soIutions of the many-body Schr6dinger equation in the subspace S, i.e., with eigenvectors restricted to this subspace.

MANY-BODY SCHR(.JDINGER EQUATION 547

Consider now a collective wave function of the Griffin-Hill-Wheeler (GHW) type:

~(x) = (x, c~)f(c~)de, (2.4)

where ~(x, e) is a many-body wave function (e.g., a single Slater determinant) in the space of single-particle coordinates x (including spin and charge variables if necessary), with the set of generator coordinates e as parameters. The wave functions 7t(x) represent a restricted class of many-body wave functions generated by taking all possible combinationsf(w) of ~b(x, v.) (here we use integration and summation symbols interchangeably) over the range of c~. In other words, the totality of wave functions of the type (2.4) forms a subspace of the Hilbert space of many-body wave functions.

In the G H W generator-coordinate (GC) theory the weight funetionf(c 0 is a well- defined function of e; and it is varied in order to minimize the energy

Ef = I f*(°OK(c~' e')f(c()dc~ de ' , (2.5) Sf*(eft(a, e')f(e')dc~ d~'

where

K(o~, o~') = J cb*(x, ~)(x[H]x')~(X' , c()dxdx' , (2.6)

= f e')dx. (2.7)

The functionf(c 0 which minimizes (2.5) is then found to be the solution of the G H W variational equation

f~EK(e, e')-Efl(~, e ' ) ] f(a ' )de ' = (2.8) 0. 1

It is possible, however, to derive eq. (2.8) in such a way that it is identically the truncated problem (2.2) in the subspace of wave functions (2.4).

For simplicity, let us suppose first that the generator coordinates • are discrete variables, so that the linear expansion (2.2) is really

n

~f(x) = Z f , ~b(x, cq), n < 0% (2.9) i = l

then we have a truncated or incomplete expansion of the many-body wave function in terms of the base functions ~bi(x ) = ~b(x, e,). The many-body Schr6dinger equation in the truncated space can then be written in the form (2.8) by replacing integrations by summations, eq. (2.8) being the standard form of the Schr6dinger equation in a non-orthogonal basis 10). There are as many distinct solutions {fi} as there are linearly independent base functions in (2.9) (the question of linear dependence of base functions is discussed later in this section); and they represent the different energy eigen- functions of the many-body problem in the chosen subspace.

It is easy to write a corresponding result for generator coordinates which are con-

5 4 8 C H U N W A W O N G

tinuous parameters, with the linear integral (2.4) replacing the linear combination (2.9). The result, if valid, is the GHW variational eq. (2.8), now interpreted as the many-body SchrSdinger equation in the subspace of wave function (2.4).

Before the question of validity is considered, it is useful to discuss an important generalization of the GC theory by using not the subspace of many-body wave functions (2.4), but a subspace of the Hilbert space of state vectors. Let [@®, or simply re), denote a vector in Hilbert subspace such "chat

(xlc~)e = g~(x, ~). (2.10)

Consider a subspaee S consisting of all such vectors ]c~). If I@ are used as a basis of S, the basis is not orthogonal since the overlap

<cqc() = I(c~, c() (2.11)

is not a delta function. Defining a complementary basis ]~) by the bi-orthogonality relations

(~1~') = (c~[g,') = 6 ( ~ - ~ ' ) , (2.12)

we obtain the usual bi-orthogonal expansions 11) in the subspace S (unless otherwise stated all subsequent results are for this subspace only, so that we need not add the subscript S explicitly to various operators):

= f (2.13)

[f) = f lc~)(~l f )de (2.14)

The many-body Schr6dinger equation

( ~ - E ~ ' ) l f ) = 0 (2.15)

in the subspace S can be written explicitly in terms of the bi-orthogonal bases as the integral equation

f ~=[-(c~'J~gC'[~)-E(c~'[~)-l(~lf)d~ = 0, (2.16) i

where the solution (~[f) should be normalized by using the scalar product

( f f f ) = ( f ]~)(c~lf)dcz (2.17a) 1

f2 = ( f [g)(e le ' ) (~ ' l f )dc~ dc~', (2.17b) 1

use having been made of the relationship

f; (~[f) = (~l~ ')(~ ' l f)dc( (2.18) 1

MANY-BODY SCHR(JDINGER EQUATION 549

between the two sets of state components. By making the correspondence

=

(c(!-2/C1@ = K(~', ~), (2.i9)

(~ l f ) = f (~ ) ,

we recover the GHW variational equation. Conversely, the GHW variational equation has apparently been generalized to a many-body Schr6dinger equation in the subspace S by replacing the weight function f(@ by the state component (~lf).

