THE INVERSE MEAN FIELD METHOD (IMEFIM) - FROM STATICS TO DYNAMICS

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THE INVERSE MEAN FIELD METHOD (IMEFIM) - FROM STATICS TO DYNAMICS

E. Hefter

To cite this version:

E. Hefter. THE INVERSE MEAN FIELD METHOD (IMEFIM) - FROM STATICS TO DYNAMICS.

Journal de Physique Colloques, 1984, 45 (C6), pp.C6-67-C6-77. �10.1051/jphyscol:1984608�. �jpa- 00224209�

JOURNAL DE PHYSIQUE

Colloque C6, supplement au n°6, Tome 45, juin 198* page C6-67

THE INVERSE MEAN FIELD METHOD UMEFIM) - FROM STATICS TO DYNAMICS E.F. Hefter

Institut fur Theoretische Physik, Universitat Hannover', Appelstrasse 2, D-3000 Hannover 1, F.R.G.

Abstract - We sketch the development of a Schrodinger-type formalism for extended particles. It encompasses the basic ingredients of HF, Fokker-Planck-type transport equations, phenomenological nonlinear Schrodinger equations and other models. Applications to static and dynamical problems are discussed.

1 - INTRODUCTION

According to the present wisdom, the many-body Schrodinger equation is the appropriate starting point for any sound analysis of (nonrela- tivistic) quantum mechanical problems like, say, the one posed by the atomic nucleus. But unfortunately we are not able to solve that equa- tion (but for computer experiments). Hence, one resorts to various ap- proximations to it. Besides (semi-) classical approaches, the most prominent attempt to circumvent the associated problems is presumably given by the physical picture of the mean field, U, (generated by the constituents of the system) in which its constituents move rather in- dependently from each other. For a system containing N bound-states with the E. (single-particle) energy eigenvalues this notion leads to

o 1

(1) Within the ubiquitious Hartree-Fock approach (HF) the mean field is determined microscopically via

(2) (where we suppressed the Fock-term). To solve (1), the traditional di- rect approach requires an a priory knowledge of the macroscopic poten- tial, U, or of the microscopic nucleon-nucleon interaction, V , which gives rise to the UH F of (2). With such an input one may then pro- ceed to evaluate the wavef unctions I|J . and the energy eigenvalues E. /I/.

Historically, Schrodinger started with (1) and applied it to atomic physics, i.e. to a system in which the (Coulomb-) interaction is very well known and in which the spatial separations between its constitu- ents (nucleus and electrons) are so large that it is an excellent ap- proximation to treat them as point-particles. For time dependent po-

Résumé - Nous esquissons le développement d'un formalisme du type Schrôdinger pour des particules étendues. Ceci entoure les ingrédients fondamentaux de HF, des équations de transport du type Fokker-Planck, des équations phënoméno-logicales nonlinéaires Schrôdinger et d'autres modèles. On discute des applications pour des problèmes statiques et dynamiques.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984608

C6-68 JOURNAL DE PHYSIQUE

tentials, U(x,t) -f , Schrodinger replaced the r.h.s. of (1) by ifia $J. (z,t)

-- a reasonable phenomenological step which is, however, by no Aeans unique / 2 / . The success of the method confirmed the suggestions of Schrodinger and lead eventually to the present-day formalism of quan- tum mechanics with the many-body Schrodinger equation. A description tailored for structureless point-particles.

However, in particular in nuclear physics we are obviously dealing with extended objects having also internal degrees of freedom. The nu- cleons within a nucleus occupy such small regions of space that they must feel the spatial extensions of each other. Hence, it would be ab-

- surd to ignore their (persistent- or) self-interactions which are physically speaking synonyms for the fact that they do have spatial extensions (internal degrees of freedom). The roles of self-interac- tions and of the associated nonlinearities have been discussed in great detail in the book by Burt / 3 / . There it is clearly shown that their inclusion does not contradict traditional considerations, it rather supplements them providing a more complete and richer structure Yet, self-interactions or persistent interactions are something concep tually "new" in so far as they are nonperturbative.

