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General semi-classical method for collective motion and its connection with the Rowe-Basserman and marumori theories, and adiabatic limits
Nuclear Physics ONorth-Holland
A441 (1985) 271-290 Publishing Company
GENERAL SEMI-CLASSICAL METHOD FOR COLLECTIVE MOTION AND ITS CONNECTION WITH THE ROWEBASSERMAN AND MARUMORI THEORIES, AND ADIABATIC LIMITS G. DO DANG
and ABRAHAM
KLEIN*
Laboratoire de Physique ThSorique et Hautes Energies +, Universitk de Paris XI, Centre d’Orsay, 91405 Orsay Cedex, France Received
10 September
1984
Abstract: In recent work, we have shown that in the adiabatic limit (large amplitude, small momentum), time-dependent Hartree-Fock theory (TDHF) yields a well-defined theory of large-amplitude collective motion which provides an essentially unique construction for a collective hamiltonian. An alternative theory, put forward by Rowe and Basserman and by Marumori is, apparently, not restricted to small momenta. We describe a general framework for the study of collective motion in the semi-classical limit without limitation on the size of coordinates or momenta, which includes all previous methods as limiting cases. We find it convenient, as in the past, to consider two general systems: first, a system with n degrees of freedom and no special permutation symmetry, and, second, a system of fermions described in TDHF. For both systems the problem can be formulated as a search for a hamiltonian flow confined to a finite-dimensional hypersurface in a phase space, which itself may be finite- or infinite-dimensional. Though, in general, there are no exact solutions to this problem, we can formulate consistent approximation schemes corresponding to both the adiabatic and Rowe-Basserman, and Marumori limits. We also show how to extend the momentum expansion, which underlies the adiabatic approximation, to higher orders in the momentum. We thereby confirm the structure of the theory found in our previous work.
1. Introduction
In recent work’,*), we have provided a new and (we believe) theoretically clean formulation of the adiabatic limit (ATDHF) of time-dependent Hartree-Fock theory (TDHF). Our most important result is that Villars’ equations 3*4),which express the content of that theory have an essentially unique solution under “suitably” imposed boundary conditions leading to a construction for an adiabatic collective hamiltonian. In work to be reported separately 5), we have shown how to solve these equations directly by a method which ties in naturally with existing technology for doing self-consistent cranking calculations. * Permanent USA. t Laboratoire
address:
Dept. of Physics, University
associe au Centre
National
of Pennsylvania,
de la Recherche
271
Philadelphia,
Scientifique.
Pennsylvania
19104,
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G. Do Dang, A. Klein / Gene4
semi-classical
method
Though the work described above has responded to some of the open questions in the previous literature6), it has hardly responded to all of them. One of the most important remaining questions is the domain of application of the Rowe-Basserman and Marumori theories7,8). In particular these authors have stressed that their work is not a small-momentum expansion and therefore is more general than the adiabatic theory. This point will be examined in the current work. There is no doubt that a theory applicable to the domain of large coordinates and large momenta is of greater generality than the usual limits recognized in low-energy nuclear physics: the large-amplitude adiabatic limit, where we expand in powers of the momentum only, and the anharmonic vibrational limit, where we expand uniformly in momenta and coordinates. In all three cases, we assume that there is at least one large number N which characterizes the degree of collectivity. The simplest case in principle is the anharmonic vibrational one, where with suitable (and consistent) scaling not only is N-l small but so also are X/N and P/N, where X, P are a canonical pair characterizing a collective degree of freedom. Next in difficulty is the adiabatic limit, where only l/N and P/N are small. Finally the fully semi-classical limit in which only l/N is available as an expansion parameter is notoriously more difficult than the previous special cases. In the present paper we shall formulate a theory of collective motion which makes use at first only of the semi-classical assumption. For this formulation it suffices to study the Wigner transform (WT) of a suitably formulated quantum theory ‘), retaining only the leading terms in the kinematics and dynamics. To illustrate our approach, we shall consider two classes of problem, following a sequence which we found useful in our previous studies2). First we shall consider a problem with n degrees of freedom and no particular permutation symmetry. Here the semi-classical limit is indistinguishable from the classical one and we find ourselves studying a problem in hamiltonian classical mechanics. Decoupling of p -Cn collective degrees of freedom from the total is then recognized as the problem of finding conditions under which a hamiltonian flow in n dimensions can actually be confined to a hypersurface of p < n dimensions. We formulate and study such a set of conditions, some of its properties, and some of its limiting cases. This study is carried out in sect. 2 where we formulate the problem of finding a hamiltonian flow in less than n dimensions. An examination of the resulting equations makes it evident that, at best, this can be done only approximately. We formulate and describe two methods of approximate decoupling. The first treats momenta and coordinates on a symmetric footing and is seen to be equivalent to the work of Rowe and Basserman. The second method admits an expansion in the momenta. Here we have pushed the theory to one higher order than previous workers in order to deepen our understanding of the structure of the theory and of its consistency. In sect. 3, we consider the second class of systems, namely a system of fermions described in TDHF approximation. The essential task of this section is to show that
G. Do Dang, A. Klein / General semi-classical
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273
the TDHF theory may be expressed in hamiltonian form and that every equation and every result found in sect. 2 has its analogue in the present case. We thus conclude with a uniform theory for the two classes of system. In a concluding section, sect. 4, we provide some discussion of the relation of the simplified approach of this paper to previous work’*2,9,‘o) of the problem “requantization”, and of methods of solution of the theory. Some algebraic details of the derivations of sect. 2 are found in the appendix.
of
2. Classical description of collective motion 2.1. FORMULATION
We are given a hamiltonian H(E, a), where (tl . . . 5”) are n canonical coordinates and ~r=(ai a.. 7~,) are the corresponding canonical momenta. Hamilton’s equations, ?i,=
-
aH/ap,
(2.1)
p= aH/aTa,
(2.2)
define an initial value problem relating to the whole of phase space. What interests us here is the following special question: The equations 5a=5a’(X,P),
“u=“,(xJ+
(2.3)
P=P1”‘Pp,
(2.4
where x=xi describe a hypersurface
. ..XP
9
E of dimension 2p (p < n) in the phase space. Are there
any solutions of eqs. (2.1) and (2.2) in which the “hamiltonian flow” is confined, approximately at least, to the hypersurface E? The conditions for such a flow to be realized are as follows: (i) x and p are a partial set of canonical coordinates for the full phase space. This can be expressed, (summation convention)
for example,
by a set of Lagrange
bracket
~a71,_~alr,=,f axiaPj
(ii) We define a collective hamiltonian K(x,
apjaxi H,(x,
J.
conditions
(2.5)
p) by the equation
P) = H(C;(x, P),+,
P)).
(2.6)
We require that .$” and or,, as expressed by (2.1) and (2.2), describe a flow on the hypersurface 8, i.e. if 5” and TT~are initially on the hypersurface, then they remain on the hypersurface. On the one hand, the right-hand sides of (2.1) and (2.2) are defined on the hypersurface by (2.3). On the other hand, the condition that [“(x, p) and ~Jx, p) be dynamical variables on 2 can be expressed by the requirement that
274 H, is the appropriate
G.Do
Dang, A. Klein / General semi-classical
method
hamiltonian to determine their time development, i.e. ?i, = [G Kl,,
aTa aHC =aar a& axiaPi axi’
(2.7)
aPi
i”=
[V, H&r,,
(2.8)
where the subscript PB denotes a Poisson bracket. Alternatively, combining (2.1), (2.2), (2.7) and (2.8) we obtain a set of partial differential equations which together with the kinematical constraints (2.5) hopefully determine the hypersurface 2, namely - g
= [7,x, H,],,,
(2.9)
[CH,],,.
(2.10)
g=
Before studying whether these equations do indeed fulfill their stated aim, let us consider more closely the meaning of eqs. (2.3) and (2.4). If indeed motion can be confined to the hypersurface E, this would appear to imply that there are n -p canonical pairs which can be ignored, where the corresponding coordinates can be frozen in value, (the corresponding momenta set to zero). Thus we imagine that underlying (2.3) is a locally invertible canonical transformation, where we can then also write for the partial inverse transformation xi=xQ,9),
(2.11)
Pi=Pi(E,+
(2.12)
This interpretation is essential if, indeed, we are to justify eq. (2.6), which in the case of a full canonical transformation, defines the transformed hamiltonian. In this limit, eqs. (2.9) and (2.10) become identities. For later convenience, we record the following relations, which are equivalent to the content of (2.5) or of the corresponding Poisson bracket conditions:
ap ap. _=I axi arm' axi aB at* api7
-=--3
aTa
z=
axi
Iap.
- aga ) --
air,=
a.p
aPi ’
(2.13)
(2.14)
For some further work, it is convenient to introduce a standard tensor notation in phase space. Let us write $=
E,n,
yi=x,p,
/.L=1...2n,
(2.15)
i=1.*.2p,
(2.16)
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G. Do Dang, A. Klein / General semi-classical method
and define the antisymmetric matrices Jpy and yjj
by the equations
E”n,
Pen,
i
i>p,
(2.17)
(2.18)
where
JpvJxv= 6,~3 Yijcfkj
=
(2.19) (2.20)
&ik.
The equations of motion (2.9) and (2.10) may be written in this notation as (2.21) The derivatives of the collective hamiltonian on the right-hand side of (2.21) may be expressed in the form aH .=aH??l!
(2.22)
ayi avpayi.
Finally, eqs. (2.13) and (2.14) can be written as (2.23) and the canonicity conditions (2.5) as aBcJ
a17” =jTii
ayi pvayj
.