The correspondence between the GC equation and the many-body Schr6dinger equation in the subspace S can be made more obvious by showing that at least one orthonormal representation for the subspace can be constructed from the GC states. Let {fz(e)} be a complete set of orthonormal functions for the interval (e,, ca), i.e.

f~*(e)/,(e)de = 6,j, (2.20) 1

Ef~(oOf*(e') = 6(~--c(), (2.21) i

then the countable set of states

I~b,> = f I@f,(e)dc~ (2.22)

can be orthonormalized by the Gram-Schmidt process with the result that there are N < oo orthonormal states

In> = ~/3i(n)1¢~>, (2.23) i = 1

where the label i has been re-ordered on orthonormalization to go over the linearly independent states only. Therefore the GC equation in this representation is

N

2 [<nl2/fln'>-E6.."]c. ' = O, (2.24) r K = l

where ""

(n]W]n') = Y, fl*(n)fi;(n' *(c0(elg/fle')f;(c4)dada'. (2.25) ~=i /=i

The result is just the SchrOdinger equation in the subspace of the N orthonormaI vectors In).

It is often convenient to carry the original bi-orthogonal notation further by defining the overlap (~1~') which is related to (~]e ') by the integral equation

qU #t 1

o r

<~'1~> = <¢ l~"><~" l~ )d¢ ' . (2.26b) 1

550 CHUN WA WONG

The overlap <~1c7'> can be expressed in terms of the orthonormal states In> as

N

<~1~'> -- ~ <~ln><nl~'> B = I

N

= E [ ~ i fi~(n)~*(n)f,(cOf*(e')3. (2.27) n = l i = 1 j = l

One can see also from the integral eq. (2.26) that since the kernel <c~'le"> is always well-behaved (usually differentiable in e' and e"), while <~'lc~> or <e'l~> is a Dirac delta function, the quantity <~1~'> must be in general less well-behaved than the Dirac delta function. To get sensible results one must usually require that the e integrations in the GC equation be performed before the summation over n, so as to ensure that the eigenvalue equation to be solved is the many-body Schr6dinger equation (2.24). This requirement must be considered part of the definition of the overlap <~f~'>, and it can be relaxed only when it is certain that in the particular case under consideration the final result is independent of the order of integration and summation. With this implicit requirement, the GC equation in the form (2.16) and the relation

(,

<sir> = j <~/~'>@'/f>d~' (2.28)

complementary to eq. (2.18) can both be written in an intuitive, if often only symbolic, notation.

We have thus shown that the replacement of the weight functionf(e) by the state component <~lf> represents a generalization of the GHW generator-coordinate theory to the many-body Schr6dinger equation in a certain subspace of the many-body state vectors if the bi-orthogonal representations are used as suggested. It then follows that the eigenstates If> have meaning independent of the representation. An important result of this generalization is that eq. (2.16) can have solutions which are not obtain- able from the original GHW variational eq. (2.8). This point will be studied in the next section, especially subsect. 3.3.

The fact that the variational equation in the GC theory has the Schr6dinger form has already been pointed out by Brink and Weiguny s) in the special case of the Gaus- sian-overlap approximation. The present section generalizes their result to arbitrary overlap functions.

3. The nature of state components in non-orthogonal representations; the Gaussian overlap approximation as an example

To illustrate the use of the bi-orthogonal notation labelled by generator coordinates in the new GC theory, we study in this section a well-known difficulty in the original GHW theory in the Gaussian-overlap approximation (GOA). This difficulty, which is reviewed in subsect. 3.1 and which can be avoided by using complex generator coordinates (see subsect. 3.2), can also be resolved in the new GC theory without

MANY-BODY SCHRODINGER EQUATION 551

using complex coordinates (subsect. 3.3). Subsect. 3.4 discusses the GOA as an example of overlap functions which are easier to handle than the general case.

