ON SOME APPLICATIONS OF TIME-DEPENDENT INVARIANTS TO THE SEMICLASSICAL FORMULATION OF QUANTUM MECHANICS

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ON SOME APPLICATIONS OF TIME-DEPENDENT INVARIANTS TO THE SEMICLASSICAL

FORMULATION OF QUANTUM MECHANICS

R. Dreizler, H. Kohl

To cite this version:

R. Dreizler, H. Kohl. ON SOME APPLICATIONS OF TIME-DEPENDENT INVARIANTS TO THE

SEMICLASSICAL FORMULATION OF QUANTUM MECHANICS. Journal de Physique Colloques,

1984, 45 (C6), pp.C6-35-C6-44. �10.1051/jphyscol:1984605�. �jpa-00224206�

JOURNAL DE PHYSIQUE

Colloque C6, supplement au n°6, Tome 45, juin 198* page C6-35

ON SOME APPLICATIONS OF TIME-DEPENDENT INVARIANTS TO THE SEMICLASSICAL FORMULATION OF QUANTUM MECHANICS

R.M. Dreizler and H. Kohl

Institut fur Theoretisahe Physik der Universitat Frankfurt, D-6000 Frankfurt/M., Robert-Mayer-Str. 8, F.R.G.

Abstract - I t i s shown, that time-dependent c l a s s i c a l and quantum mechanical i n v a r i a n t s (TDI) may be of considerable value in semiclassical formulations of time-dependent quantum mechanical problems. We demonstrate, that the a p p l i c a - tion of TDI allows the application of several ground-state technologies as for instance the Wigner-Kirkwood and WKB-expansion to the TD s i t u a t i o n . Possible a p p l i c a t i o n s to semiclassical wave packet motion and to (low energy) nuclear hydrodynamics are o u t l i n e d .

I . Introduction

The semiclassical treatment of time-dependent (TD) quantum mechanical problems a t t a i n s continuing i n t e r e s t . In t h i s contribution we show, that some i n t e r e s t i n g r e s u l t s can be obtained, if time-dependent i n v a r i a n t s (TDI) are employed. A more detailed paper on t h i s subject i s in preparation to which we r e f e r the i n t e r e s t e d reader for d e t a i l s .

In a recent note [1] we have shown, t h a t the density matrix p ( q | q ' ; e ) for a system of 2N fermions, evolving from the ground-state in a TD p o t e n t i a l can be represented as

(1)

where I(p,q,t) is the Weyl transformation of the solution of the following operator equation, defining the TDI I(p,q,t),

(2) Here f . . .DpDq denotes the phase space path integral, which is needed because of the possible complicated p-dependence of the TDI, e is the Fermi energy of the q

ground-state. Weyl transforming equation (2) one obtains an equation for I(p,q,t) (3) Résumé - On montre que les invariants mécaniques classiques et quantiques dépendant du temps (TDI) peuvent avoir une valeur importante pour les formulations semi-classiques des problèmes mécaniques quantiques dépendant du temps. On démontre que leur utilisation permet l'application de plusieurs technologies d'état normal, par exemple les développements Wigner-KirKwood et WKB. On indique des applications possibles au mouvement de paquets d'ondes semi-classiques et à l'hydrodynamique nucléaire à basse énergie,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984605

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As usual, this form is most useful, if the classical limit of eq. (2) is to be considered. It is straight-torward to find the formal solution of eq. (2). Intro- ducing the evolution operator U(t) of the system, one finds

Lt js easy to deduce [2] , that

a) I has a time-kdepevdent eigenvalue spectrum, A

b ) the eigenvalues of I(t) coincide with those of H(t=o),

c) the eigenfunctions of the TDI differ from the solution of the corresponding TD Schrodinger equation only by a TD phase.

