ELSEVIER

15 August 1994

Physics Letters A 191 (1994) 265-274

PHYSICS LETTERS A

Motional Stark effect on Rydberg states in crossed electric and magnetic fields

David Farrelly Department of Chemistry and Biochemistry, Utah State University, Logan, UT84322-0300, USA

Received 18 February 1994; revised manuscript received 2 June 1994; accepted for publication 7 June 1994 Communicated by A.R. Bishop

Acknowledging the finite size of the nucleus in a Rydberg atom subjected to a magnetic field introduces an additional term into the Hamiltonian. Though often neglected or negligible in atoms, this term, which produces the motional Stark effect, may be important, e.g., in the solid state to an exciton interacting with a strong magnetic field. The consequences of this term in the problem of a two-dimensional Rydberg atom or exciton in crossed external electric and magnetic fields are investigated. A rich dynamics is uncovered: depending on the mass ratio and external field strengths, new types of internal motion may arise as compared to the dynamics possible in the infinite nuclear mass approximation.

1. Introduction

The problem of a Rydberg atom interacting with a strong magnetic field-the quadratic Zeeman effect (QZE)- has, for the past fifteen years or more, been one of the foremost problems in the study of chaos in atomic physics [ 1-3 1. It is, therefore, nothing short of remarkable that new dynamical features of the problem continue to be discovered. Recently, a number of researchers have started to study the effects arising from the incomplete separation of the center-of-mass (CM) and internal degrees of freedom for neutral or charged N-particle systems interacting with magnetic fields [ 4-61. Specifically, if the nuclear mass is taken to be finite, then, in a magnetic field, for a neutral system it is possible to achieve only a so-called pseudoseparation of the CM. In the pseudo- separated problem the internal degrees of freedom completely determine the CM motion, but the CM itself does not directly influence the internal dynamics. Thus, it has been shown by Cederbaum and Schmelcher [ 41 that a transition to chaos in the internal degrees of freedom can spark unbounded, chaotic motion in the CM dynamics. However, the emphasis here is somewhat different being concerned, not with the CM motion, but with the effect of relaxing the infinite nuclear mass assumption on the internal dynamics.

For concreteness, consider the hydrogen atom: in a homogeneous constant magnetic field an often neglected motional Stark term appears in the Hamiltonian if the nucleus is taken to have finite mass [ 7 1. The effective motional Stark field lies perpendicular to the magnetic field, thus breaking the axial symmetry of the QZE, and producing a fully three-dimensional system. The problem is formally equivalent to a hydrogen atom interacting with crossed electric and magnetic fields in which the nucleus is taken to have infinite mass - herein referred to as the EX B problem. It might seem, at this point, that little is to be learned about the internal dynamics by studying the motional Stark effect beyond what can be gleaned from a consideration of the hydrogenic ExB

0375-9601/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDZO375-9601(94)00473-3

266 D. Farrelly /Physics Letters A I91 (1994) 265-274

problem. It is worth pointing out, however, that the EX B problem itself is only partially understood. Further, incomplete separation of the CM will be especially important, e.g. in exotic atoms - sometimes termed atom- cules or polycules [ 8 ] - where the masses of the constituent particles may be comparable (as in the positronium (Ps) atom) [ 9 1. In solid state physics similar considerations apply to excitonic Rydberg states in which, in the effective mass model (EMM ), the effective masses of the electron and hole may be comparable or equal - the CM motion in these types of problem can be influenced profoundly by coupling to the internal dynamics [ 10 1. In this Letter the motional Stark effect is included in a study of two-dimensional (2D) Rydberg systems sub- jected to crossed external electric and magnetic fields. As compared to the 3D system, the 2D model is consid- erably more amenable to theoretical and computational study but is far from being simply a toy problem: the famous quasi-Landau (QL) resonances in the QZE were originally accounted for in a 2D planar model [ ll- 131, and 2D models of excitons interacting with magnetic fields have a long and respectable history [ 14-l 61. In fact, study of the 2D problem reveals that the motional Stark term can have a dramatic impact on the internal dynamics, including inducing new types of motion and transitions to chaos. It is reasonable to expect similarly profound consequences in the 3D problem, although the 3D problem is amenable to analytical study in the high field limit [ 171. The effect of varying the external field strengths, the pseudomomentum K and the relative masses of the two particles on the internal motion is investigated: a consideration of the CM motion itself is avoided.