3.1. DIFFICULTIES IN USING REAL GENERATOR COORDINATES IN THE GIIW TI-I.EORY

For the interval (e~, c~2) = ( - 0 % oo)let

(c~lc~'} = e -~('-"')2, s > 0

= ~ ~e -~2 (x/2s--~)"] Fe -s,'~ ('j2sc()"] (3.1) ,,:oL _.1 L _1"

Eq. (3.1) can be considered as an expansion in terms of the complete set of orthonormal states Ln}, n = 0 to 0% such that

(c~In} = e = . (3 .2) .jnt

If (nit/}, where It/} is an orthonormal representation for the interval (t h, I12) = ( - c o , co), is chosen to be the nth orthonormalized Hermite polynomial

e-2SC2~ z / 2 ¼ ( n i l } = ( . I n ) - x/2~- ' [ ~ - ) H~,(,J4sctt), (3.3)

the ~ --+ tl transformation function is then

(~1~) =const e -2s(e-en)z. (3.4)

Given a normalizable wave function in the orthonormal representation I~}

@If} = const e - r ~ , 7 > 0, (3.5) and using (3.4) we get

(a l f} = const exp _ _ 1 y~2. (3.6) 1 + 712s

One possible transformation to the 15} representation is that considered by Griffin and Wheeler z):

(oTis/) = c o n s t j dt exp [(t2/8s) + i t ( e - ct/)], (3

for which

and

(~[f> = const exp 1 y~2 (3.8) 1 -~ /2s

1" (~15'} = cons t J exp [(t2/4s) + i t ( ~ - c~')]dt. (3.9)

The three wave functions (3.5), (3.6), and (3.8) have some interesting properties characteristic of the respective representation. The wave function (t/If} in the ortho-

552 CHUN WA WONt3

normal representation is intermediate in "width" 2) between the other two wave functions. For ? > 0, 2s > 7 the wave function (czlf) is broader than (Y.[f} [ref. 2)]. In addition, we note that the finite norm of If} can be verified from formula (2.17) only when 7 < 2s. For ? > 2s the wave function (3.8) is no longer satisfactory since its use with (3.6) in (2.17) no longer gives a finite norm.

3.2. COMPLEX GENERATOR COORDINATES

The above mentioned difficulty of the weight function (3.8) has been noticed by many people 1, 3, 4, 8). Brink and Weiguny, and also Ui and Biedenharn s), suggest a way to resolve the difficulty by generalizing the generator coordinates to complex values with imaginary parts which are the momenta conjugate to the original generator coordinates. Complex generator coordinates have already been used by Jancovici and Schiff 6) in the context of the GC formulation of the particle-hole random-phase approximation.

The use of complex generator coordinates relies on the observation of Bargmann 12) that the Hilbert space of configuration-space wave functions can be mapped onto the vector space of entire analytic functions through the hermitian scalar product

( f ig ) = fd#(z)[f(z)-l*g(z), (3.10)

where * represents complex conjugation and d/z(z) is the real measure

d# (z) = i e_~, 2 dx dy. (3.11) 7Z

Let the complex generator coordinate (it is sufficient to take only one coordinate) be denoted by z, whose real part is the original generator coordinate •, here denoted by x. Measuring z in units of (2s) -~ we get for the Gaussian-overlap approximation considered in subsect. 3.1 the generalization:

<z'lz> = e -~(~'*-~)2, (3.12)

(Z,*'~n (z'ln) = e -~z'*2 ~ ) (3.13)

(zlq) = e -(~*-c02, (3.14)

where the real variable q is now used in place of the real variable t/ [also in eqs. (3.5) and (3.7)], and

(zlf} = const exp - 1 7(z,)2. (3.15) 1+7

Now if the state component (zig} is related to the function g(z) of the scalar product (3.10) as