In formulae these statements read as

These are just the properties, that make the TDI valuable in the treatment of TD quantum mechanical many-particle systems. It looks strange, that this point has been overlooked for such a long time. Only recently Lewis and Symon pointed out, that the classical TDI can be of some use to treat a certain TD Vlassov-Poisson problem exactly [ 3 ] . As a general introduction into the literature on TDI one could take ref. [4]and the references thereiy. In this note the class of classical systems with a Hamiltonian of the form H = p 12 + V(q,t) has been determined, which admits a classical TDI linear or quadratic in p.

11. Treating systems with a known global TDI

For reasons that will become clear later we call a system to have a global TDI, if the solution of eq. (2) (or equivalently of eq. ( 3 ) ) is known. Unfortunately the class of such system, both, classical and quantum mechanical, is rather limited up to now. As we have recently pointed out [ I ] , applying Weyl transformations, it is straight-forward to generalize the classical results of ref. [4]to quantum mechanics. For illustrational purposes, we refer to this example. As has been shown, a TD system characterized by the Hamiltonian

admits the TDI

Here U is an arbitrary function of its argument, ~ " t ) is also arbitrary up to the initial condition n7(t=o) = o, which is necessary to satisfy the initial condition for I(t). Furthermore, ~ ( t ) is the solution of the following differential equation

Now eq. (1) contains the path integral representation of the Bloch series for the TDI I(t) and we note, that its defining equation is

So we see that, contrary to common reasoning, partition function techniques can be applied to treat TD systems. To be brief, we give only a short outline of possibi- lities to find semiclassical solutions of eqs. (1) and (9).

1. The method of 'effective potential': Concerned with the time-independent case, March and coworkers published a series of papers, where they attempted to extract

the one-particle fermion density matrix from the canonical density matrix (see e.g. [ 5 ] and ref. therein). The developments indicated above, suggest to employ the same ideas in the TD situation, if, of course, the global TDI is known for the system under study. To make this explicit for the TDI of eq. (7) we write the TDI

A A

as I = I t ~ ( 2 ) , where I (t) contains the first three terms of that equation. One is then ?ed toPthe ansatz O

which defines the field @

(qlq';<).

Introducing the expression for K ~ ( ~ < ~ ~ ' O ) into eq. (9) we end up with an eq. for the field 6 (qlq';<) which, for our example, reads

t

One could now start with a detailed discussion of this equation, especially one could investigate several limiting situations, as for instance a small 6 during the time interval1 of interest. Such studies are then required for any special system under investigation.

2. WKB-expansion of the density matrix: A general discussion of the semiclassical expansion of phase space path integrals has been given by Mizrahi [ 6 1 . The

essential ingredient to such approximations (in the lowest order) is the classical action integral

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where the integration is over T , as defined in eq. (I), not over t. It would be interesting to see, if there is a relation between this kind of expansion and the WKB-expansion of the Wigner function as suggested by Berry and Balazs [ 7 ] .

3. TD Wigner-Kirkwood expansion: A further possible semiclassical treatment of eq. (1) is opened by an application of the well-establishedwigner-Kirkwood expansion. As a starting point one could choose the following representation of the density operator

c+i-

where E is again the Fermi energy of the ground state. This method is reviewed in ch. 13 of ref. [8] and we can avoid to go into the details here, since one would proceed as in the stationary case, which is, of course, contained in eq. (14) for t=o. Again, such expansions could lead to quite different results for different TD systems, if the corresponding TDIs are different in essential terms.

Although studies along these lines have just started, we believe, that they could contribute to the understanding of TD quantum mechanical many-fermion systems in an external TD (mean-field) potential.In m n y cases, however, it will prove impossible to find the global TDI. In these situations one can perhaps rely on a slightly modified concept, which will be suggested in the next two sections.