The Letter is organized as follows: the 2D Hamiltonian is derived in Section 2 after a pseudoseparation of the center of mass. In Section 3 the technique of normalization is applied to the planar problem in order to uncover the various bifurcations that can occur in the (averaged) dynamics as a function of magnetic and electric field strengths and variation of the relative masses of the two particles. In Section 4 the classical dynamics is studied by direct integration of the equations of motion and computation of Poincare surfaces of section. Comparison is made with results obtained using the normal form. Brief conclusions are brought together in Section 5.

2. The Hamiltonian

An excellent description of the procedure for treating both neutral and charged N-particle systems in constant electric and magnetic fields is given by Johnson et al. [ 71. In the case of a neutral two-body Coulombic system (masses m, and m2, charges -e and Ze) interacting with constant electric (E) and magnetic (B) fields, after pseudoseparation of the CM, the Hamiltonian may be written

H= rk (K-eBXr)2+ & (p-eJ3BXr)2+E.r- $, (1)

wherer=(r],~=m,m2/M,M=m,+m2,and~=(m2 - ml ) /2M. The pseudomomentum K is a constant-of-the- motion. Atomic units are adopted: the unit of length is the Bohr radius 6 for a two body system of reduced mass ,u and “nuclear” charge Ze, i.e., ao=h2/pZe2; the particle of mass m, is taken to be the nucleus: In these units ,U = +i = a0 = e= 1. Suitable measures of the electric and magnetic field strengths are E. = eZ/a f and B. = fic/ea f and it is convenient to work with the dimensionless electric and magnetic fields F=E/Eo and y= B/B,. Note that the fields are measured in units that depend explicitly on the masses. The coupling between the internal and CM motion, i.e., the motional Stark term is given by ( 1 /M) (Bx r) ‘r. In 2D, without loss of generality, K and P may be taken to lie along arbitrary directions in the x-y plane: it simplifies the discussion to assume that they point along orthogonal axes. Assuming further that the magnetic field lies along the z-axis and setting Z= 1 leads to the effective internal Hamiltonian [ 71

H=E=f(p;+p;)-; - zl+s y l -6 (WY -YP,) + (F+yK/M)x+ iy2(x2+y2) ) (2)

where 6= m, /m2. This Hamiltonian with S= 0 is equivalent to the 2D EX B Hamiltonian for the hydrogen atom

D. Farrelly /Physics Letters A 191(1994) 265-274 261

with an (effective) electric field term (F+ yK/M)x [ 18-201. Confronted with so many parameters it is essential to determine which are fundamentally relevant to the dynamics. Simple scaling arguments show that the dynam- ics depends on the scaled energy b= El yzf 3 and the parameter e = (F/y + K/M) / y ‘13. Alternatively, one may fix the energy and vary the parameters y and f= (F+ yK/M) separately. To simplify the discussion further only the results for two mass ratios are reported: 6= 0, and 6= 1 with E < 0. For intermediate values of 6 the dynamics is sufficiently similar to 6= 0 not to require a separate discussion. Thus, at each mass ratio, only y and 9 (or E and b) need be varied which reduces the study to a manageable size. In the absence of an external electric field, the importance of the motional Stark term will thus depend on the relative magnitudes of K, M and y.

At this point it is worth making a few general comments about Eq. (2). In the absence of the effective electric field term the problem is integrable. This can most easily be seen by rewriting the Hamiltonian in cylindrical coordinates x=p cos +, y=p sin # which gives the planar QZE for arbitrary masses,

P: 1 yl-6 H=!P;+ 2pz - -j - 21+6~c+ly2p2.