(2.24)
2.2. GENERAL METHOD OF SOLUTION - THE ROWE-BASSERMAN THEORY AS AN APPROXIMATION
For each value of i, the quantities (aq@/ayi) define a tangent vector to the hypersurface at the point in question. Given the set of tangent vectors, then eqs. (2.21) and (2.22), by trivial counting provide enough equations to determine the hypersurface 7” = v”(u). Therefore we need additional equations for the determination of the tangent vectors. We can attempt to find these by differentiating eq. (2.21) with respect to the collective variables y’. We notice immediately that we will not, in general, succeed in our aim because the resulting equations will also contain the double array of vectors (a*+/‘/ay’ayj) which measure the curvature of the collective hypersurface. We return to this essential point below. But first let us notice that the process of differentiation leads us to the introduction of a natural affine connection in phase
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G. Do Dang, A. Klein / General semi-classical method
space, namely the quantity ppaf
a2yi a$
w ayi
’
which under a change of coordinates q + 11 has the correct transformation
(2.25) proper-
ties, namely (2.26) With the help of (2.25), we derive from (2.21) (2.27) where Hp.y is the covariant derivative (2.28) Eqs. (2.27) are a set of local RPA equations subject to the normalization (2.24). But as remarked above, these equations cannot be implemented because the operator HP; y is not fully known. If there were some metric structure in the phase space, given a priori, the riV could be expressed in terms of it and the problem closed. But this is possible only in very special (and often uninteresting) cases. In the general case the problem does not terminate at this point: we have to seek equations for
(a2qyayia_+), etc.
A natural local linearization procedure does suggest itself, however. This is to assume that the curvature of the collective hypersurface itself is small on the average and thus to ignore it. This leads to a closed theory which consists of (2.21), (2.22), (2.24) and (2.28) with H,;.+
a2H/avpaqy.
(2.29)
Within the present class of problems, we have thus found the theory of Rowe and Basserman7). The main characteristic of this theory is that it treats coordinates and momenta on an absolutely equal footing. Therefore, it may be applicable to large momenta and large coordinates i.e. to the full semi-classical limit. On the other hand, we see that it is not an exact theory: indeed it is exact only when H is globally quadratic in 7”. Still, it is an interesting theory which, to our knowledge, has not so far been applied consistently to any problem. [The application found in ref. 7, is an application of the Rowe-Basserman theory in the adiabatic limit. In this limit, except for special cases, it is only an approximate version of the full adiabatic theory.]
G. Do Dang, A. Klein / General semi-classical 2.3. EXPANSIONS
method
271
IN MOMENTA
We have just
seen how, in principle,
a theory of collective
motion
can be
developed without making the assumption of either small velocities or small displacements. Next we consider the situation that an expansion in the momenta is of interest. In a canonical description, this is a conventional choice. Furthermore, we assume the coordinates to be even under time reversal and the momenta to be odd. We thus take the initial hamiltonian H(& a) as a power series in even powers of v~, namely H( E, lr) = V(E) + $r&K”~(
{) + $7rJrf17rY7r&~@+(6) )
(2.30)
where we have chosen to go to fourth order in the momenta in order to show how the adiabatic approximation, now well established up to second order in the momenta’*2), can be extended systematically to the next order. It suffices to consider a single collective coordinate, so that we are looking for a collective hamiltonian in the form H,=V-(x)++pzX(x)+~p%?(x).
(2.31)
To solve the proposed exercise, it stands to reason that we should make a corresponding expansion in momenta for the description of the hypersurface (2.3). For present purposes, we write 6” = +;b,(x> + &#$,(X)P2,
(2.32)
Ir, = @(x)p
(2.33)
+ )~~‘(x)p?
The aim is to show if we apply the canonicity condition (2.5) and the equations of motion (2.9), (2.10) consistently to fourth order in p, we thereby obtain a set of conditions
which suffice to determine the four sets of functions I#$,, cp&, @)
and
+&‘). As a special case, we must regain the usual adiabatic theory, which is expressed in terms of the functions $J&) and @, when we specialize to the usual quadratic hamiltonians. Thus the extension of the theory by one more order serves to clarify the connection between the structure of the hamiltonian and that of the hypersurface on which we seek a description of the collective motion. The first terms of (2.32) and (2.33) describe a point transformation, provided the first of the canonicity conditions given below is satisfied. Turning to the actual machinery, the application to (2.5) of (2.32) and (2.33) yields two consistent conditions of zero and second order in p, namely (2.34) (2.35) Next consider eq. (2.9). Comparison of (2.30)-(2.34)
shows that eq. (2.9) yields three
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G. Do Dang, A. Klein / General semi-classical
method
possibly consistent conditions of zero, second, and fourth order in p, namely V,, = $‘,)(dV/dx), +@$lP’K”@,, +( l@$
+ $b”@)
+ $v,,+#$,=
@)&T+
:#:“‘+;,
(2.37)
K”p,Y + +lJ’,“‘@J+;i,K”@,E,Y
+ t+;,&YX,~,Y
+ ~~~~@)@j/(O)~+J 0 =
Correspondingly follows:
d#r) - TX+
(2.36)
d,&
I d,& Jy+ dx
-+%
#X&T+
~~~&$~.
(2.38)
eq. (2.10) yields two conditions of first and third order in p, as
In deriving these equations, it is well to recall that the p-dependence arising from (2.32) must be recognized in the arguments of the functions V and Kafl. Eqs. (2.36)-(2.40) must be consistent with the conditions which follow from the equality of H and H, on the collective hypersurface. These are ~((PP(x)) @J,tp)K”~ $~‘,o’@~~~g/py~~
= V(x)
(2.41)
3
(2.42)
+ V,&c*1, =.Y >
+ a( ~~‘~~’ + @#‘)
K””
’ (‘)‘clj3 (‘)$(l) -Y Ka’B,Y+ + da
++;;,q$,V,p = $9.
(2.43)
What must follow now is a certain amount of unenlightening algebra, necessary, however, to the understanding of the structure and use of the preceeding equations. Such algebraic details as we feel are absolutely necessary are given in the appendix. Here we will be satisfied with a summary of the results. Consider first the usual adiabatic limit, in which we work only to quadratic order in momentum. In that theory, studied previously, we omitted the functions 4:) and +;i,, i.e. we worked with a point canonical transformation. The relevant equations are (2.34), (2.36), (2.39), and (2.37) (the p2 term). We mention the latter last because under the approximation cited - point transformation - we showed that (2.37) is a consequence of (2.34), (236) and (2.39) and may therefore be dropped. The remaining equations and the equation Q(
x> =
ax/a.p,
P-44)
G. Do Dang, A. Klein / General semi-classical method
219
consistent with (2.14) and necessary for the consistency of (2.36) with (2.41), are sufficient to determine the unknowns +&(x), @j(x), V(x) and X(x>. This is not trivial to prove and represents the most important result of ref. ‘). Finally (2.41) and (2.42) are consequences of eqs. (2.34), (2.36) and (2.39). The generalization of this structure when we go to fourth order in momentum is almost straightforward. Now, of eqs. (2.34)-(2&l), it is (2.38), which is the condition of order p4 which follows from the other equations, but only to the first order in $2) and c#& But this is all we can expect at this level of accuracy. If we examine our equations, for example, the Lagrange bracket condition, we see that in order to have second-order accuracy in the quantities $2) and +fi,, we must also carry out our expansion of 5” and or, to the next order of momentum beyond that written in (2.32) and (2.33). The remaining equations plus (2.44) should, in analogy with the results of ref. ‘), provide a sufficient number for the determination of the unknownst: $I&, c#&, $J’,“‘,I/J&‘),V(x), X(X), Z(x) and imply eqs. (2.41)-(2.43). We have thus provided the tools for a consistent theory to order four in the momentum. It should also be apparent that further expansion in the coordinates will provide a theory of anharmonic vibrations.
3. study of TDHF 3.1. FORMULATION
The strategy throughout this section - whether we deal with an approximation which treats coordinates and momenta on an equal footing or one in which we expand in momenta - will be to show that we can map the TDHF problem onto the classical problem studied in the previous section. It is well known7*“) that TDHF is equivalent to a problem of classical hamiltonian dynamics, but for our convenience, we require a special form, which we proceed to develop. In this development, we shall consider the elements discussed in sect. 2 in roughly the same order and seek to uncover the corresponding elements of TDHF. The first point is to show that the TDHF equations can be put into hamiltonian form. In consequence, we shall be able to identify the quantity which plays the role of “classical hamiltonian”. Working with only a single collective canonical pair x, p, we seek a density matrix p(x, p) satisfying the condition P2(% P) = Ph and thus describing a Slater determinant, equations ifi&,
P) = is,
P)
(3.1)
which is also the solution of the TDHF &,
P)l
nbv
t The proofs of ref. ‘) have, however, not yet been extended to this case.
(3.2)
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G. Do Dang, A. Klein / General semi-classical method
where &’ is the Hartree-Fock hamiltonian -%b(X, P> = h,, + L&&
P> 7
(3.3) hob, Vacbdare the one- and (anti-symmetrized) two-particle matrices characterizing a standard nuclear hamiltonian, and a, b, c, d refer to arbitrary complete sets of single-particle states. The dependence of p on x, p will be sought in connection with the problem in collective motion. But first we consider the problem of showing that the TDHF eqs. (3.2) can be written in hamiltonian form (under more general circumstances than the restriction to the collective hypersurface). We recall that according to one of the theorems of Thouless’2) p(x, p) is completely characterized by its particle-hole ph and hp matrix elements in an arbitrary set of single-particle states. Eq. (3.1) effectively determines the remaining elements. If we utilize the representation in which p(x, p) is diagonal, eqs. (3.2) may be written ifir,, =srr, = ~Wu&&,
= aH/&+,
,
iphp = -.X?hp = - 6 Wm/GPph = - 6H/6Pph,
(3.4)
a form which resembles Hamilton’s equations with H(P) = K&P) = h,~p,, + f
(3.5)
KbcdPcnPdb
playing the role of hamiltonian, except that pph and pg are not canonical variables. They can be related to a set of classical canonical variables
(3.6)
Tph’ -rhp,
(3.7)
by means of the equations13,‘4) Pph= P&
=~((~+is)[l-~(~2+n2)]“2)ph
(3-g)
[Ph= &{(CY-1’2)ph+(ry-1’2),p}?