111. The concept of a local TDI

As a first attempt to treat systems with an unknown global TDI one could try to make a kind of local approximation and proceed as follows. Expand the Hamiltonian up to second order in (q - 6) around the arbitrary point 6 , to obtain a local

Hamil tonian

To find a solution of eq. (3) we make the ansatz

hereby defining the

local

i k

with b(q,t) and c(q,t) defined as

The propagator corresponding to the quantized form of 1' can be found by evaluating the path integral with eq. (16) or, perhaps more readily, by solving Schwingers equation for the action operator for our problem. The result is

Kn2,0(q5(q'~) = 1 4nfiy n shm~lr, I

'I2

2

0 2

with 6. R, A defined as

It turns out, that = o

.

n2

a2 a2pt(q1q';~)

~ ( ~ , t ) = ?;;;-- lim

q+q' aq aq'

It should be appearent from these considerations, that the treatment above, if applied to the ground-state (t=o), is identical with the so called 'partial fi-resumation technique' as originally suggested by Bhaduri [91 and further

developed by Durand, Bartel, Brack and Schuck [ l o , 111.

It turns out, that there are still some open questions in our TD situation. We attempt, of course, to evaluate the remaining Laplace integrals by the saddle point method, as has been also originally suggested by Bhaduri for the time-independent case 191. Quite recently it has been shown, that good results for the density and energy for the harmonic oscillator and Woods-Saxon system are obtained, if one includes the first two corrections in the saddle point approximation. It is obvious, however, that this cannot be applied straight-forwardly to our situation.

The main reason for using the saddle point method at all, is the necessity to eliminate the spurious shell oscillations 1101. So one could say, that the saddle point method is employed as a kind of smoothing procedure. This works well for the systems studied up to now in the time-independent case, namely essentially the

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harmonic oscillator and the Woods-Saxon potential. Whether this works in any case in the TD situation, is a question still to be studied.

In the appendix we present the results of of some preliminary calculations, where the smoothing has been performed with the first order saddle point method. The TD potentials are defined as follows.

In Fig. 2:

-2(q-10+10t) 2

4.5q2 + toe , o ~ t < l V(q,t) =

2

4. 5q2 + , t > l

As can be seen from figure 2, the approach is not completely energyconserving.

Whether this can be corrected by using a higher order saddle point approximation and/or another local TDI remains an open question.

IV. Some additional applications of the (local) TDI

In this section we report on some further possibilities to apply (local) TDIs.

1. E. Hellers wave packet motion: It turns out, that the local TDI as constructed in the last section can be employed to formalize the (semiclassical) wave packet approach as originally suggested by E. Heller [12], who also applied it in several papers. In two recent papers this original approach has been generalized by Heller himself [I31 and independently by Coalson and Karplus (141. This approach has attained considerable interest in quantum chemistry. Consider a wave packet in a potential at time t=o. To describe the time-evolution of the prepare? wave: one could proceed as follows. Calculate first the expectation values of p and q

Now, guided by Ehrenfests theorem, Meller suggested [12,13] that the center of mass of the wave packet moves according to the classical equations of motion, however, with the full Hamiltonian replaced by the local one, which coincides exactly with the one given in eqs. (15118) and q now identified with the solution of the corresponding Hamilton equation. Because of this dependence on = q(t), the center of mass moves in a TD potent'ial even if no explicit time-dependence is present. Up to here, we just repeated Hellers point of view. To continue, however, we avoid to rely on a representation of the solution of the Schr6dinger equation by a series of ad hoc chosen products of Gaussians and Hermite polynomials [13,141, but rather use the concept of the TDI and proceed as follows.

a) Determine

p(o)

^1 1 1 1

b) Solve the eigenvalue problem H (t=o) I$ (t=o)> = I$ (t=o) > to find complete basis I$l(t=o)>

.

"1 1 1

C) Instead of solving the Schrijdinger eq. H (t) l$n(t)>=i6a (t)>for every basis t n

state, rely on the local TDI as defined in eq. (16) in ~ t s quantized form and solve the problem

d) Expand the initial state I$(t=o)> = Za 16 1 (t=o) > and determine the initial

values of the a

.

e) There are then two possibilities to represent the time-evolution of the initial wave packet / $ (t=o) z

.

m

semiclassical: I$(t)> =

gZo

exact :

0 )

with ihi = _{rn=o rn} Z a <qn(t) l~~(t)l$~(t) >

.