The Hamiltonian is one-dimensional and thus, per force, integrable. Potentially, the motional Stark effect can, therefore, destroy the integrability of the planar QZE. Moreover, in the Stark limit ( y= 0) the problem is also integrable. It is also noteworthy that, apart from the diamagnetic term, Eq. (2) is the same (in the 2D approxi- mation) as the Hamiltonian for a Rydberg electron interacting with a circularly polarized (CP) radiation field or the Hamiltonian describing the effect of radiation pressure on dust orbiting a planet. Both of these problems are of considerable interest in planetary [ 2 1 ] and atomic [ 241 physics.

2.1. Regularization

As written Eq. (2 ) is inconvenient for numerical studies because of the Coulomb singularity at r= 0. This hindrance can, however, be overcome easily by using the Levi-Civita regularization procedure. First a transfor- mation to semiparabolic coordinates is made,

x=i(u2-u2), y=uv, r=i(u2+u2) (4)

which yields the Hamiltonian

&‘=E=f(u2+u2)(PZ,+P:)- -& - g$ (u2+2?)(uP,-uP,)+~~(uGI~)+~y2(u~+2?)~.

(5)

The next step is to make the transformation to a new time s,

dt z=2r.

An overall multiplication of the new Hamiltonian by u2+ Y’ results in the following pseudo-Hamiltonian [ 221,

9=2=f(P2+P2) E(u2+v2)- I’-’ u ” - 41+6(u~+v~)(uP”-vp,)+~~(u4-v4)+~y~(u~+Y~)~. (7)

Re-scaling of the coordinates and momenta as follows,

u-->u Js2’ v-> -$> Pll-> @Pu, P,-> J3zP,, (8)

where 8= m finally gives

268 D. Farrelly /Physics Letters A 191 (1994) 265-274

xx=; =j(P2,+P;)+f(U2+v2)- &+yj (u2+v2)(uP,-VP,)+ gj (u4--v4)+ & (u2+v2)3.

(9)

An important difference now emerges between the planar CP system (in which the diamagnetic term is absent) and the current problem: it is apparent that the diamagnetic term in Eq. (9) prevents ionization in the plane, whereas, in the CP problem ionization is possible depending on the magnitudes of y and 9. In both problems the dynamics, as shown in the following sub-section, exhibits a complicated dependence on these parameters.

2.2. Critical point analysis

The topology of the potential energy surface depends on y and s” in a non-trivial way. Critical points lie along the x-axis, i.e., along either the U- or *axes in the regularized coordinates. The effective potential

V(u,v)=~(u2+v2)+ & (u4-v4)+ & (u2+v2)3 (10)

has real critical points lying exclusively on the v-axis determined by the solution of

V;=v-297v3+ j$syV=O . (11)

In solving for the critical points it must be kept in mind that the pseudopotential corresponds to the original problem only for the fixed value of the pseudoenergy X= 2. Thus a full critical point analysis of the pseudopo- tential is unnecessary. It turns out that there are four basic regimes relevant to the real dynamics:

Regime (i): 0 < 9~ k, 0 <y< 4Y/fi. In this regime the potential has a single well at the origin and the motion is always localized within the well, for all values of y as shown in Fig. la. The system in this limit resembles a perturbed isotropic oscillator.

Regime (ii): &<Y<&, (72~-~~-4)/54~~~4~/~. In thisrangethepotentialagainhasasin- gle well at the origin and the saddle points lie at or above Xx= 2. At the lower limit, p= k, if y= 0, the pseudo- energy corresponds to motion coincident with the two saddle points as shown in Fig. 1 b: this is the same as the zero frequency threshold in the CP problem. At the upper limit in 9 the motion is constrained to the central well only if y also takes its upper limit. For intermediate cases with y # 0 the potential resembles that shown in Fig. 1 c except that the saddles lie at or above X= 2. Note that in Fig. 1 c two additional wells have appeared and,

i -6-4-Z 0 2 4 6 -6-4-2 0 2 4 6

Fig. 1. Effective potentials along u=O in the four regimes defined in the text. The horizontal line shows the pseudoenergy Xx=2: (a) 9=&i, y=49/,/3, (b) ip=& y=O, (c) 9=&, y=24c/fi, (d) .S==&, y=845.