Our aim is to try to formulate a concept having the potential to deal in a consistent way with the self-interactions/spatial extensions/in- ternal degrees of freedom of quantum mechanical objects. Since it can not be disputed that (1) provides a highly successful description for most of the problems encountered in nonrelativistic quantum mechanics, we go back to this historical starting point and use it as a basis for our considerations. Of course, we have to expect that at least some of the traditional interpretations associated with ( 1 ) will have to be modified as we progress. Since our journey leads us into terra icog- nita, it is understood that we might encounter objects, connections, etc. for which we will not immediately have generally accepted inter- pretations. But the landmarks which we have to use for our orientation are the traditional results/approaches (for point-like objects) which we should recover if we suppress the internal structures of our ob- jects. The (nonlinear) HF approach / I / and the material related to non- linear classical and quantized fields / 3 , 4 / are to serve as an addi- tional guidance and consistency check.

(1) is so successful that it has to contain the most prominent ingre- dients required by nature. Accepting its basic form as reasonable, it would be illegal to introduce "by hand" any additional terms or modi- fications. Hence, there is no way to play around with the qi or Ei.

The latter are given to us by the experimentalist (basically that holds for the true many-body Ei's, but let us ignore this detail for the time being). Since the potential -- i.e. the input required for the direct approach -- is not too well determined and the only quantity whose "interior" we do not quite understand, it is most suggestive to

apply inverse methods / 5 - 9 / to the solution of (1). The implications are that we utilize information on the discrete and continuous parts of the energy spectrum of the Schrodinger operator to solve (1) for wavefunctions and potential. Thus we do not touch the formal structure of ( I ) , we are only going to solve it in a different way. In doing so, the formalism should provide us by itself with indications whether ex- tended objects -- the Ei being the keys to their internal structure --

do require a different treatment from point-like particles and how one should proceed from statics to dynamics.

At this point the place of our approach in respect to other applica- tions of inverse methods to nuclear physics has to be clarified: E- ditionally inverse methods are applied to extract from scattering

phase-shifts and/or differential cross-sections the respective scat-

tering potentials. 1.e. information on the continuous part of the spec-

trum is utilzed to obtain these potentials. In contrast to this usage

- we believe that the most important information on the atomic nucleus is - contained in the discrete part of its spectrum. Hence, our primary in- terest will be focussed on this part of the spectrum, i.e. on (1).

We give below the rudiments of the emerging inverse mean field method (Imefim) /9/ drawing attention to common features and structures with well-established approaches. A novel point of Imefim is that (for head- on collisions) the mathematics yield automatically (in their spatial components uniquely specified) dynamical evolution equations for the mean field U. Imefim has not just points of contact with traditional

formal theories but also with fluid dynamical (semi-) classical ap- proaches, transport theory and field theory. The common structures of Imefim and HF lead to an almost self-contained evaluation of the re- quired input, the Ei, see section 3.. Section 4. contains a rather short sketch of applications and predictions related to nuclear physics.

The final part is devoted to a short summary. -- Because of the severe limitations in length, we are not always as explicit and detailed as desirable. We hope that the reader will forgive us for not being able to shorten the presentation in a more concise and self-contained way.

2 - THE STRUCTURE OF IMEFIM

To obtain (at least in part analytically) managable results we limit ourselves to spherically symmetric systems. Consequently the appropri- ate radial SchrBdinger equations corresponding to (1) may be cast into the form of 1 D equations (with )J(~)+)~(r) /r; r-tx>O; u(x)-W(r)+blt(ll+l )/r2) :

- ~ a _XX ) . (x)+U ₁ (x) q i = E ~ ; $ ~"=Il 2,. . . ^{,N; b1x2/2m} .

To keep the number of uncertain parameters to a minimum we include for the time being only the N occupied ground-state levels in our consider- ations and use the idealized single-particle model. (If desired, then the formalism nay also be extended to accomodate collective states.) Applying inverse methods, we have to evaluate the function K(x,y) by solving the (Gelfand-Levitan-Marchenko) integral equation

Q)

K(~,y)+B(x+y)+~/ B(y+y') .K(x,yl) .dyl=O ; y>x ( 4 ) /5-9/. The kernel B is (in terms of the associated ID problem) deter- mined by the reflection coefficients R(E), E>O, and by the N bound- state energy eigenvalues Ei:

-

The coefficients ci are uniquely specified by the boundary conditions and the wanted potential U(x) is given by

The general solution, U(x), should naturally contain both, contribu- tions due to the continuum of the spectrum, say UC(x), and to its dis- crete part, say UN(x) . There is no way to obtain the general solution U(x) in a closed form. (4) has to be solved numerically.