(3.9)
and their inverses
=ph
=
(3.10)
-i~{(ry-1’2)ph-(ly-1’2)hp}.
Here rph =
Pph 3 y
rhp = =
$ [
Php,
rpp, =
1 + (1 - 4rtr)“2]
rhh’
)
--0,
(3.11)
and it is understood that t and II have the same support as r, namely ph and hp indices only. Eq. (3.8) is a classical version of the Holstein-Primakoff transformation relating generalized “spin” variables to canonical pairs. Once we establish, see below, that
G. Do Dang, A. Klein / Generalsemi-classical
the 4 and II are canonical,
method
281
then, of course, any further set obtained by canonical
transformation will serve equally well. In the representation in which p is diagonal, we have Q, = Err = qp,, = 0, similarly for the hp elements, and from (3.8)-(3.10) JPphm
p’h’c
we therefore find L
‘6
pp’
&&,
(3.12) (3.13) (3.14) (3.15)
By applying the chain rule to the first of eqs. (3.4), changing the variables from as well as (3.6) and (3.1) we (Pph> Php) + why rph) and utilizing eqs. (3.12)-(3.15) find that the first of eqs. (3.4) may be written aH aH * i-6Ph- Tph=a5ph+i~.
(3.16)
Similarly, the second of eqs. (3.4) yields the same result except for a change in the sign of i. We thus arrive at Hamilton’s equations. It is important to remark again that the demonstration just given has nothing to do with the problem of collective motion. We have shown that the TDHF problem for an arbitrary Slater determinant may be expressed in terms of Hamilton’s equations. Returning to the problem of collective motion, we may take over intact the remaining discussion of subsect. 2.1. In essence we seek to satisfy eqs. (2.9) and (2.10) which express the requirement that a hamiltonian flow is confined to the hypersurface (2.3). That this is not in general possible means that certain approximations will be necessary. Again we will consider two general possibilities, namely that in which we treat coordinates and momenta on an equal footing and second, the result of expanding systematically in momenta.
3.2. THE ROWE-BASERMAN
THEORY
With the identifications made earlier in this section we can, in principle, simply refer to the material presented in subsect. 2.2. Nevertheless, we choose here to present a brief alternative version, using the better-known density matrix variables. In this manner, we shall have more immediate contact with the work of previous authors 6,7). Thus we seek a two parameter family of Slater determinants p(x, p) which we shall characterize at each point of the phase space x, p by its diagonal representation. In this representation the equations of motion analogous to eqs (2.9) and (2.10)
G. Do Dang, A. Klein / General semi-classical method
282
namely
*)
ibob=LKdob=GLhJPB =iaH, --&?zb i JHCaPab ax
t3p
ap
(3.17)
’
ax
have the non-trivial matrix elements ~ph=j~--i~-
aH
ap,,
ax -~hp=i~_-i~-aH ax
aH
ap,,
ap
ap
ax
aPbp
aH
aphp
ap
ap
ax
9
.
To see the meaning of these equations let us define the real functions A(x, ~(x, p) by the equations
(3.18) p)
and
x = aHJap,
(3.19)
p= -aHJax.
(3.20)
These appear below in eq. (3.24) in the form of Lagrange multipliers. We define further appdax
uph =
-
rph
-iap,h/ap
=
(3.21)
= a;p, = -T&.
(3.22)
Thus the second of eqs. (3.18) appears as the complex conjugate of the first. Together they are now equivalent to the equation (3.23)
[%Pl=o,
where 2
is a generalized cranking hamiltonian, namely .9=Jf?-p7-iihts,
(3.24)
involving two, initially unknown, cranking operators u and r, determined by the local tangent plane to the hypersurface. (Thus it appears that, in general, 2 will not be a real operator.) As explained previously, to close the description, we seek equations for the operators T and u which, consistent with eq. (3.1) have only ph and hp non-vanishing elements. As also explained previously, this closure will not be achieved at this level unless we assume in the resulting equation that the derivatives of 7 and u with respect to x and p may be ignored. We furthermore assume that our choice of x and p satisfies a*HJax
ap = 0,
(3.25)
and define, in analogy with a harmonic oscillator, ahlap
= a*HJap*
- acL/ax = a2HJax2
= m-l(x)
,
= w*(x)m-l(x).
(3.26) (3.27)
G. Do Dang, A. Klein / General semi-classical method
283
Thus differentiating (3.23) (in turn) with respect to x and p and utilizing the various assumptions and definitions of this subsection we find a version of the “local” RPA equations as -[%?,a]+[a.&/ax,P]=
(3.28)
-io%l[r,p],
(3.29)
i[~,7]+[a~/ap,p]=im-l[a,p].
Given p, (3.28) and (3.29) are linear equations in u and 7 of which only the ph elements need be considered. It remains only to specify a normalization condition for the solutions of these equations. This can be done with the help of the canonicity condition (2.5) and eqs. (3.9) and (3.10). We find straightforwardly, using the hermiticity of p and of its various derivatives, that (3.30) This completes the specification of the theory, since we have enough equations for all unknowns. There remains one matter - the actual choice of collective variables x,p. The discussion of this point is best left to actual cases, except to remark that the freedom of canonical transformation on the hypersurface remains.
3.3. EXPANSION IN MOMENTA
The remaining new technical task which we have set for ourselves in this paper is that of carrying out the momentum expansion for the solution of the TDHF theory. We believe that here clarity will be served if we make the translation of the results of subsect. 2.3 in the most literal way. This means first that we expand eq. (3.5) which we write more explicitly as NE/+
(3.31)
eI,(Pka
in the form (2.30) and identify thereby the elements V( [), KaP(E) and LUByS(E) which provide the input into the theory. As the second step, we wish to identify the ingredients c#$&and $i, of eqs. (2.32) and $c) and #(,1) of eq. (2.33). The theory is then fully defined by substituting these expressions, to be given below, into eqs. (2.34)-(2.43). To carry out the first of our tasks we first expand p([, a) in a formal power series in II including fourth-order terms, namely P(b)
= P’“‘(&)
+ P(la)(E)%
+ $p(3Qfy
+ $P’2”%)%~/j
E) 7rJrp?ry + fp@“f+y
~)7rJr~7r~~8.
(3.32)
Here, of course, (Y,/3.. . run over all ph indices in some representation. We shall carry out all calculations in the first part of our task in the representation in which p(‘)(E) is diagonal. This entails a considerable simplification in the ensueing formulas. First by investigating the consequences flowing from the joint requirement of
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G. Do Dang, A. Klein / General semi-classical method
(3.32) and the condition p = p*,
we obtain the consequences
p(o)= (p(o))*, P P
w
= p’o)pu*’ + pwpK9 )
(3.34)
cw3) = p(o)pw) + pw7)p(o) + pwp(‘8)
p(3”BY)
=
p(0)p(3aBY)
+3[{p P
(3.33)
(4Mbv
=
+
)
(3.35)
p(3w3Y)p(o)
UN, pwv)}
p(o)p(4d3YQ
+ pwpw
+
+ { pw), pm}
+ { p(‘Y), pw)}]
,
(3.36)
p(4”8YQp(o)
+f[{p(‘a’),p(38us)}+(,t,P)+(,oy)+(at,6)]
+i[{p
W),
p(*Y”)}
+
{ p(*aY),
p(*8~)}
+
{ p(*“f9,
p(*/3Y)}]
.
(3.37)
Here the curly brackets are anti-commutators. These equations have the following consequences: in the representation in which p(O) is diagonal, p (‘) has only ph and hp elements, and the pp’ and hh’ elements of
. . are determined in terms of the non-vanishing elements of the lower-order parts of p. Next consider eq. (3.8). A formal expansion, which is understood to refer only to the ph and hp parts of p, has the form p(2), p(3), p@).
fip
= [(l
- +t*)l’*
-i$s3(1
+ ir(l-
_ $[*)-l’*
$t*)l’* - &$a*(1
- $&*(I
- f<*)-l’*
- $~2)-1772(1 - ft2)-1’2
+ ... . (3.38)
In the representation
in which p(O) is diagonal, we have (P(“))ph=
(3.39)
[EC1 -f5*)1’2]ph=0’
and &ll = [,,r = 0, as previously remarked prior to eq. (3.12). It follows from the examination of (3.38) that in this representation it reduces to &pPh
=
i9rph- i+rpgpJWrpW.
(3.40)
We conclude, upon comparison with (3.32) that (lPW
=
-
(2p’h’p”h”)
=
0,
“’ h “’ ) =
_
Pph Pph (3p’h’p”b”p
Pph
=
(3.41)
= i + app,aw, d-
(3.42) pf;‘h’p”h”p
“’ h “’ )
i~&~3(8pp~8p?~p~~~
p(;p,h’p”,,“pIISh 0,r,VV,,‘P’) = o. P
pf;‘U
8,,,,
8,w))
(3.43) (3.44)
G. Do Dang, A. Klein / General semi-classical method
Here S, means the sum over all 3! pe~utations
285
of the pairs p’h’, p’h” and p “‘h “’
considered as single indices, divided by 3! It now follows from (3.34) that p$? = pf& =
0. Fimilly from (3.35) and (3.37), respectively, we can calculate &Y”’
= S, (&,
(3.45)
Isp’r*St&> ,
p~~p’h’)= -S,(s,s,,zsPP~), 12 pE;v”‘h”‘)=
(3.46)
~S~(spp~sp’p~~~“g,~’ 6,.!,
p~kyh’fFp 12
?”h”) = - qs4fshh, ah’& s,,
a,,,),
(3.47) (3.48)
The definitions of S, and S, are the analogues of that given for S,. We are in possession now of all the results necessary to compute the elements of eq. (2.30). One substitutes the expansion (3.32) into the Hartree-Fock functional and collects terms of different order in 7~~.The odd orders are guaranteed to vanish owing, for example, to the antisymmetry properties of (3.41) and (3.43). The nonvanishing coefficients K and L are then easily evaluated, using (3.41)-(3.48). We find 1y PhP’h’
L= 16 2 hh lsPpF -
; vhh’pp’
+xp’p] +
-
5 vhp,pht
Lpbp’wp”h”p “h’m= fS,$ xp,p2 6ppzs
(
p’pI
~spp+%h’ +
6
p,,p,,,
+3Eah’hl
+ vph’hp’
8,
-
,.,
f F/Pp’fi’,
(3.49)
&W)
It is also evident that V(E) = ~,,(P’“‘(E)).