(zig) = h(z)g(z), (3.16)

MANY-BODY SCHX6DINGER EQUATION 553

the equivalalence of the scalar products (3.10) and (2.17) requires that <z”Jf> must be related tof(z) as

<z”(f) = G [h(z)“]-lJp(z)

e -==I (zlf) =----, 7~ Ih(

(3.17)

The function h(z) in eq. (3.16) which connects the state component <zig) in the non-orthogonal representation lz) introduced in sect. 2 with the entire analytic function of g(z) of eq. (3.10) can be determined by comparing Bargmann’s complex delta function relation

1 [; e~z-zwj d;(z’)dz’ = F(z), (3.18)

[where I?(z) is any entire analytic function of zj with the Dirac relation

s <Z[z’)(Z’jf>dz’ = <z”[f), (3.19)

where

(Z/z’> = e ___ -zz* (z\z’>

n Ih(z

according to eqs. (3.17) and (3.12). The result is

[h(z)12 = e-+(22+2*2)_

Therefore

(Zlf> = go3 (zlf). Z

Finally we write down by inspection

and

(al.?) = e+(Z-Z*)2 ~ <zlz’>

e+cZ,-“,*)2

7c 7c

(3.20)

(3.21)

(3.22)

In repeating the complex method we have not specified the physical meaning of the imaginary part of z. Nor is such a specification necessary. However, we may, if we wish, specify that the one-to-one correspondence between the original complete set of harmonic-oscillator states In,) of eq. (3.3) and the complete set of orthonormal states In,) of eq. (3.13) is a mapping of harmonic-oscillator states upon themselves by

554 CHUN WA WONG

choosing Z = q+½i(1/i)(8/Oq), i.e., a Fock representation tz) in q. The Fock representation for the complex method has been used by Brink and Weiguny ~); and it gives a particularly simple picture of the effect of using complex generator coordinates. But such an interpretation of the imaginary part of z is not necessary. The validity of the method is not reduced if we choose to view the states fn~> in the abstract.

3.3. A SOLUTION FOR REAL GENERATOR COORDINATES

The re-derivation of the complex method given above suggests that its success is connected with the fact that the Bargmann complex delta function is better behaved than the Dirac delta function in real generator coordinates. (The unphysical variable y has the effect of spreading out the original Dirac delta function which would otherwise appear.) It is not completely clear, however, why the transformation (3.7) is useless whenever ? > 2s. To answer this question, we solve the problem with real generator coordinates using the new GC theory.

The one-parameter Gaussian-overlap approximation first considered by Griffin and Wheeler z) assumes that in addition to an overlap of the form (3.1) the Hamil- tonian kernal has matrix elements

<~IRI~'> ~ + ~ [2+-4++(~-¢)~+2+~M ~__2' +, ~ > o. (3.23)

Since

_(e_e,)e_~(=_~,)= - 1 8 e_S(~_=,>2, (3.24) 2s 8e

the integral eq. (2.16) for the kernel (3.23) can be reduced to a differential equation, now for the state component <~]f = n>:

qt+ q+ Mh z 271 E++-Eg+ --4Ms - 2 ~M <rain+> --- 0, (3.25)

where

• 1 [ ( ~ 2 - 1 ) d ~1 (3.26a)

q+ = ~ + +2 ~ +27+ , (3.26b)

[q+, q~+] = 2. 0 .27)

The solution of eq. (3.25) is similar to that for the harmonic-oscillator potential. As usual, we may write the result in operator form:

__1 q~+ln-l> = In), (3.28a)

1 x/- ~ q+ln> = I n - l > , (3.28b)


q+[0> = 0, (3.29)

where explicit forms of the operators q+, q+* in the representation j c~> have already been given in eq. (3.26). The solutions are

h2~ 2 h27 : , lx En+ -~ E g - - - + 2--~-~,n-e~), (3.30)

4Ms and

1 <~10+> = No+ exp - - - y~2 (3.31)

1+7/2s

for the so-called positive-energy solutions. There are also negative-energy solutions with energies

h272 hZy E._ = Eg 2 - - (n+½) (3.32)

4Ms M

in which we are not interested. Since we consider here only the positive-energy solutions we may now drop the subscript + .