D. Farrelly /Physics Letters A I91 (I 994) 265-274 269

depending on the initial conditions, trajectories may or may not be trapped in one or more of the three wells. Regime (iii) : &<F<&, yc(725-d--4)/54 or &<.?F, O~y<4F./fi. In these cases the pseu-

dopotential saddle points lie below the pseudoenergy Xx=2 and the motion is expected to be complex. Fig. lc is typical of this regime. For y=O the potential resembles Fig. lb except that the saddles lie below X= 2.

Regime (iv): y> 4f/$. In this regime the potential has a single minimum at the origin for all 9. Unlike in regions (i)-( iii) the potential increases strictly monotonically away from the origin in both the positive and negative pdirections. This is illustrated in Fig. Id.

3. Normalization

The Hamiltonian X is in an especially convenient form for numerical and analytical investigations. First, classical perturbation theory and symbolic computer algebra is used to compute the normal form. The normal form, XNF, is generated in the {u, v} coordinates but is most conveniently and compactly re-expressed in terms of the generators of the Lie algebra of the group SU ( 2 ) . To 2nd order the normal form is given by

X NF=2rz+ !..?$!! (3j;+g)+ 2 (5n2-3jz-2A2jz)- qjxjy

where

A- ‘-’ -i3

(12)

(13)

and n is the principal quantum number and

j=n=t(PZ,+u2)+$(Pt+v2), jx={(Pi+u2)-t(P2+v2), jY=t(uPU-uPV),

j,=f(P,P,+uu).

These quantities satisfy the same Poisson bracket relations as angular momentum, namely,

(14)

{j& > = $k.ik , (15)

where tju is equal to 1 for even permutations of its subscripts, to - 1 for odd permutations and to 0 otherwise. Together with the relation

n2=j2=jz+js +jf , (16)

the { ji, i= 1, 2, 3) generate the Lie algebra of the group SU (2 ): hence the notation j,, j,,, j,. Actually, the { ji, i = 1, 2, 3) constitute the Lie algebra for both SO ( 3 ) and SU (2 ) which are locally isomorphic, but SU (2 ) is the appropriate symmetry group of the isotropic oscillator. Eq. ( 12 ) illustrates an important geometric property of these quantities: they define a constant energy sphere whose radius is half the unperturbed energy (2n). The other j act like Cartesian coordinates on the surface of the sphere. Thus the orbit space in the averaged approx- imation can be represented by level curves of the normal form on the surface of a sphere of radius n. Some prellminary comments are in order: It is apparent that two integrable limits of the problem exist, the Zeeman limit when the normal form is purely a function of j, and the Stark limit when the normal form depends only on j, Dependence of the normal form on a single generator is often the mark of integrability [ 231. In intermediate regimes the normal form will, in general, be a function of all three generators. The case with 6= 1 differs consid- erably from all other values of 6 in that the linear Zeeman term in Eq. (2) vanishes. For 6# 1 the first order term in the normalized Hamiltonian has the form constx (a1 jx+a2 j,,) where (Y,, and a2 both depend on 9 and y and a2 also depends on the mass ratio. A simple rotation on the SU (2 ) sphere can thus be made to convert the lowest order term to the form const’xjx,, where the direction x’ is defined by the field strengths and 6: thus

210 D. Farrelly /Physics Letters A 191 (I 994) 265-2 74

on passage from the Stark to the Zeeman limits for 6# 1 a gradual, field dependent, re-orientation of the quan- tization axis is expected. Based on the critical point analysis described above, the normal form is expected to be a reasonable approximation only for motion confined to the central well shown in Fig. 1 when a harmonic approximation for the unperturbed Hamiltonian is valid.