(3.51)
Of these results (3.50) is new and (3.49) and (3.51) are equivalent to results given in our previous work’p2). Unfortunately in the latter we chose to define K on a doubled space, allowing hp as well as ph elements so that the detailed identifications are in superficial disagreement. The dynamical equations which follow are the same, however. Turning to our second task, that of finding the elements of (2.32) and (2.33) this can be accomplished by a calculation which is in several respects the inverse of the calculation performed above. We expand eqs. (3.9) and (3.10) in powers of r. It will
286
G. Do Dang, A. Klein / General semi-classical method
be clear below that we require at most cubic terms. Using rr = r, we find ~~ph~((rph+Thp)+f[(TTT)ph+(TTT)hp],
(3.52)
fi7rph= -i(rph-rhp)-i$[(rrr)ph-(rrr)hp].
(3.53)
The next step brings in the description of the collective motion in terms of p(x, p) which we subject to the adiabatic expansion P(x,p)=p(O,(x)+p(‘)(x)p+3p(*)(x)P*+tp(3)(X)P3+aP(4)(x)p4.
(3.54)
Here all the p(“) are hermitian matrices. For n even the matrices may be chosen, consistently, as real and symmetric, for n odd as imaginary and antisymmetric. Substituting the ph and hp elements of the expansion (3.54) into (3.52) and (3.53) and collecting terms, it suffices for our purposes to retain terms of the zero and first order in p(O). This is because we shall finally choose the representation in which p(‘)(x) is diagonal, i .e . p(O) ph = p$” = 0. Since, for the theory of subsect. 2.3 we require at most the first derivatives of .$Ph and VP,,,the terms of higher order in p(O) will simply not contribute. Thus in a general representation, but to linear terms in p(‘), we have \/26%
(2r~)+p*[~~~)+(r(l)r(l)r(0))ph + (r%%(r))p~
+,(l)&.(*)
+ (r%W))ph])
+ rwr(*)rm
)
+ r(*)rmr(l)
(3.55)
+
r(*~r(l)r(09ph) .
(3.56)
Finally comparing (3.55) and (3.56) with the equations
in the representation derivative, we have @$=O,
EPh= G&t+
$P”$Jtf$,
(3.57)
Tph= J1$ P
+ +P3+Icl(plh) >
(3.58)
in which p(O) is diagonal, and computing the required first (3.59) (3.60) (3.61)
(3.62)
287
G. Do Dang, A. Klein / General semi-classical method #g
dJ/$ -=.
dx #$3
=
-
la
ifip$ -dp$)
dx
,
(3.63) (3.64)
’
- ifi p$,
(3.65) dp”’ zp(2++1)
&$‘ii -= dx
d (0) dpCo) +p”‘L dx PC2)+ P(2)yjyp(‘)+
P
(1)P (2)dx dp(O)+ p(*)p(‘)-dp”’ dx
1 pi
(3.66)
This completes the structure of the theory insofar as we wish to develop it in this paper. Together with refs. ly2) we have attempted to provide a complete theory of large-amplitude collective motion. Basically what remains is the very essential task of showing that one or more of the alternatives described can be put into service. A start on this venture has been made ‘). 4. Summary and conclusion This paper is based on the assumption that the problem of large-amplitude collective motion may be studied by means of an expansion about the classical limit of the many-body problem. We have referred to this. somewhat loosely as the semi-classical approximation in order to keep in mind that the theory presented in this paper can be derived from a fully quantum theory by the method of Wigner transformation*), that therefore requantization is straightforward, and that the inclusion of quantum corrections is also, in principle at least, straightforward. However, in this paper, we have superficially studied a problem in classical mechanics. It should be pointed out that somewhat similar classical ideas can be found in the works of Brito and Sousa9) and of da Providencia and Urbana”), which are equivalent in content to subsect. 2.1 and part of subsect. 2.3, though the presentation of the ideas of subsect. 2.1 differs considerably. Where our work has wider scope is first, in the realization, that the canonical framework is so general that it provides, finally, the basis for understanding from a unified starting point all the viable literature of large amplitude collective motion. From this point of view there seem to be only two rational alternatives worth discussing in detail+: either one treats coordinates and momenta on a symmetrical footing, or one assumes that the coordinates are “large” and the momenta “small”. We have shown that the Rowe-Basserman theory is an approximate version of the former and the adiabatic theory an approximate version of the latter. We have carried the adiabatic approximation to one higher order than previous work. + Of
course, if there is more than one collective coordinate, mixed cases can and do occur.
G. Do Dang, A. Klein / General semi-clossical
288
method
The second major result of this paper is a new demonstration theory
may be rewritten
so that the full machinery
of how TDHF
of the classical
theory is
applicable to it. With respect to the matters described in detail in this work, the presentations may be taken to supercede the corresponding material of refs.‘,‘). Of these papers, we continue to refer to ref. ‘) for a relatively exhaustive theoretical study of the solution of the equations of the adiabatic limit. Further discussion of this question is nevertheless of interest, in particular in connection with the relationship between the Rowe-Basserman and adiabatic approaches and under what circumstances they may overlap. However, we have decided to postpone this discussion for inclusion with future applications. We also refer to both refs. ‘y2) for our “best shot” at providing a fully quantum foundation for the classical limit considered here. Further work will be devoted to the development of the practical consequences of our theory. One of stimulating hospitality part under
the authors (A.K.) would like to thank Dr. B. Giraud for several discussions. He is also grateful to K. Chadan and B. Jancovici for the which made his stay at LFTHE so pleasant. This work was supported in contract number 40132-5-20441 of the US Department of Energy.
Appendix The purpose of this appendix is to provide some additional details of the consistency and structure of eqs. (2.34)-(2.43). We first show that the conditions (2.41)-(2.43) which define the collective hamiltonian are consequences of the previous kinematical and dynamical conditions. Differentiating (2.41) we find V,, = (dY(x)/dx)(
8x/&$“)“‘.
(A.1)
This agrees with (2.36) provided ( ax/ag”)(o’ = I@.
(A-2)
This equation is consistent with the canonicity condition (2.14). Thus (2.41) is an integral of (2.36). Turning to the proof of (2.42), starting from (2.39), we form the sum necessary to apply (2.34) and remember (A.l) and (A.2). The result is (2.42). Similarly we try to derive (2.43) from (2.40). Again, forming the sum necessary to apply (2.34), we derive from (2.40) the equation
64.3)
G. Do Dang, A. Klein / General semi-classical
method
289
We wish the difference between (A.3) and 4 times (2.43) to vanish. This wish is satisfied if in the difference in question we insert (2.35) and (2.39). Next we consider the role played in the theory by the dynamical condition of order p2, eq. (2.37). In the adiabatic limit, this equation is not independent of the conditions of order zero and one in p, but can be derived by properly combining (2.42) and its first derivative with (2.39) and setting 4:) and $yi, to zero. In the event that Ir/c) and +Fi, are retained, the same combination is seen to be consistent with (2.37) provided the new kinematical condition (2.35) is utilized. In the latter case (2.37) provides one set of equations for the additional variables J/t) and $&. A second set is provided by the condition of order p3, namely eq. (2.39). One extra unknown, Z’, is “determined” by the additional kinematical constraint (2.35). It remains only to consider the condition of order p4, eq. (2.38). In analogy to what was found for the order p2 equation in the adiabatic limit, we expect to be able to show that this condition is here not independent of conditions of order zero to three in p. However, a little extra care is necessary because an examination of the kinematical conditions, for example, shows that the present theory is consistent only to the first power of $c) and $$‘i, and therefore the proof of dependence of (2.38) should be expected to hold only to that order. The detailed demonstration involves the following steps: (i) Calculate the derivative with respect to [” of eq. (2.43). (ii) In the resulting equation substitute (2.40), (2.39) and (2.37). (iii) Take note of the identity (‘4.4) Proof:
Utilize
(A-5)
and (A.2). (iv) Use the equation
(A-6) which can be verified by multiplying by (d+To/dx), Use the identity
summing and using (2.34). (v)
which is an analogue of (A.5). (vi) Finally if we drop the terms depending on the square of @pi,and $k), we find (2.38).