But since we believe the discrete part of the ground-state spectrum of the respective nuclei to contain the most relevant information, we ignore for the moment all contributions from other parts of the spec- trum. It is natural to expect now solutions U(x)=UN(x) of the form

N N

-

UN (x) = X U . (x) = X a . (E . ) - P . (x;E.) with P . :$f (x)

j=1 NJ j=l 3 3 3 3 3 3 ( 7 )

C6-70 JOURNAL DE PHYSIQUE

where the aj(Ej) are state dependent coefficients. In terms of the re- lated ID problem, (7) implies that we consider the case of reflection- less potentials with R(E)=O for all E>O. From (5) it is readily appre- ciated that (7) is to simply drastically the solution of (4). Indeed, it is well known /5-9/ to lead to the analytical Bargmann potentials:

N N 2

UN(x)= Z U (x)= C [-4--14. (x)]=-2~a~~[ln(detF)]

j =I ~j j=l 3 3 (8)

F _{i J} . (x) =6 _ij +Jf (x) f (XI'/ (q-) ^{; f} _I ^{. (x) =2=-} _I ^exp ( 2 m - x ) _I .

A glance at these equations shows that U (x) is uniquely determined by the N bound-state energy eigenvalues E j . N ~ n the simple case N=l we have

where we have introduced the abbreviation P ~ - J T M . For small argu- ments of the sech, the formfactor of (9) is almost exactly given by a Gaussian; for large arguments it is very close to a Saxon-Woods fun- tion. (a), (9) hold for k=O; for higher k-values more complicated ex- pressions of.the same structure are obtained /lo/. The one for R=l re- duces e.g. for N=l to -

C; 2

(XI = -2aXxln[x(l+ - - ( I 2K1 +-I K ~ X -exp(-ZK~ ax)] with K:=-E~/M. (10) We show this expression only to demonstrate that such results may be e p tracted from the literature, however, below we concentrate for the sake of simplicity only on the case k=O, i.e. (8).

Numerical and formal studies j6-9, 11,12/ show that the approximation U(X)~U~(X), i.e. (a), yields in a lot of cases results ranging in their quality from good to excellent. The tentative conclusion is that it is justified to use the analytical solution (a), a point which facilitates very much the discussion of the concepts involved. If more accurate re- sults are required, then one may always proceed to solve (4) numeri- cally.

Does the solution of the inverse problem lead to a view at (3) which differs from the traditional one? -- It does. Let us insert its general solution, U(x)=Uc (x) +UN (x) , into the initial equations (3) to obtain

N 2

-wa xx + . ¹ (x)+[ ^j=1 c [ - 4 ~ . + ~ (x)]+uC(x)

I

This is a nonlinear Schrodinger equation (NOSE) ; i. e. inverse methods tell us that the appropriate Schrodinger-type equations have cubic non- linearities (Uc(x) is also proportional to the density, but it in- volves integrals rather then sums, see, e.g. (15) below). From the more general discussions of /3,4/ and references it is already known that such nonlinearities do arise in various ways as soon as one accounts explicitly for the spatial extensions and/or self-interactions of the quantum mechanical objects.

In some recent contributions (/13/ and further papers by the same au- thors) the relations between classical Newtonian-type equations and their corresponding (nonlinear) Schrodinger-type counterparts are dis- cussed in a systematic way. In the case of dissipative Schrodinger e- quations (where the additional dissipative terms do influence the mo- tion of the particles) one obtains additional terms for the respective classical equations of motion of their mean values. According to pre- liminary results (/13/ and D. Schuch, private communication) a cubic NOSE like (11) does not give rise to such additional terms implying that this particular nonlinearity can at most have an effect on the internal degrees of freedom of the objects modelled by them. Such a

finding is fully in line with the above answers given by the inverse method.