Now the representation le> is not an orthogonal representation; therefore the normalization constant No cannot be determined without knowing <c~10 ). To obtain this state component we must solve the integral eq. (2.18), which in our case reads

No exp 7 ~2 _- :°~ e-S(~-~,)2<~,10>d:~," (3.33) 1 + 7/2s

By expanding <~]0> in Hermite polynomials H,(%/so 0 we obtain the solution

y " 1 <~10> = No,~V ~- ~=o ~ [ 4s(l~-7/2s)l ~. H2"(x/se)' (3.34a)

which can also be written as a power series in c~

V! ) <~10> = lVo eS~2y, Y 1 ~ 1 ( 2 n + 2 l - 1 ) ! [ (-s)tcc2k (3.34b) .=o {s 1+)/2s .~ (21-1)!~ l! l=0

The solution (3.34) is unique. For 7 < 2s, the infinite series on the right is convergent and can be summed to

V~ ~2, ~<2s. (3.35) [1+7/2s'~ ~ <c~[0> =fo(~) = No zc ~l-~?/2s ) exp 1-7/2s

If ~ > 2s, the infinite series is divergent; but the scalar product

V s ,=~o(_¼) . (1-7/2s]"(2n)! (3.36) <010> = N~ • = U-T~r/2~: (.!)2

converges for all positive values of 7 and s. This result shows that although the state component (710) is an ordinary function in the example studied, being the solution

5 5 6 C H U N WA W O N G

of a differential equation, the state component (8If) in the other bi-orthogonal representation does not always appear as an ordinary function. This apparently un- satisfactory result can be, and has actually been, avoided by following the requirement stated near the end of sect. 2 that the use of the bi-orthogonal notation is always valid only if the ~ integration is taken before the n summation.

The study of the GOA also suggests that the existence of (~lf> as a well-behaved function for a non-orthogonal representation l~) should depend not only on the nature of the states ]~), but also on the nature of the eigenstates If). The requirement is that the summation in

(~[f> = ~ <~[n>(n[f> (3.37) n

over the components (n] f> in an orthonormal representation In> should be convergent. This requirement should be satisfied for both real and complex generator coordinates.

It is perhaps relevant to give here the reasons for suggesting the use of generator coordinates as state labels in spite of the restriction which must be put on this usage: it simplifies formulas, it refers to the physical nature of the generator coordinates, and it symbolizes the fact that the GC equation is now identically the many-body Schr6- dinger equation in the Subspace under consideration.

3.4. GOA AS AN EXAMPLE OF SEPARABLE OVERLAP FUNCTIONS

It is shown in subsect. 3.1 that the overlap function in the GOA is a sum of separable terms which can be written as

I(ct, cz') = ~ (ct[n><n]ct'), (3.38) n

where (win> = c,~F,,(e), (3.39)

F~(c 0 being a well-behaved function for the interval (cq, c~2) and cn is a complex constant. The form of eq. (3.38) shows that In> form an orthonormal and linearly independent basis for the subspaee and that the quantity (gin} can be found by solving the integral eq. (2.18). This is the main reason why the GOA is easy to handle. (The form of the Hamiltonian kernel (3.23) permits further simplification.) The point to be made here is that the GOA is not the only type of overlap functions of the form (3.38), and that the form (3.38) might suggest other useful approximations for solving the GC equation.

4. The nature of generator coordinates: Redundant variables and spurious states

The choice of generator coordinates can be quite flexible. Familiar candidates 5 -s) include the size constants of the common single-particle potential well (phenomeno- logical or self-consistent, spherical or deformed), the state labels (i.e., quantum num- bers) of linear and nonlinear vibrations, the coefficients of quasiparticle transformations, the parameters or state labels of nucleon clusters as sub-units of the nucleus, or a


combination of several of these. In general, it is not necessary for the generator coordinates to refer to an entire nucleus or a large cluster in it. They can be particle-hole variables 6) or even single-particle ones. They can also be parameters appearing in the wave function. The power of the GC theory lies in the conceptual ease with which all these effects, single-particle as well as collective ones, can be treated on the same footing.