290
G. Do Dang, A. Klein / General semi-classical method
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
A. Klein, Nucl. Phys. A410 (1983) 74 A. Klein, Nucl. Phys. A431 (1984) 90 F. Villars, Nucl. Phys. A285 (1977) 269 K. Goeke and P.G. Reinhard, Ann. of Phys. 112 (1978) 328 G. Do Dang and A. Klein, Phys. Lett., submitted K. Goeke and P.G. Reinhard, ed., Time-dependent Hartree-Fock and beyond, Lecture Physics, vol. 171 (Springer, Berlin 1982) sect. VI D.J. Rowe and B. Basserman, Can. J. Phys. 54 (1976) 1941 T. Marumori, Prog. Theor. Phys. 57 (1977) 112 L.P. Brito and C.A. Sousa, J. of Phys. Al4 (1981) 2239 J. Da Providencia and J.N. Urbana, in ref.6), p. 343 P. Kramer and M. Saraceno, Lecture Notes in Physics, vol. 140 (Springer, Berlin 1981) D.J. Thouless, The quantum mechanics of many-body systems (Academic, NY, 1981) E.R. Marshalek and J. Weneser, Phys. Rev. C2 (1970) 1682 J.P. Blaizot and Y. Orland, Phys. Rev. C24 (1981) 1740
Notes
in
A441 (1985) 271-290 Publishing Company
GENERAL SEMI-CLASSICAL METHOD FOR COLLECTIVE MOTION AND ITS CONNECTION WITH THE ROWEBASSERMAN AND MARUMORI THEORIES, AND ADIABATIC LIMITS G. DO DANG
and ABRAHAM
KLEIN*
Laboratoire de Physique ThSorique et Hautes Energies +, Universitk de Paris XI, Centre d’Orsay, 91405 Orsay Cedex, France Received
10 September
1984
Abstract: In recent work, we have shown that in the adiabatic limit (large amplitude, small momentum), time-dependent Hartree-Fock theory (TDHF) yields a well-defined theory of large-amplitude collective motion which provides an essentially unique construction for a collective hamiltonian. An alternative theory, put forward by Rowe and Basserman and by Marumori is, apparently, not restricted to small momenta. We describe a general framework for the study of collective motion in the semi-classical limit without limitation on the size of coordinates or momenta, which includes all previous methods as limiting cases. We find it convenient, as in the past, to consider two general systems: first, a system with n degrees of freedom and no special permutation symmetry, and, second, a system of fermions described in TDHF. For both systems the problem can be formulated as a search for a hamiltonian flow confined to a finite-dimensional hypersurface in a phase space, which itself may be finite- or infinite-dimensional. Though, in general, there are no exact solutions to this problem, we can formulate consistent approximation schemes corresponding to both the adiabatic and Rowe-Basserman, and Marumori limits. We also show how to extend the momentum expansion, which underlies the adiabatic approximation, to higher orders in the momentum. We thereby confirm the structure of the theory found in our previous work.
1. Introduction
In recent work’,*), we have provided a new and (we believe) theoretically clean formulation of the adiabatic limit (ATDHF) of time-dependent Hartree-Fock theory (TDHF). Our most important result is that Villars’ equations 3*4),which express the content of that theory have an essentially unique solution under “suitably” imposed boundary conditions leading to a construction for an adiabatic collective hamiltonian. In work to be reported separately 5), we have shown how to solve these equations directly by a method which ties in naturally with existing technology for doing self-consistent cranking calculations. * Permanent USA. t Laboratoire
address:
Dept. of Physics, University
associe au Centre
National
of Pennsylvania,
de la Recherche
271
Philadelphia,
Scientifique.
Pennsylvania
19104,
272
G. Do Dang, A. Klein / Gene4
semi-classical
method
Though the work described above has responded to some of the open questions in the previous literature6), it has hardly responded to all of them. One of the most important remaining questions is the domain of application of the Rowe-Basserman and Marumori theories7,8). In particular these authors have stressed that their work is not a small-momentum expansion and therefore is more general than the adiabatic theory. This point will be examined in the current work. There is no doubt that a theory applicable to the domain of large coordinates and large momenta is of greater generality than the usual limits recognized in low-energy nuclear physics: the large-amplitude adiabatic limit, where we expand in powers of the momentum only, and the anharmonic vibrational limit, where we expand uniformly in momenta and coordinates. In all three cases, we assume that there is at least one large number N which characterizes the degree of collectivity. The simplest case in principle is the anharmonic vibrational one, where with suitable (and consistent) scaling not only is N-l small but so also are X/N and P/N, where X, P are a canonical pair characterizing a collective degree of freedom. Next in difficulty is the adiabatic limit, where only l/N and P/N are small. Finally the fully semi-classical limit in which only l/N is available as an expansion parameter is notoriously more difficult than the previous special cases. In the present paper we shall formulate a theory of collective motion which makes use at first only of the semi-classical assumption. For this formulation it suffices to study the Wigner transform (WT) of a suitably formulated quantum theory ‘), retaining only the leading terms in the kinematics and dynamics. To illustrate our approach, we shall consider two classes of problem, following a sequence which we found useful in our previous studies2). First we shall consider a problem with n degrees of freedom and no particular permutation symmetry. Here the semi-classical limit is indistinguishable from the classical one and we find ourselves studying a problem in hamiltonian classical mechanics. Decoupling of p -Cn collective degrees of freedom from the total is then recognized as the problem of finding conditions under which a hamiltonian flow in n dimensions can actually be confined to a hypersurface of p < n dimensions. We formulate and study such a set of conditions, some of its properties, and some of its limiting cases. This study is carried out in sect. 2 where we formulate the problem of finding a hamiltonian flow in less than n dimensions. An examination of the resulting equations makes it evident that, at best, this can be done only approximately. We formulate and describe two methods of approximate decoupling. The first treats momenta and coordinates on a symmetric footing and is seen to be equivalent to the work of Rowe and Basserman. The second method admits an expansion in the momenta. Here we have pushed the theory to one higher order than previous workers in order to deepen our understanding of the structure of the theory and of its consistency. In sect. 3, we consider the second class of systems, namely a system of fermions described in TDHF approximation. The essential task of this section is to show that
G. Do Dang, A. Klein / General semi-classical
method
273
the TDHF theory may be expressed in hamiltonian form and that every equation and every result found in sect. 2 has its analogue in the present case. We thus conclude with a uniform theory for the two classes of system. In a concluding section, sect. 4, we provide some discussion of the relation of the simplified approach of this paper to previous work’*2,9,‘o) of the problem “requantization”, and of methods of solution of the theory. Some algebraic details of the derivations of sect. 2 are found in the appendix.
of
2. Classical description of collective motion 2.1. FORMULATION
We are given a hamiltonian H(E, a), where (tl . . . 5”) are n canonical coordinates and ~r=(ai a.. 7~,) are the corresponding canonical momenta. Hamilton’s equations, ?i,=
-
aH/ap,
(2.1)
p= aH/aTa,
(2.2)
define an initial value problem relating to the whole of phase space. What interests us here is the following special question: The equations 5a=5a’(X,P),
“u=“,(xJ+
(2.3)
P=P1”‘Pp,
(2.4
where x=xi describe a hypersurface
. ..XP
9
E of dimension 2p (p < n) in the phase space. Are there
any solutions of eqs. (2.1) and (2.2) in which the “hamiltonian flow” is confined, approximately at least, to the hypersurface E? The conditions for such a flow to be realized are as follows: (i) x and p are a partial set of canonical coordinates for the full phase space. This can be expressed, (summation convention)
for example,
by a set of Lagrange
bracket
~a71,_~alr,=,f axiaPj
(ii) We define a collective hamiltonian K(x,
apjaxi H,(x,
J.
conditions
(2.5)
p) by the equation
P) = H(C;(x, P),+,
P)).
(2.6)
We require that .$” and or,, as expressed by (2.1) and (2.2), describe a flow on the hypersurface 8, i.e. if 5” and TT~are initially on the hypersurface, then they remain on the hypersurface. On the one hand, the right-hand sides of (2.1) and (2.2) are defined on the hypersurface by (2.3). On the other hand, the condition that [“(x, p) and ~Jx, p) be dynamical variables on 2 can be expressed by the requirement that
274 H, is the appropriate
G.Do
Dang, A. Klein / General semi-classical
method
hamiltonian to determine their time development, i.e. ?i, = [G Kl,,
aTa aHC =aar a& axiaPi axi’
(2.7)
aPi
i”=
[V, H&r,,
(2.8)
where the subscript PB denotes a Poisson bracket. Alternatively, combining (2.1), (2.2), (2.7) and (2.8) we obtain a set of partial differential equations which together with the kinematical constraints (2.5) hopefully determine the hypersurface 2, namely - g
= [7,x, H,],,,
(2.9)
[CH,],,.
(2.10)
g=
Before studying whether these equations do indeed fulfill their stated aim, let us consider more closely the meaning of eqs. (2.3) and (2.4). If indeed motion can be confined to the hypersurface E, this would appear to imply that there are n -p canonical pairs which can be ignored, where the corresponding coordinates can be frozen in value, (the corresponding momenta set to zero). Thus we imagine that underlying (2.3) is a locally invertible canonical transformation, where we can then also write for the partial inverse transformation xi=xQ,9),
(2.11)
Pi=Pi(E,+
(2.12)
This interpretation is essential if, indeed, we are to justify eq. (2.6), which in the case of a full canonical transformation, defines the transformed hamiltonian. In this limit, eqs. (2.9) and (2.10) become identities. For later convenience, we record the following relations, which are equivalent to the content of (2.5) or of the corresponding Poisson bracket conditions:
ap ap. _=I axi arm' axi aB at* api7
-=--3
aTa
z=
axi
Iap.
- aga ) --
air,=
a.p
aPi ’
(2.13)
(2.14)
For some further work, it is convenient to introduce a standard tensor notation in phase space. Let us write $=
E,n,
yi=x,p,
/.L=1...2n,
(2.15)
i=1.*.2p,
(2.16)
215
G. Do Dang, A. Klein / General semi-classical method
and define the antisymmetric matrices Jpy and yjj
by the equations
E”
Pen,
i
i>p,
(2.17)
(2.18)
where
JpvJxv= 6,~3 Yijcfkj
=
(2.19) (2.20)
&ik.