A glance at (11) and at (1),(2) indicates immediately that HF and the above NOSE have the same mathematical structure, i.e. cubic nonline- arity. This may be illustrated by recalling that the famous Skyrme forces, which are employed in numerous static and dynamical HF calcula- tions / I / , may be shown to yield the mean field uSk=-ap+~u with AU con- taining higher order terms in the density /14/. Insertion of this ex- pression into (1) or into its radial counterpart (3) clearly brings about the close correspondence between the two approaches. It is even more obvious if we interprete (3) as a 1D equation, write the nucleon- nucleon interaction in terms of a Delta-function ~ l u s a rest. AVnn. ----.

namely Vnn=-Vod (x,x1 ) +AVnn, and insert this into the HF equations to arrive at

/IS, 16/ which is essentially the same as (1 1 ) . Possible differences between the two may only arise due to the Uc- and ~ v ~ ~ - t e r m s (the lat- ter denotes the appropriately weighed integral over AVnn); or due to the AU-term in the mean field generated by the Skyrme forces. Hence, Imefim and HF have almost identical structures, however, within HF the simplified version retaining only the summed nonlinearity (i.e. AVnn=O and/or AU=O) arises as a result of rather drastic simplifying as- sumptions (Delta-forces); within Imefim, on the other hand, neglect of Uc can be shown to give rise to rather small deviations from the exact results (i-e. at least for most cases of interest to us /6-9/). Hence, we believe Imefim to give rise to a richer and more general structure

(and dynamics) than H F -- a notion which still has to stand its test before it can be accepted.

..

Lct us return to (11) and follow /16/ in defining a row matrix E (E1,E2, ..., EN) and a one column wavefunction Y with the N elements to wrlte the N equations (11) in the compact form Tj -uaXxy(x)-vor2.y, = E Y with Jj=ylj [J-E~M-~/v,]-'/~; j=l ,.. ^,N . ⁽¹³⁾

(Alternatively the respective NxN matrices could be used.) This equa- tion has the same form as nonlinear Schrodinger-type equations e- merging from the application of fluid dynamical concepts to quantum mechanics (see e. g. / I , 17/ and references) .

The above considerations show clearly that the relations given by the inverse method are not just compatibel with rather general findings arrived at in field theoretical /3/ and stochastic approaches /4/ to quantum mechanics, but also with the HF method and with phenomenlogical NOSES.

Now we would like tq try to find out whether this approach is also to lead us towards a consistent and unbiased (by the historical develop- ment for point-particles) dynamical description. To assess the ema- nating concepts, we take the simple case in which our initial equation

(3) is interpreted as a 1D Schrodinger equation, i.e. -m<x<+m . ^{In the}

real 3D world our results are therefore of relevance for head-on col- lisions and for spherically symmetric (equilibration) processes. In all other situations they may or may not be realistic.

To obtain the "nucleus" of a realistic dynamical evolution equation for the potential, U, we concentrate first on the search for a related con- servative equation (with atEi=O). From the discussion of / 1 8 / it ap- pears very suggestive to expect that the spatial dependence of the

C6-72 J O U R N A L DE PHYSIQUE

UN(x) of (8) is to help us very much to determine the spatial part of the wanted evolution equation. However, it appears almost unbelievable to expect Ul (x) alone to serve that purpose. Yet, it is possible to provide a mathematical proof for the statement that Ul(x) determines uniquely the spatial part of the desired conservative equation (B.

Fuchssteiner, private communication). The time evolution is left un- certain in respect to a scaling of the time variable. But the physics as discussed in /18/ eliminate this ambiguity leading to

N

Without loss of generality the v -term could be omitted since (14) --

like the SchrBdinger equations ( 9 ) themselves -- is invariant under Galilean transformations. For a system containing two nuclei with Np and NT=N-N bound-states and with the speeds vp, vT, respectively, the constants P ~ i are given by Li=vp/4Ei (for i=1 ,. . ,Np) and L.=vT/4Ej

(for j=Np+l, ..., ^{N=N +N} _{P T} ^{) . I}

In view of the preceeding discussion the appearance of a nonlinear term in (14) does not constitute too much of a surprise. However, the third- derivative term is rather unusual. It is interesting (though it may be accidental) that such a term emerges also in the discussion of stochas- tic electrodynamics and quantum mechanics /4/, which may be related to Imefim /19/. (In applications of (14) to fluid dynamics the a,,,U-term describes dispersion.) In the rather general gauge theorles such terms do not arise (see e.g. / 2 0 / and references), yet, their absence is in no way substantiated by mathematical or physical arguments -- ^it

is only motivated by the desire to keep the formalism as simple as possible. It can not be excluded that such terms might turn out to be of significance in non-Abelian gauge theory (E. Kapugcik, private com- munication).