There are other attractive aspects in the GC formulation of the many-body Schr/5- dinger equation. We discuss in this section two nontrivial examples: (i) the removal of redundancy of generator coordinates, and (ii) the removal of spurious states of c.m. motion.

4.1. REDUNDANT GENERATOR COORDINATES AND LINEARLY DEPENDENT BASE VECTORS

The generating function q~(x, v.), or (x] @, which defines the subspace S causes the replacements of x by c~ as state labels. To a certain extent, this replacement is similar to the classical transformation to generalized coordinates. The important difference is that a generator coordinate can be any parameter appearing in ~, the latter being actually a many-body wave function. Because of this, the above replacement of state labels has meaning only in the subspace S.

The GC theory also contains a problem of redundant variables, if by redundancy we mean the use of both the original s.p. variables (or their equivalents) and the additional generator coordinates. Variations of the generator coordinates can generate states which are already present at different values of the generator coordinates, just as classical redundant variables are already contained in the original s.p. variables. The removal of such redundancy in the GC theory is clearly achieved by the elimina- tion ofatl linearly dependent states from the chosen basis for the subspace S. The result should be considered the GC equivalent of the imposition of constraints in the usual problem of redundant variables.

The redundancy considered here is a more general, or less clear cut, concept than that in classical mechanics. The specification of ~ involves a set of many-body states dependent on the original s.p. variables. If this set of many-body states is a complete set, the generator coordinates can be said to be completely redundant. If the set is incomplete, but there are linearly dependent base vectors, the generator coordinates can be described as partly redundant. With GC transformations for which the base vectors are linearly independent, the generator coordinates are complementary rather than redundant. Use of the term "redundancy" to describe the complicated situation discussed above serves to underline the similarity with the classical situation that the generating function qs, as the equation of constraint, sets up a correspondence between the original s.p. coordinates and the additional generator coordinates in the subspace S. Once this point is made, the alternative term "the linear dependence of the chosen bases in the subspaze S" may be used to make clear the mathematical problem involv- ed. This problem has already been mentioned in sect. 2.

558 CHUN ~VA WONG

4.2. C O N S T R U C T I O N OF I N T E R N A L STATES

"['he construction of internal wave ffmetions (or more generally state components) by removing the dependence on the c.m. motion in the general case of an arbitrary many-body wave function can be considered as a problem of redundant coordinates involving the c.m. degrees of freedom.

Following Peierls and Thouless 4), let us consider first the requirement of translational and Galilean invariances. The total wave function having c.m. momentum K must be of the form

~'K, ~t(X, x - X) = e~"' xT~ot(x- X), o r

(X, x - X I K , int) = e ~ ' x ( x - X l i n t ) , (4.1)

in the coordinate representation with the c.m. coordinate X and the set of internal coordinates r ~ - X denoted here by x - X . Galilean invariance requires that Tin t be independent of K in the coordinate representation, or be dependent on k - K only in the momentum representation. Now any function of argument x - X can be written as an integral over an auxiliary variable t:

f ( x - X ) = f,]~(t)h(x-X-t)dat. (4.2)

A change of variable to p = t + X gives

~t2-X

Tint (x- X) = J , l -x g(p - X)~o(X- p)d3p. (4.3)

The dependence of the limks of integration on X can now be removed by taking t~¢ = -oo , t2i = 0% i = x , y , z . On going to the Fourier transform of g we get finally

~'int(x-X) = )Je i~"(° -x~ f (K ' )~o(X-p)d3pd3K' , (4.4)

which is the Peierls-Thouless wave function 4). Here ~0 is a many-body (e.g., a shell- model) wave function and f (K ' ) is a weight function which according to ref. *) is to be determined by minimizing the total energy. In the terminology of this paper we may regard the factor

f eiK"(v-X)~o(x--p)da p -~ ( x - X [ K ' ) (4.5)

in eq. (4.4) as a GC transformation and (K ' l f ) as the generalization (in the sense of sect. 2) o f f (K ' ) , i.e., as a state component of [f). Thus the calculation of (/~'[f) is equivalent to the solution of the Schr6dinger equation in a certain subspace of many- body state vectors.