The equations of motion (2.9) and (2.10) may be written in this notation as (2.21) The derivatives of the collective hamiltonian on the right-hand side of (2.21) may be expressed in the form aH .=aH??l!
(2.22)
ayi avpayi.
Finally, eqs. (2.13) and (2.14) can be written as (2.23) and the canonicity conditions (2.5) as aBcJ
a17” =jTii
ayi pvayj
.
(2.24)
2.2. GENERAL METHOD OF SOLUTION - THE ROWE-BASSERMAN THEORY AS AN APPROXIMATION
For each value of i, the quantities (aq@/ayi) define a tangent vector to the hypersurface at the point in question. Given the set of tangent vectors, then eqs. (2.21) and (2.22), by trivial counting provide enough equations to determine the hypersurface 7” = v”(u). Therefore we need additional equations for the determination of the tangent vectors. We can attempt to find these by differentiating eq. (2.21) with respect to the collective variables y’. We notice immediately that we will not, in general, succeed in our aim because the resulting equations will also contain the double array of vectors (a*+/‘/ay’ayj) which measure the curvature of the collective hypersurface. We return to this essential point below. But first let us notice that the process of differentiation leads us to the introduction of a natural affine connection in phase
216
G. Do Dang, A. Klein / General semi-classical method
space, namely the quantity ppaf
a2yi a$
w ayi
’
which under a change of coordinates q + 11 has the correct transformation
(2.25) proper-
ties, namely (2.26) With the help of (2.25), we derive from (2.21) (2.27) where Hp.y is the covariant derivative (2.28) Eqs. (2.27) are a set of local RPA equations subject to the normalization (2.24). But as remarked above, these equations cannot be implemented because the operator HP; y is not fully known. If there were some metric structure in the phase space, given a priori, the riV could be expressed in terms of it and the problem closed. But this is possible only in very special (and often uninteresting) cases. In the general case the problem does not terminate at this point: we have to seek equations for
(a2qyayia_+), etc.
A natural local linearization procedure does suggest itself, however. This is to assume that the curvature of the collective hypersurface itself is small on the average and thus to ignore it. This leads to a closed theory which consists of (2.21), (2.22), (2.24) and (2.28) with H,;.+
a2H/avpaqy.
(2.29)
Within the present class of problems, we have thus found the theory of Rowe and Basserman7). The main characteristic of this theory is that it treats coordinates and momenta on an absolutely equal footing. Therefore, it may be applicable to large momenta and large coordinates i.e. to the full semi-classical limit. On the other hand, we see that it is not an exact theory: indeed it is exact only when H is globally quadratic in 7”. Still, it is an interesting theory which, to our knowledge, has not so far been applied consistently to any problem. [The application found in ref. 7, is an application of the Rowe-Basserman theory in the adiabatic limit. In this limit, except for special cases, it is only an approximate version of the full adiabatic theory.]
G. Do Dang, A. Klein / General semi-classical 2.3. EXPANSIONS
method
271
IN MOMENTA
We have just
seen how, in principle,
a theory of collective
motion
can be
developed without making the assumption of either small velocities or small displacements. Next we consider the situation that an expansion in the momenta is of interest. In a canonical description, this is a conventional choice. Furthermore, we assume the coordinates to be even under time reversal and the momenta to be odd. We thus take the initial hamiltonian H(& a) as a power series in even powers of v~, namely H( E, lr) = V(E) + $r&K”~(
{) + $7rJrf17rY7r&~@+(6) )
(2.30)
where we have chosen to go to fourth order in the momenta in order to show how the adiabatic approximation, now well established up to second order in the momenta’*2), can be extended systematically to the next order. It suffices to consider a single collective coordinate, so that we are looking for a collective hamiltonian in the form H,=V-(x)++pzX(x)+~p%?(x).
(2.31)
To solve the proposed exercise, it stands to reason that we should make a corresponding expansion in momenta for the description of the hypersurface (2.3). For present purposes, we write 6” = +;b,(x> + &#$,(X)P2,
(2.32)
Ir, = @(x)p
(2.33)
+ )~~‘(x)p?
The aim is to show if we apply the canonicity condition (2.5) and the equations of motion (2.9), (2.10) consistently to fourth order in p, we thereby obtain a set of conditions
which suffice to determine the four sets of functions I#$,, cp&, @)
and
+&‘). As a special case, we must regain the usual adiabatic theory, which is expressed in terms of the functions $J&) and @, when we specialize to the usual quadratic hamiltonians. Thus the extension of the theory by one more order serves to clarify the connection between the structure of the hamiltonian and that of the hypersurface on which we seek a description of the collective motion. The first terms of (2.32) and (2.33) describe a point transformation, provided the first of the canonicity conditions given below is satisfied. Turning to the actual machinery, the application to (2.5) of (2.32) and (2.33) yields two consistent conditions of zero and second order in p, namely (2.34) (2.35) Next consider eq. (2.9). Comparison of (2.30)-(2.34)
shows that eq. (2.9) yields three
278
G. Do Dang, A. Klein / General semi-classical
method
possibly consistent conditions of zero, second, and fourth order in p, namely V,, = $‘,)(dV/dx), +@$lP’K”@,, +( l@$
+ $b”@)
+ $v,,+#$,=
@)&T+
:#:“‘+;,
(2.37)
K”p,Y + +lJ’,“‘@J+;i,K”@,E,Y
+ t+;,&YX,~,Y
+ ~~~~@)@j/(O)~+J 0 =
Correspondingly follows:
d#r) - TX+
(2.36)
d,&
I d,& Jy+ dx
-+%
#X&T+
~~~&$~.
(2.38)
eq. (2.10) yields two conditions of first and third order in p, as
In deriving these equations, it is well to recall that the p-dependence arising from (2.32) must be recognized in the arguments of the functions V and Kafl. Eqs. (2.36)-(2.40) must be consistent with the conditions which follow from the equality of H and H, on the collective hypersurface. These are ~((PP(x)) @J,tp)K”~ $~‘,o’@~~~g/py~~
= V(x)
(2.41)
3
(2.42)
+ V,&c*1, =.Y >
+ a( ~~‘~~’ + @#‘)
K””
’ (‘)‘clj3 (‘)$(l) -Y Ka’B,Y+ + da
++;;,q$,V,p = $9.
(2.43)
What must follow now is a certain amount of unenlightening algebra, necessary, however, to the understanding of the structure and use of the preceeding equations. Such algebraic details as we feel are absolutely necessary are given in the appendix. Here we will be satisfied with a summary of the results. Consider first the usual adiabatic limit, in which we work only to quadratic order in momentum. In that theory, studied previously, we omitted the functions 4:) and +;i,, i.e. we worked with a point canonical transformation. The relevant equations are (2.34), (2.36), (2.39), and (2.37) (the p2 term). We mention the latter last because under the approximation cited - point transformation - we showed that (2.37) is a consequence of (2.34), (236) and (2.39) and may therefore be dropped. The remaining equations and the equation Q(
x> =
ax/a.p,
P-44)
G. Do Dang, A. Klein / General semi-classical method
219
consistent with (2.14) and necessary for the consistency of (2.36) with (2.41), are sufficient to determine the unknowns +&(x), @j(x), V(x) and X(x>. This is not trivial to prove and represents the most important result of ref. ‘). Finally (2.41) and (2.42) are consequences of eqs. (2.34), (2.36) and (2.39). The generalization of this structure when we go to fourth order in momentum is almost straightforward. Now, of eqs. (2.34)-(2&l), it is (2.38), which is the condition of order p4 which follows from the other equations, but only to the first order in $2) and c#& But this is all we can expect at this level of accuracy. If we examine our equations, for example, the Lagrange bracket condition, we see that in order to have second-order accuracy in the quantities $2) and +fi,, we must also carry out our expansion of 5” and or, to the next order of momentum beyond that written in (2.32) and (2.33). The remaining equations plus (2.44) should, in analogy with the results of ref. ‘), provide a sufficient number for the determination of the unknownst: $I&, c#&, $J’,“‘,I/J&‘),V(x), X(X), Z(x) and imply eqs. (2.41)-(2.43). We have thus provided the tools for a consistent theory to order four in the momentum. It should also be apparent that further expansion in the coordinates will provide a theory of anharmonic vibrations.
3. study of TDHF 3.1. FORMULATION
The strategy throughout this section - whether we deal with an approximation which treats coordinates and momenta on an equal footing or one in which we expand in momenta - will be to show that we can map the TDHF problem onto the classical problem studied in the previous section. It is well known7*“) that TDHF is equivalent to a problem of classical hamiltonian dynamics, but for our convenience, we require a special form, which we proceed to develop. In this development, we shall consider the elements discussed in sect. 2 in roughly the same order and seek to uncover the corresponding elements of TDHF. The first point is to show that the TDHF equations can be put into hamiltonian form. In consequence, we shall be able to identify the quantity which plays the role of “classical hamiltonian”. Working with only a single collective canonical pair x, p, we seek a density matrix p(x, p) satisfying the condition P2(% P) = Ph and thus describing a Slater determinant, equations ifi&,
P) = is,
P)
(3.1)
which is also the solution of the TDHF &,
P)l
nbv
t The proofs of ref. ‘) have, however, not yet been extended to this case.