In any case, the terms in square brackets, (14), contain all the in- formation on the internal degrees of freedom which may be extracted from the Ei. Suppressing these two terms would reduce (14) and (11) to their linear counterparts, i.e. in the force free case we obtain then

a t U=-vOaxU and the free Schrodinger equations (if we still want to extract structure information from the latter, then we have to insert

"by hand" appropriate model potentials, say, the harmonic oscillator).

As in the static case, the general solution of (14) is given in terms of Uc (dispersive waves) and of UN (stationary waves or N-soliton solu- tions). The individual potential-contributions/potential-bags/solitons, UNi(xlt), collide purely elastically with each other -- after all we constructed so far only a conservative equation. It is of note that the integral over U(x,t) is a conserved quantity. It is given by

The contributions from the continuum vanish (for R(E)=O or) for pure N-soliton solutions, U~(x,t); in this case the solution of (14) is given by (8) with the substitution x ⁺ (x-vit) E (x-Li4E. t) . (For more de- tailed studies into the mathematical structure of (14$ and hence for more general physical connections it is of note that (14) is via B a c k - lund transformations related to other nonlinear evolution equations, say the Sine-Gordon equation, /21/ and references. This equation in turn has been shown to be equivalent to the massive Thirring model.)

(14) is already a very useful and interesting equation, yet, to make it more realistic, we have to supplement it by further terms:

From analogies to other applications of (14) and from rather general arguments we take the (at present only heuristic) justification for

adding the term f(E)U to account for external forces /9,22/ (equiva- lently we could use (14) with an energy dependent effective mass, say M(E)--K~/~~(E)). On a similar basis the term D(x;E) axxU is added to ca- ter for binding and dissipation /9,19,23/:

Instead of aktempting to provide microscopic derivations of (16), we would like to find out whether there are some connections with w?ll- established approaches. To that end we employ (8) to re-write (16) in terms of its densities, to obtain a multi-component nonlinear Fok- ker-Planck-type equationPjr 19/; an equation which turns out to be the simplest possible quantum mechanical master equation (/24/ and A. Uhl- mann, private communication). For the sake of argument we give just the resulting equation for N=l (we drop the index "1") :

The difference between this transport equation for extended objects and the traditional Fokker-Planck equation for structureless particles is contained in the terms in square brackets. And the latter, i.e. (17) with M=O is fairly successful in the discussion of deep inelastic col-

lisions of heavy ions. According to /9,19,23/ (17) (i.e. its multi- channel version) should be used instead.

Summarizing the contents of this section, we considered spherically symmetric 3D systems, applied inverse methods to the solution of the bound-state problem (3) to obtain an analytical expression (8) for the mean field (which then includes the self-interactions of the different energy levels). Mathematics and physics cooperate to provide us di- rectly with the related ( I D ) nonlinear dynamical evolution equation for the mean field. The cases discussed explicitly refer to L=0, yet, the respective results for RfO are also available in the literature /lo/

(i.e. including the respective evolution equations). From the above discussion it is inferred that the emerging formalism is apparently consistent with conventional (semi-) classical models, with rather general (quantum mechanical and field theoretical) considerations and

-- in the appropriate limit -- also with traditional approaches for structureless point-particles.

3 - EVALUATION OF THE ENERGY EIGENVALUES

The only quantity in nuclear physics which we really know to a very good precision is the total binding energy (or mass), Bt=A-B(A), of the nucleus containing A nucleons. Yet, within the traditional direct ap- proach it stands at the end of a procedure starting with Vnn (or U) and going via the evaluation of the spectrum (i.e. the Ei) eventually over to Bt. Within the spirit of inverse methods it appears most sensible to invert the procedure, i.e. to follow the scheme

In section 2. we dealt largely with the 'second half of (18). At present we find it difficult to state with confidence and accuracy that the

first part of (18) can be done within Imefim itself. The possibility that this might ( ? ) be the case arouse in view of the fact that its structure is very much the same as the one of H F . And within H F it is possible to do the first step in (la), see / 2 5 / and to be

Assuming that the nucleon-nucleon forces are only two-body forces, one can derive for the H F approach the relation