-

where V (t) is defined by H - H = VR(t). In any case, the time-dependence of the R-

states I$ (t)> is given by a TD parametrization via the solutions of the eqs.(l7) which are2ow coupled to the Hamilton eqs. for p and q . The eventual value of such an expansion lies, of course, in its n-dimensional generalization, which is needed in several applications to quantum chemistry.

2. Hydrodynamical approach: The densit.y operator p(t) has the exact form

~ ( t ) = 0(c-~(t)) for the systems treated in this note. Weyl transforming this equation one ends up with the expression for the Wigner function f(p,q,t).

Expecially, the 'TF-limit' of f (p,q,t) is given by fTF1(p,q,t)= G(c-I(p,q,t))

.

From this relation it is obvious, that I(p,q,t) contains information about the local Fermi surface. For systems with an unknown global TDI one could employ the concept of *-expansion and solve only the simpler classical part of eq. (3) (*=o).

We feel, that this would essentially be equivalent to make the transition from W-space (as Balazs c~$;s it) to the classical phase space. In that case an

approximation like f (p,q,t) = 0 (c-~~~(~,~,t))would result. So it is clear, that the TDI as treated in this note should also be of interest in the hydrodynamical approach to the many-fermion problem. Another question present y under investiga- tion is, if a representation of the Wigner function as f (p,q,t)=FJ(~-~f

1

(that is introducing the local TDI, or a slight modification, of section 111) could be of use to close the set of hydrodynamical equations.

3. Relations to density-functional theory: It is still an open question how to construct a TD kinetic energy functional T E D ] . Using the well-known gradient expansion technique or, equivalently the TD Wigner-Kirkwood expansion of section I1 one would be able to derive expressions for this quantity if the global TDI is known for the system under study. Such investigations for a restricted class of systems, as for instance for the TDI of eq. (7) should give hints, which could be useful even in more general situations.

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Try:: - 0.0 ENLRCY - 95.924

Fig. 1

-

The density and kinetic energy density (TAU) are shown and compared with, the exact results (wavy lines).

T l Y E - 1.1 E N E R G Y = 352.92989

T I Y t

-

T I Y L 1.5 ENERGY ; 405.3563)

Fig. 2

-

JOURNAL DE PHYSIQUE

R e f e r e n c e s :

[ I 1 Kohl H . a n d D r e i z l e r R.M., P h y s . L e t t . 9&(1983)95

[ 2 1 L e w i s H . R . a n d R i e s e n f e l d W.B., J . Math. P h y s . E ( 1 9 6 9 ) 1 4 5 8 [ 3 1 L e w i s H.R. a n d Symon K.R., P r e p r i n t L o s Alamos 1 9 8 3

[ 4 I L e w i s H.R. a n d L e a c h P.G.L., J . Math. P h y s . 2 3 ( 1 9 8 2 ) 2371 [ 5 1 Lawes G.P. a n d March N.H., P h y s i c a S c r i p t a g1980) 4 0 2

16 I M i z r a h i M . M . , J . Math. P h y s . E ( 1 9 7 7 1 7 8 6 and J-Math. Phys. z ( 1 9 8 1 ) 1 0 2 ( 7 1 B e r r y M.V. a n d B a l a z s N.L., J . P h y s . E ( 1 9 7 9 ) 6 2 5

18 1 R i n g P . a n d S c h u c k P., The n u c l e a r many-body p r o b l e m , S p r i n g e r 1 9 8 0 19 1 B h a d u r i R.K., P h y s . Rev. L e t t . 3 9 ( 1 9 7 7 ) 3 2 9

[ l o ] Durand M., B r a c k M . a n d S c h u c k

K,

[ I 3 1 L e e S . Y . a n d H e l l e r E . J . , J.?hem. P h y s . s ( 1 9 8 2 ) 3 0 3 5 [ I 4 1 C o a l s o n R.D. a n d K a r p l u s M . , Chem. P h y s . L e t t . z ( 1 9 8 2 ) 301