4

3

2

qJ:

-1

-2

-3

-4

(a)

-4 -3 -2 -1 0 1 2 3 4

V 4

3 I 2 1 1

P V0

-1

-2

-3

-4

Fig. 2. poti& surfaces of section (SOS), defmed by u = 0, and level curves of the normal form on the unit sphere for 6= 0: (a) and (b ) ~~0.01, F=O, (c) and (d) ~~0.01, F=0.002, and (e) and (f) y=O, F=0.0005. Front (left sphere) and back (right sphere) views of the spheres are shown for clarity.

D. Farrelly /Physics Letters A I91 (I 994) 265-2 74 271

4. Dynamics

Eq. ( 10) suggests that the topologies of the trajectories will reflect the dependency of the normal form on the generators: e.g., if the normal form is purely a function of j, (Zeeman limit) then the trajectories are expected to be, adopting the language of molecular vibrational spectroscopy, precessional-mode or P-type in character. Similarly, pure dependence on jX implies normal-mode or N-type behavior and on j, local-mode or Ltype be- havior. In the Stark limit the problem is separable and thus the trajectories resemble normal modes. The origin and character of the three types of modes in Rydberg atoms has been studied in detail by Milligan and Farrelly [25]. In the present example, when 6# 1 more complex behavior is possible since the appropriate zero-order mode is neither j, nor jY but a field-dependent linear combination of the two generators: thus the presence or absence of the linear Zeeman term is critical. The level structure and spectra of these systems is expected to be sensitive to the particular kinds of motion displayed by the classical system [ 19,16 1.

The two mass ratios are now considered.

4.1. 6=0

The problem in this limit reduces to the planar EX B system for the hydrogen atom. Poincare surfaces of section and level curves of the normal form on the constant energy sphere n= 1, !& 1 are exhibited in Fig. 2 showing a typical transition from the Zeeman to the Stark limit. In the Zeeman limit - Figs. 2a and 2b - the level curves consist of simple circulations around the jY axis, while in the Stark limit - Figs. 2e and 2f - the circulations are about the j, axis. Thus as 9 is increased at fixed y the axis of rotation tilts in the plane j,=O starting at the j,, axis and ending at the j, axis. This process is similar to the CP problem in that, through first order in y and 9, no bifurcations occur during the re-orientation of the quantization (rotation) axis, as reflected in the surfaces of section shown in Fig. 2. In the Zeeman limit the surfaces consist of two sets of crescent shaped nested curves corresponding to precessional motion: the two families differ in the sense of their precessional motion about the origin and thus in the sign of the angular momentum L,. In the Stark limit the surface of section consists of nested curves surrounding an elliptic fixed point at the origin. The transition from the Zee-

-4 -3 -2 -1 0 1 2 3 4

V

Fig. 2. Continued.

212 D. Farrelry /Physics Letters A I91 (1994) 265-2 74

man to the Stark limit exhibits a rich variety of motion. For small fields - Regime (i) - starting from the Zeeman limit, as 9 is increased the right hand family of curves in Fig. 2a swells and creeps toward the origin at the expense of the left hand family. Eventually, in the Stark limit, the right hand family in Fig. 2a has centered itself on the origin, and, without a bifurcation having occurred, the left hand family has disappeared [ 2 11. In this regime the normal form thus proves capable of providing a qualitative understanding of the dynamics, and, in

4

2

pvO

-2

-4

(a)

-4 -2 0 2 4

V

4

2

P v O

-2

-4

-4 -2 0 2 4

V Fig. 3. Poinc& surfaces of section (SOS), defined by u=O, for 6=0: (a) y=O.5, F=O.O575, (b) Y=O.5, F=O.O975.

2-

Py:-

-1 -

-2 -

j-----J x -4 -3 -2 -1 0 1 2 3 4

V

Fig. 4. As Fig. 2 but for S= 1: (a) and (b) y=O. 1, F= 0.0005. Note the bifurcation that has occurred on the unit sphere. In the Zeeman and Stark limits the level curves of the normal form resemble those shown in the corresponding limits in Fig. 2.