C6-74 J O U R N A L D E PHYSIQUE

with n.=2j+l (1 9)

between the total binding energy and the average kinetic energy of the nucleon in the i-th level, ti, and the respective energy eigenvalue, Ei. In /25/ the virial theorem has been used to eliminate the kinetic

enerqy from (1 9) to obtain thus

The very r.h.s. in (20) is due to a numerical comparison of the in- homogeneity indices, k, for different nuclei with 4<A(56 indicating that k is to a good approximation a constant, i.e. k=-0.461. Besides the degeneracies ni, the Ei are the only unknowns in (20); the Bt are well established and tabulated. Using (20) and Koopmans' theorem (that the separation energy of the least bound nucleon is just given by the difference in the total binding energies of the respective neighbouring nuclei) the single-particle spectra have been evaluated for nuclei with mass numbers ranging from 4 to 100 (without a must for discontinuing the procedure) /25/. The quality of the resulting Ei is of the same order as the one of self-consistent calculations. But the above pro- cedure does not require a computer -- if desired then all the necessa- ry calculations may be done with a pencil! In a related study (E.F.

Hefter, to be published) it has been demonstrated that the above sum- rule is also consistent with the rms radii determined experimentally.

In other words: HF with two-body forces 2 consistent with Bt, the Ei and with the rms radii -- provided it is applied in the spirlt of in-

- verse methods.

Our tentative conclusion is that there are within the traditional di- rect approach to the problem two main sources for possible errors which lead to the impression that HE' is not able to reproduce all these data simultaneously: (i) In contrast to "appropriate" nucleon-nucleon inter- actions, phenomenologically adjusted Vnn's can reproduce all these data at the same time. It might be necessary to reconsider the definition of

"appropriate", i.e. in particular the transition from the Vnn within a nucleus to the free Vnn. (ii) Preliminary calculations of ours per- formed within Imefim indicate that the deep-lying hole-states in nuclei are rather important for evaluating the density distributions and rms radii. To our knowledge HF-calculations treat them often

(mostly? always?) as adjustable parameters.

To our present understanding point (ii) might be the most important one.

4 - APPLICATIONS OF IMEFIM

In the expression for the total potential, the UN(x) of ( 8 1 , each of the individual contributions UNi has the peculiar property that its amplitude appears again in the argument of the respective formfactor, see in particular the U1 of (9). Hence, U1 is the simplest possible ap- proximation to UtUN which still contains this prominent feature of the complete expression. ~onsequently'a study of U1 should already provide us with first estimates on some global prop rties of the atomic nucleus.

The relative nuclear (charge rms) radii, R$1rr2 (A) >, are rather well determined by isotope shift measurements. From (9) we may readily e-

where we suppress the index " 1 " which in (9) indicates that we deal with a one-bag or one-soliton solution. The indices 1,2 in (21) refer to the nuclei with mass numbers A,, A2, respectively. If we use (9) in such an averaged sense, then it would be most unreasonable to employ still a specific eigenvalue Ei in (9) or (21); we should rather take

an appropriate average of all the Ei of the respective nucleus. From (19), (20) we take it for granted that the binding energy per nucleon, B(A)=Bt/A, serves that purpose. The left half of (21) as derived from (9) contains only the effect of the collective binding energy and can be used to obtain estimates for re tive radii. However, if we apply the simple relation R ( A ) = ~ ~ T S - ~ - ~ with ro=l .2 fm, then we may ~ 4 ~

combine it with the 1.h.s. of (21 ) to arrive at the expression glven on the very r.h.s. of (21). It contains both, the effects of binding and of the saturation of nuclear forces. A more detailed discussion of

(21) may be taken from /26/ where also applications are presented. --

It is of interest to note that the very r.h.s. of (21) emerges (with a slightly different constant) in a more direct and completely indepen- dent way from a discussion of the sum rule (19) (E.F.Hefter, to be published).

Fig.1: From the figure it may be taken that HF (with Skyrme forces /27/;

dotted line) and liquid drop model (broken line; /27/) yield basically only tangents to the curve indica- ted by the experimental points / 2 7 h However, the r.h.s. of (21) qives

I I a , , , - the correct bending required-by the

- 190 200 the experimental data. To our know-

/

4

The work outlined in section 3. provides us with a consistent and complete input for e- valuating UN and p . Since this work has been completed rather recently, we did not yet manage -0.8 to do the respective computations with UN. (Yet,

usicg a different approach (see /28/ and refs.) we looked together with P.E. Hodgson at the density and

O h

/

Furthermore, we found that it is a nice approximation to use instead of (15) a much simpler relation containing only the binding energy per nucleon, i . ^e.,

8n "

J ( A ) = - - - C n 1-EifiL/2m - 1 . 6 2 - 8 n J ~ ( ~ ) ~ ~ / 2 m ' .