(3.2)
280
G. Do Dang, A. Klein / General semi-classical method
where &’ is the Hartree-Fock hamiltonian -%b(X, P> = h,, + L&&
P> 7
(3.3) hob, Vacbdare the one- and (anti-symmetrized) two-particle matrices characterizing a standard nuclear hamiltonian, and a, b, c, d refer to arbitrary complete sets of single-particle states. The dependence of p on x, p will be sought in connection with the problem in collective motion. But first we consider the problem of showing that the TDHF eqs. (3.2) can be written in hamiltonian form (under more general circumstances than the restriction to the collective hypersurface). We recall that according to one of the theorems of Thouless’2) p(x, p) is completely characterized by its particle-hole ph and hp matrix elements in an arbitrary set of single-particle states. Eq. (3.1) effectively determines the remaining elements. If we utilize the representation in which p(x, p) is diagonal, eqs. (3.2) may be written ifir,, =srr, = ~Wu&&,
= aH/&+,
,
iphp = -.X?hp = - 6 Wm/GPph = - 6H/6Pph,
(3.4)
a form which resembles Hamilton’s equations with H(P) = K&P) = h,~p,, + f
(3.5)
KbcdPcnPdb
playing the role of hamiltonian, except that pph and pg are not canonical variables. They can be related to a set of classical canonical variables
(3.6)
Tph’ -rhp,
(3.7)
by means of the equations13,‘4) Pph= P&
=~((~+is)[l-~(~2+n2)]“2)ph
(3-g)
[Ph= &{(CY-1’2)ph+(ry-1’2),p}?
(3.9)
and their inverses
=ph
=
(3.10)
-i~{(ry-1’2)ph-(ly-1’2)hp}.
Here rph =
Pph 3 y
rhp = =
$ [
Php,
rpp, =
1 + (1 - 4rtr)“2]
rhh’
)
--0,
(3.11)
and it is understood that t and II have the same support as r, namely ph and hp indices only. Eq. (3.8) is a classical version of the Holstein-Primakoff transformation relating generalized “spin” variables to canonical pairs. Once we establish, see below, that
G. Do Dang, A. Klein / Generalsemi-classical
the 4 and II are canonical,
method
281
then, of course, any further set obtained by canonical
transformation will serve equally well. In the representation in which p is diagonal, we have Q, = Err = qp,, = 0, similarly for the hp elements, and from (3.8)-(3.10) JPphm
p’h’c
we therefore find L
‘6
pp’
&&,
(3.12) (3.13) (3.14) (3.15)
By applying the chain rule to the first of eqs. (3.4), changing the variables from as well as (3.6) and (3.1) we (Pph> Php) + why rph) and utilizing eqs. (3.12)-(3.15) find that the first of eqs. (3.4) may be written aH aH * i-6Ph- Tph=a5ph+i~.
(3.16)
Similarly, the second of eqs. (3.4) yields the same result except for a change in the sign of i. We thus arrive at Hamilton’s equations. It is important to remark again that the demonstration just given has nothing to do with the problem of collective motion. We have shown that the TDHF problem for an arbitrary Slater determinant may be expressed in terms of Hamilton’s equations. Returning to the problem of collective motion, we may take over intact the remaining discussion of subsect. 2.1. In essence we seek to satisfy eqs. (2.9) and (2.10) which express the requirement that a hamiltonian flow is confined to the hypersurface (2.3). That this is not in general possible means that certain approximations will be necessary. Again we will consider two general possibilities, namely that in which we treat coordinates and momenta on an equal footing and second, the result of expanding systematically in momenta.
3.2. THE ROWE-BASERMAN
THEORY
With the identifications made earlier in this section we can, in principle, simply refer to the material presented in subsect. 2.2. Nevertheless, we choose here to present a brief alternative version, using the better-known density matrix variables. In this manner, we shall have more immediate contact with the work of previous authors 6,7). Thus we seek a two parameter family of Slater determinants p(x, p) which we shall characterize at each point of the phase space x, p by its diagonal representation. In this representation the equations of motion analogous to eqs (2.9) and (2.10)
G. Do Dang, A. Klein / General semi-classical method
282
namely
*)
ibob=LKdob=GLhJPB =iaH, --&?zb i JHCaPab ax
t3p
ap
(3.17)
’
ax
have the non-trivial matrix elements ~ph=j~--i~-
aH
ap,,
ax -~hp=i~_-i~-aH ax
aH
ap,,
ap
ap
ax
aPbp
aH
aphp
ap
ap
ax
9
.
To see the meaning of these equations let us define the real functions A(x, ~(x, p) by the equations
(3.18) p)
and
x = aHJap,
(3.19)
p= -aHJax.
(3.20)
These appear below in eq. (3.24) in the form of Lagrange multipliers. We define further appdax
uph =
-
rph
-iap,h/ap
=
(3.21)
= a;p, = -T&.
(3.22)
Thus the second of eqs. (3.18) appears as the complex conjugate of the first. Together they are now equivalent to the equation (3.23)
[%Pl=o,
where 2
is a generalized cranking hamiltonian, namely .9=Jf?-p7-iihts,
(3.24)
involving two, initially unknown, cranking operators u and r, determined by the local tangent plane to the hypersurface. (Thus it appears that, in general, 2 will not be a real operator.) As explained previously, to close the description, we seek equations for the operators T and u which, consistent with eq. (3.1) have only ph and hp non-vanishing elements. As also explained previously, this closure will not be achieved at this level unless we assume in the resulting equation that the derivatives of 7 and u with respect to x and p may be ignored. We furthermore assume that our choice of x and p satisfies a*HJax
ap = 0,
(3.25)
and define, in analogy with a harmonic oscillator, ahlap
= a*HJap*
- acL/ax = a2HJax2
= m-l(x)
,
= w*(x)m-l(x).
(3.26) (3.27)
G. Do Dang, A. Klein / General semi-classical method
283
Thus differentiating (3.23) (in turn) with respect to x and p and utilizing the various assumptions and definitions of this subsection we find a version of the “local” RPA equations as -[%?,a]+[a.&/ax,P]=
(3.28)
-io%l[r,p],
(3.29)
i[~,7]+[a~/ap,p]=im-l[a,p].
Given p, (3.28) and (3.29) are linear equations in u and 7 of which only the ph elements need be considered. It remains only to specify a normalization condition for the solutions of these equations. This can be done with the help of the canonicity condition (2.5) and eqs. (3.9) and (3.10). We find straightforwardly, using the hermiticity of p and of its various derivatives, that (3.30) This completes the specification of the theory, since we have enough equations for all unknowns. There remains one matter - the actual choice of collective variables x,p. The discussion of this point is best left to actual cases, except to remark that the freedom of canonical transformation on the hypersurface remains.
3.3. EXPANSION IN MOMENTA
The remaining new technical task which we have set for ourselves in this paper is that of carrying out the momentum expansion for the solution of the TDHF theory. We believe that here clarity will be served if we make the translation of the results of subsect. 2.3 in the most literal way. This means first that we expand eq. (3.5) which we write more explicitly as NE/+
(3.31)
eI,(Pka
in the form (2.30) and identify thereby the elements V( [), KaP(E) and LUByS(E) which provide the input into the theory. As the second step, we wish to identify the ingredients c#$&and $i, of eqs. (2.32) and $c) and #(,1) of eq. (2.33). The theory is then fully defined by substituting these expressions, to be given below, into eqs. (2.34)-(2.43). To carry out the first of our tasks we first expand p([, a) in a formal power series in II including fourth-order terms, namely P(b)
= P’“‘(&)
+ P(la)(E)%
+ $p(3Qfy
+ $P’2”%)%~/j
E) 7rJrp?ry + fp@“f+y
~)7rJr~7r~~8.
(3.32)
Here, of course, (Y,/3.. . run over all ph indices in some representation. We shall carry out all calculations in the first part of our task in the representation in which p(‘)(E) is diagonal. This entails a considerable simplification in the ensueing formulas. First by investigating the consequences flowing from the joint requirement of
284
G. Do Dang, A. Klein / General semi-classical method
(3.32) and the condition p = p*,
we obtain the consequences
p(o)= (p(o))*, P P
w
= p’o)pu*’ + pwpK9 )
(3.34)
cw3) = p(o)pw) + pw7)p(o) + pwp(‘8)
p(3”BY)
=
p(0)p(3aBY)
+3[{p P
(3.33)
(4Mbv
=
+
)
(3.35)
p(3w3Y)p(o)
UN, pwv)}
p(o)p(4d3YQ
+ pwpw
+
+ { pw), pm}
+ { p(‘Y), pw)}]
,
(3.36)
p(4”8YQp(o)
+f[{p(‘a’),p(38us)}+(,t,P)+(,oy)+(at,6)]
+i[{p
W),
p(*Y”)}
+
{ p(*aY),
p(*8~)}
+
{ p(*“f9,
p(*/3Y)}]
.
(3.37)
Here the curly brackets are anti-commutators. These equations have the following consequences: in the representation in which p(O) is diagonal, p (‘) has only ph and hp elements, and the pp’ and hh’ elements of
. . are determined in terms of the non-vanishing elements of the lower-order parts of p. Next consider eq. (3.8). A formal expansion, which is understood to refer only to the ph and hp parts of p, has the form p(2), p(3), p@).
fip
= [(l
- +t*)l’*
-i$s3(1
+ ir(l-
_ $[*)-l’*
$t*)l’* - &$a*(1
- $&*(I
- f<*)-l’*
- $~2)-1772(1 - ft2)-1’2
+ ... . (3.38)
In the representation
in which p(O) is diagonal, we have (P(“))ph=
(3.39)
[EC1 -f5*)1’2]ph=0’
and &ll = [,,r = 0, as previously remarked prior to eq. (3.12). It follows from the examination of (3.38) that in this representation it reduces to &pPh
=
i9rph- i+rpgpJWrpW.