Most of the arguments given above for the translational problem are identical to those given by Peierls and Thouless 4). We see, however, that the requirement for internal wave functions to depend only on the internal degrees of freedom will auto-


matically ensure that these wave functions are also Galilean invariant. The reason is simply that Galilean invariance is a property of the c.m. degrees of freedom, not of the internal degrees of freedom. This is an important point in considering the rotational case because there is now no invariance principle analogous to Galilean invariance; but the requirement for internal wave functions to depend only on internal degrees of freedom must still hold. The consequence is that, quite apart from the problems of angular-momentum projection, which is formally straightforward, the construction ofinternaI states from a general many-body state for both c.m. and internal degrees of freedom can be achieved by simply integrating over the c.m. degrees of freedom completely. From the point of view of constructing correct internal states, it is immaterial how the c.m. variables are eliminated [e.g., the choicef(K') = 8(K'-Ko) is acceptable in eq. (4.4)]. But as in the translational case, the internal states can be improved by solving a GC problem with the c.m. variables as (part of the set of) generator coordinates. ]'he c.m. variables under consideration are any set of state labels, continuous or discrete, which can be used to completely label the c.m. states. It might be expected that different choices of these variables will give different, but equivalent, methods for constructing internal states.

Since the projection of angular momenta does cause some complication, we would like to set up in detail the GC problem for the rotational case with the choice of the c.m. momentum Q (or rather IQ[, the c.m. orbital angular momentum Lc and its projection Me) as the c.m. variable to be eliminated. Consider a state [g) of the many- body system which in general has the expansion

Ig) = ~ aLj,jKI(LoJ~)JK, g), (4.6) LcJIJK

where I(LoJs)JK, g) is an orthonormalized state with orbital angular momentum Lc for the c.m. state, angular momentum J~ for the internal state and total angular momentum J and projection K (along a suitable direction fixed to the "body") and azcs~s:~ the corresponding expansion coefficient. States of total angular momentum J, projection M (along a suitable direction in space) can be projected out of Ig) by the usual projection operator P~M:

P rg) - 1 f (4.7)

f2 being the set of Euler angles eft 7 by which the state ]g) has been rotated. The result of operating with P~M on [g) is clearly

PJKMIg) = 2 aLoj~JKI(Lc, JI) JM, g) LeJI

= 2 aLjIsK(LcMcJIMIIJM)ILcMeJIMJ, g). (4.8) LcJI

M~MI

Suppose there is a wave function q~n~r.~uo (X) of the c.m. coordinate X having the

560 cn-tJN WA WON~

orbital angular momentum L~ and projection Mr, the integral

f ~voLogo(X--p)(x-- PlP]:~lg)dap = const ~ aLos~jK(L~M, JIMx[JM)(x-X[JIMI, g; (N~LoJK)), (4.9)

J I M I

(where the displacement p of the c.m. in (x--PIP~M[g} does not affect the internal state in P/~M[g) and where the p integrations go from - oo to + oo so that the changes of variables to X - p can be made without affecting the limits of integration) gives termes proportional to the wave function for the internal degrees of freedom only. The constant on the r.h.s, of eq. (4.9) depends on the choice of CNoLoMo (X) and on the state fg). Finally each internal wave function is isolated with the help of Clebsch- Gordan coefficients:

(x-X[JIMI, g; (N~L~JK))

const ~o~ M'JIMIIJM' ) Y£~M,o(~)e~e'(P-X)(x--plP~M, lg)d3pd2~

xfNoLc(Q)dQ, (4.10)

where we have gone over to the momentum representation for ~:

c~ L~M~(X ) = [ e-~Q'xIN~Lo(Q) YLoMo(O)daQ, (4.11) ,o

the c.m. wave functionfNCL~ ({2) in momentum space depending on Arc (representing one or more state labels as yet undefined) and L¢.