A i=l i (22)

For more details /12/ should be consulted. If we use the resulting J(A) as a basis for estimating the radius occupied by a spherical model nu- cleon within a nucleus, then we find that it is for medium and heavy nuclei on the average by about 20 % larger than the radius of a nucleon within the deuteron. This prediction is fully in line with the recent determinations of the nucleon formfactors in different nuclei. We find in some cases also differences in the respective radii for neutrons and protons which go up to about 5 % to 1 0 % of the ratio R(A)/R(d) between the radii of nucleons in nuclei with mass number A and in the deuteron, res ectively. We have no idea whether it is possible to verify this Preziction experimentally.

Due to the f (E)U-term in (16) Imefim predicts a specific (projectile) energy dependence for the amplitude of the real central nuclear part of the optical model potential (say, for nucleon-nucleus scattering);

i.e. v ( ~ , E ) = v ( ~ , o ) . f ( ~ ) with

C6-76 JOURNAL DE PHYSIQUE

(The right part of (23) is discussed further below.) The constant e in (23) is for the time being determined heuristically, its microscopic derivation is still to be given. The energy dependence of (23) compares

favourable with heuristic and formal results (see /22/ and to be pub- lished). Application of conservation laws that hold within Imefim (but not within traditional approaches) leads to the energy dependence of the volume term of the imaginary optical model potential, W=W , ^as

given by the second relation of (23) (/19/ and in preparationY. Its global predictions compare better to heuristic data than the ones based on dispersion relations.

The dynamics of (14) have been studied for elastically colliding po- tential-bags, model nucleons and model alpha-particles /29/ showing that the main characteristics of such collisions are very much the same as the ones of TDHF (in view of the discussion of section 2. this is no surprise). But distinct differences between the two are related to the fact that consequent application of the dynamical description due to Imefim yields also the correct (s-wave) phase-shifts for elastically colliding nucleons and alpha-particles (/29/ and to be published; in view of the point that these results are still based on a single ad- justable parameter, they are only to be interpreted as qualitative ones) .

In a rather tentative note /30/ it has been suggested that Imefim should also yield the appropriate description for the equilibration of quantum mechanical systems. A specific prediction in respect to the charge equilibration in deep inelastic collisions of heavy ions is e.

g. that there should be oscillations in the variance of the charge as a function of the energy loss. There are some data which provide an overshooting of the variance as also predicted (at intermediate and lower energies) but the oscillations (which should manifest themselves at lower energies) are not yet uniquely confirmed by experiment (i-e.

the uncertainties in data which seemingly do exhibit them are still to large to draw that conclusion with confidence).

Further work attempting to develop and understand the formal structure of Imefim and to test its applicability to problems in nonrelativistic quantum mechanics is still in progress.

5 - ^SUMMARY

Going back to the historical starting point of quantum mechanics, the Schrodinger equations ( 1 ) as designed for structureless point-particles, we attempted to construct. an associated formalism for extended objects having internal degrees of freedom, see section 2.. The input required for actual computations is given by the (single-particle) energies Ei.

The sum rule (19) as established within HF has then been used to e- valuate these E . ₁ in a consistent and simple, yet, fairly accurate pro- cedure. In sectlon 4. we drew attention to first applications of Imefim to static and dynamical situations in nuclear physics and to some of the emerging predictions. The formalism appears to be very interesting and useful, but since its development is still in progress, it is dif- ficult to give a qualified judgement on its actual value. It is easier to give the lower and upper bounds instead.

Lower bound: With the formal structures provided by Imefim we have an approach/model providing us with some formal results and representing a rich source for phenomenological relations. 1.e. its value is of the same order as the one of a good number of accepted heuristic approaches.

Upper bound: Imefim provides a sound basis for a deeper understanding of the atomic nucleus and a unified theory of its properties. It is

also to help to bridqe the qap between classical and quantum mechanics _{- -} /3,4,19/-

A lot more of work is required to determine the propper place of Imefim between these two extreme limits.

Acknowledgements: For collaborations and helpful discussions I am very much indebted to M. de Llano, B. Fuchssteiner, K.A. Gridnev and I.A.

Mitropolsky. I would also like to thank H.G. Kummel for the time spent on discussions related to the subject.

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