(3.40)
We conclude, upon comparison with (3.32) that (lPW
=
-
(2p’h’p”h”)
=
0,
“’ h “’ ) =
_
Pph Pph (3p’h’p”b”p
Pph
=
(3.41)
= i + app,aw, d-
(3.42) pf;‘h’p”h”p
“’ h “’ )
i~&~3(8pp~8p?~p~~~
p(;p,h’p”,,“pIISh 0,r,VV,,‘P’) = o. P
pf;‘U
8,,,,
8,w))
(3.43) (3.44)
G. Do Dang, A. Klein / General semi-classical method
Here S, means the sum over all 3! pe~utations
285
of the pairs p’h’, p’h” and p “‘h “’
considered as single indices, divided by 3! It now follows from (3.34) that p$? = pf& =
0. Fimilly from (3.35) and (3.37), respectively, we can calculate &Y”’
= S, (&,
(3.45)
Isp’r*St&> ,
p~~p’h’)= -S,(s,s,,zsPP~), 12 pE;v”‘h”‘)=
(3.46)
~S~(spp~sp’p~~~“g,~’ 6,.!,
p~kyh’fFp 12
?”h”) = - qs4fshh, ah’& s,,
a,,,),
(3.47) (3.48)
The definitions of S, and S, are the analogues of that given for S,. We are in possession now of all the results necessary to compute the elements of eq. (2.30). One substitutes the expansion (3.32) into the Hartree-Fock functional and collects terms of different order in 7~~.The odd orders are guaranteed to vanish owing, for example, to the antisymmetry properties of (3.41) and (3.43). The nonvanishing coefficients K and L are then easily evaluated, using (3.41)-(3.48). We find 1y PhP’h’
L= 16 2 hh lsPpF -
; vhh’pp’
+xp’p] +
-
5 vhp,pht
Lpbp’wp”h”p “h’m= fS,$ xp,p2 6ppzs
(
p’pI
~spp+%h’ +
6
p,,p,,,
+3Eah’hl
+ vph’hp’
8,
-
,.,
f F/Pp’fi’,
(3.49)
&W)
It is also evident that V(E) = ~,,(P’“‘(E)).
(3.51)
Of these results (3.50) is new and (3.49) and (3.51) are equivalent to results given in our previous work’p2). Unfortunately in the latter we chose to define K on a doubled space, allowing hp as well as ph elements so that the detailed identifications are in superficial disagreement. The dynamical equations which follow are the same, however. Turning to our second task, that of finding the elements of (2.32) and (2.33) this can be accomplished by a calculation which is in several respects the inverse of the calculation performed above. We expand eqs. (3.9) and (3.10) in powers of r. It will
286
G. Do Dang, A. Klein / General semi-classical method
be clear below that we require at most cubic terms. Using rr = r, we find ~~ph~((rph+Thp)+f[(TTT)ph+(TTT)hp],
(3.52)
fi7rph= -i(rph-rhp)-i$[(rrr)ph-(rrr)hp].
(3.53)
The next step brings in the description of the collective motion in terms of p(x, p) which we subject to the adiabatic expansion P(x,p)=p(O,(x)+p(‘)(x)p+3p(*)(x)P*+tp(3)(X)P3+aP(4)(x)p4.
(3.54)
Here all the p(“) are hermitian matrices. For n even the matrices may be chosen, consistently, as real and symmetric, for n odd as imaginary and antisymmetric. Substituting the ph and hp elements of the expansion (3.54) into (3.52) and (3.53) and collecting terms, it suffices for our purposes to retain terms of the zero and first order in p(O). This is because we shall finally choose the representation in which p(‘)(x) is diagonal, i .e . p(O) ph = p$” = 0. Since, for the theory of subsect. 2.3 we require at most the first derivatives of .$Ph and VP,,,the terms of higher order in p(O) will simply not contribute. Thus in a general representation, but to linear terms in p(‘), we have \/26%
(2r~)+p*[~~~)+(r(l)r(l)r(0))ph + (r%%(r))p~
+,(l)&.(*)
+ (r%W))ph])
+ rwr(*)rm
)
+ r(*)rmr(l)
(3.55)
+
r(*~r(l)r(09ph) .
(3.56)
Finally comparing (3.55) and (3.56) with the equations
in the representation derivative, we have @$=O,
EPh= G&t+
$P”$Jtf$,
(3.57)
Tph= J1$ P
+ +P3+Icl(plh) >
(3.58)
in which p(O) is diagonal, and computing the required first (3.59) (3.60) (3.61)
(3.62)
287
G. Do Dang, A. Klein / General semi-classical method #g
dJ/$ -=.
dx #$3
=
-
la
ifip$ -dp$)
dx
,
(3.63) (3.64)
’
- ifi p$,
(3.65) dp”’ zp(2++1)
&$‘ii -= dx
d (0) dpCo) +p”‘L dx PC2)+ P(2)yjyp(‘)+
P
(1)P (2)dx dp(O)+ p(*)p(‘)-dp”’ dx
1 pi
(3.66)
This completes the structure of the theory insofar as we wish to develop it in this paper. Together with refs. ly2) we have attempted to provide a complete theory of large-amplitude collective motion. Basically what remains is the very essential task of showing that one or more of the alternatives described can be put into service. A start on this venture has been made ‘). 4. Summary and conclusion This paper is based on the assumption that the problem of large-amplitude collective motion may be studied by means of an expansion about the classical limit of the many-body problem. We have referred to this. somewhat loosely as the semi-classical approximation in order to keep in mind that the theory presented in this paper can be derived from a fully quantum theory by the method of Wigner transformation*), that therefore requantization is straightforward, and that the inclusion of quantum corrections is also, in principle at least, straightforward. However, in this paper, we have superficially studied a problem in classical mechanics. It should be pointed out that somewhat similar classical ideas can be found in the works of Brito and Sousa9) and of da Providencia and Urbana”), which are equivalent in content to subsect. 2.1 and part of subsect. 2.3, though the presentation of the ideas of subsect. 2.1 differs considerably. Where our work has wider scope is first, in the realization, that the canonical framework is so general that it provides, finally, the basis for understanding from a unified starting point all the viable literature of large amplitude collective motion. From this point of view there seem to be only two rational alternatives worth discussing in detail+: either one treats coordinates and momenta on a symmetrical footing, or one assumes that the coordinates are “large” and the momenta “small”. We have shown that the Rowe-Basserman theory is an approximate version of the former and the adiabatic theory an approximate version of the latter. We have carried the adiabatic approximation to one higher order than previous work. + Of
course, if there is more than one collective coordinate, mixed cases can and do occur.
G. Do Dang, A. Klein / General semi-clossical
288
method
The second major result of this paper is a new demonstration theory
may be rewritten
so that the full machinery
of how TDHF
of the classical
theory is
applicable to it. With respect to the matters described in detail in this work, the presentations may be taken to supercede the corresponding material of refs.‘,‘). Of these papers, we continue to refer to ref. ‘) for a relatively exhaustive theoretical study of the solution of the equations of the adiabatic limit. Further discussion of this question is nevertheless of interest, in particular in connection with the relationship between the Rowe-Basserman and adiabatic approaches and under what circumstances they may overlap. However, we have decided to postpone this discussion for inclusion with future applications. We also refer to both refs. ‘y2) for our “best shot” at providing a fully quantum foundation for the classical limit considered here. Further work will be devoted to the development of the practical consequences of our theory. One of stimulating hospitality part under
the authors (A.K.) would like to thank Dr. B. Giraud for several discussions. He is also grateful to K. Chadan and B. Jancovici for the which made his stay at LFTHE so pleasant. This work was supported in contract number 40132-5-20441 of the US Department of Energy.
Appendix The purpose of this appendix is to provide some additional details of the consistency and structure of eqs. (2.34)-(2.43). We first show that the conditions (2.41)-(2.43) which define the collective hamiltonian are consequences of the previous kinematical and dynamical conditions. Differentiating (2.41) we find V,, = (dY(x)/dx)(
8x/&$“)“‘.
(A.1)
This agrees with (2.36) provided ( ax/ag”)(o’ = I@.
(A-2)
This equation is consistent with the canonicity condition (2.14). Thus (2.41) is an integral of (2.36). Turning to the proof of (2.42), starting from (2.39), we form the sum necessary to apply (2.34) and remember (A.l) and (A.2). The result is (2.42). Similarly we try to derive (2.43) from (2.40). Again, forming the sum necessary to apply (2.34), we derive from (2.40) the equation
64.3)
G. Do Dang, A. Klein / General semi-classical
method
289
We wish the difference between (A.3) and 4 times (2.43) to vanish. This wish is satisfied if in the difference in question we insert (2.35) and (2.39). Next we consider the role played in the theory by the dynamical condition of order p2, eq. (2.37). In the adiabatic limit, this equation is not independent of the conditions of order zero and one in p, but can be derived by properly combining (2.42) and its first derivative with (2.39) and setting 4:) and $yi, to zero. In the event that Ir/c) and +Fi, are retained, the same combination is seen to be consistent with (2.37) provided the new kinematical condition (2.35) is utilized. In the latter case (2.37) provides one set of equations for the additional variables J/t) and $&. A second set is provided by the condition of order p3, namely eq. (2.39). One extra unknown, Z’, is “determined” by the additional kinematical constraint (2.35). It remains only to consider the condition of order p4, eq. (2.38). In analogy to what was found for the order p2 equation in the adiabatic limit, we expect to be able to show that this condition is here not independent of conditions of order zero to three in p. However, a little extra care is necessary because an examination of the kinematical conditions, for example, shows that the present theory is consistent only to the first power of $c) and $$‘i, and therefore the proof of dependence of (2.38) should be expected to hold only to that order. The detailed demonstration involves the following steps: (i) Calculate the derivative with respect to [” of eq. (2.43). (ii) In the resulting equation substitute (2.40), (2.39) and (2.37). (iii) Take note of the identity (‘4.4) Proof:
Utilize
(A-5)
and (A.2). (iv) Use the equation
(A-6) which can be verified by multiplying by (d+To/dx), Use the identity
summing and using (2.34). (v)
which is an analogue of (A.5). (vi) Finally if we drop the terms depending on the square of @pi,and $k), we find (2.38).
290
G. Do Dang, A. Klein / General semi-classical method
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Notes
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