To summarize: we have constructed internal wave functions of good angular momentum by first projecting out a state of total angular momentum or, projections K (body axes) and M (space axes) and then removing the c.m. degrees of freedom by suitably averaging over state components displaced by various distances from each other. Each of the above two steps may be called a projection: the first projection is the usual angular momentum projection, while the second projection removes the spurious or redundant motion of the center of mass. (Alternatively, we can first in- tegrate over x-p and then project out internal angular momenta. The crucial step is always the removal of the c.m. coordinates from internal wave functions.) The double-projection method presented here differs from that of Peierls and Thouless 4) in that the angular velocity does not appear.

Depending on the symmetry properties of the state Ig) there will be one or more linearly independent states of the type (4.10) with different values of L¢, J and K. Therefore we should in general consider the quantities {2, L¢, J and K as (continuous or discrete) generator coordinates,

' ~ PJ d 3 d 2 2 (LcM~JIMtIJM') Y*oM'c(Q)ele"(°-X)(x-Pl nM'lg) P M % M ' d

-~ (x-X[QLoJK,(J~M~)) (4.12)

MANY-BODY SCHR(JDINGER EQUATION 561

as the GC transformation, and (QLcJKIfNo) as the state component of IfNo) to be obtained by solving the GC Schr6dinger equation. (No is clearly the additional state label needed to specify the distinct solutions of the equation.) This completes the formal solution of the problem of constructing internal states of good angular momentum. Once these internal states have been obtained, it is a simple matter to con- struct the states I(L~JI)JM . . . . ) for the whole nucleus having the correct behavior under rotation.

It is a practical question as to when the spurious states of c.m. motion cannot be tolerated and must be projected out by applying the GC treatment in a suitable approximation. In the problem of angular momentum projection the GC treatment is actually not unfamiliar in that even in the single-projection method a mixture of linearly independent K-states is to be treated by means of a similar equation 13, 14).

5. Further discussion

In this paper arguments are given for generalizing the variational equation in the generator-coordinate (GC) theory to the many-body Schr6dinger equation in a certain subspace S of the Hilbert space of many-body state vectors. This subspace S consists of all those states Ie ) , = le) such that (xlc~) is just the many-body wave function ~b(x, e) (the generator coordinates c~ being parameters in this wave function) used as a generating function in the Griffin-Hill-Wheeler (GHW) wave function. One consequence of this result is the appearance of solutions which cannot be obtained from the original GHW variational equation. The solution of the new GC equation is formally independent of the choice of well-behaved bases, but the states If) bi-orthogonal to the original states I~) do not always form a well-behaved representation. Because of this, it has been found necessary to specify how the formalism should be interpreted if I~) are used intuitively as basis states for the subspace.

Although the GC equation is just the many-body Schr6dinger equation, the concept of generator coordinates plays an important role not only in the choice of the subspace S but also in suggesting useful points of view. For example, the study of the redundancy of the generator coordinates suggests a double-projection method for internal angular momenta by removing the c.m. coordinates as redundant generator coordinates. Another possibility is the simultaneous use of several generating functions corresponding to different types of generator coordinates to obtain a micro- scopic description of the coupling between different dynamical effects. It would be interesting to see how these possibilities work out in practice.

The author wishes to express his gratitude and appreciation to Professor Sir Rudolf E. Peierls for the hospitality of the Department of Theoretical Physics, Oxford University, where this work was initiated. He acknowledges several profitable con- versations and correspondence with Prof. Peierls, Dr. D. M. Brink, and Dr. A. Weiguny.

562 crItrN WA WONG

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11) J. N. Franklin, Matrix theory (Prentice-I-Iall, Englewood Cliffs, N.J., 1968) p. I87; W. Schmeidler, Linear operators in I-Iilbert space (Academic Press, New York, 1965) Sect. 5.2; A. I. Mel'cev, Foundations of linear algebra (Freeman and Co., San Francisco, 1963) Sect. 110; P. M. Morse and H. Feshbach, Methods of theoretical physics (McGraw-Hill, New York, 1953) Vol. 1, Sects. 7.5 and 8.3

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the generator-coordinate theory as a flexible formulation of the many-body schrödinger equation

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