D. Fame//y /Physics Letters A 191 (1994) 265-274 273

turn the semiclassical energy level structure. For larger field strengths bifurcations may arise in the normal form and are due to the enhanced importance

of the diamagneG?term. However, it turns out that the dynamics does not respect the normal form in this regime. In particular, the transition from the Zeeman limit to the Stark limit in the actual dynamics may prog- ress through a chaotic regime in which the left hand set of curves in Fig. 2a self-destructs through the develop- ment of resonances and ultimately chaos. This cannot, of course, be predicted using normal form methods and is illustrated in Fig. 3a. Interestingly the asymmetry in the states which succumb to chaos is similar to the CP problem in hydrogen where trajectories with initial angular momenta of opposite signs exhibit markedly differ- ent ionization thresholds [ 26,271. This asymmetry is due to the presence of the linear Zeeman term and break- down of an adiabatic separation in time scales for trajectories, that roughly speaking, move opposite to the sense of the field. This adiabatic separation was first pointed out in Ref. [ 241. In summary, the motional Stark term, or an applied electric field, or both, can provoke a transition to chaos in the internal dynamics. The rich structure exhibited by the normal form at larger field strengths includes a sequence of bifurcations. However, this is not reflected in the dynamics where the system progresses from the Zeeman to the Stark limit through a transition to chaos.

4.2. S= 1

This is the opposite case to the infinite nuclear mass limit. In this case the first order normal form depends only on j, and no tilting of the rotation axis is expected. Instead, because the sphere is pinned to a fned set of axes, progression from the Zeeman limit to the Stark limit must proceed through a bifurcation. Surfaces of section and level curves of the normal form, in the low field Zeeman and Stark limits - i.e., regime (i) - are indistinguishable from the corresponding limits in the case 6~0 and are therefore not shown. A typical inter- mediate example is shown in Fig. 4. The actual progression can be seen by replacing Figs. 2b and 2c with Figs. 4a and 4b. As 9 is increased a set of N-type states appear around a center located on the jX axis - this signals the spawning of the Stark states. As F is increased further these states grow in importance, and eventually in the Stark limit, they have supplanted the Zeeman states altogether. A second major difference between this case and 6= 0 is that very little, if any, of phase space seems to be chaotic - even in regimes (ii) and (iii) the motion is extremely regular, although it remains an open question if the motion is actually integrable in this limit - cer- tainly, the Hamiltonian (5) is integrable if 6= 1 and XXT= 0 but this does not correspond to the problem in hand.

5. Conclusions

This Letter has presented a comprehensive study of the classical dynamics of a planar model of a neutral two- particle Coulomb atom subjected to crossed electric and magnetic fields. The results indicate that the motional Stark effect can have profound consequences for the classical dynamics, including inducing a transition to chaos. The planar model is not only expected to be relevant to the dynamics of two-dimensional exciton models in the EMM but also should provide insight into the more general 3D problem of Rydberg atoms interacting with crossed electric and magnetic fields.

Acknowledgement

I thank an anonymous referee for pointing out Ref. [ 171. This work was supported, in part, by a grant from the donors of the Petroleum Research Fund administered by the American Chemical Society.

274 D. Farrelly /Physics Letters A 191 (1994) 265-274

References

[ 1 ] A. Holle, G. Wiebusch, G. Main, B. Hager, H. Rottke and K-H. Welge, Phys. Rev. Lett. 56 ( 1986) 2594. [2] H. Hasegawa, M. Robnik and G. Wunner, Prog. Theor. Phys. Suppl. 98 (1990) 198. [3] K-D. Krantzman, J.A. Mill&m andD. Farrelly, Phys. Rev. A 45 (1991) 3093. [ 41 P. Schmelcher and L.S. Cederbaum, Z. Phys. D 24 ( 1992 ) 3 11. [ 51 I. Dzyaloshinskii, Phys. Lett. A 165 (1992) 69. [6] D. Baye, N. Clerbaux and M. Vincke, Phys. Lett. A 166 (1992) 135. [7] B.R. Johnson, J.O. Hirschfelder andK.H. Yang, Rev. Mod. Phys. 55 (1988) 109. [8] Europhysics News 24 (1993) 165. [9] P. Schmelcher, J. Phys. B 25 (1992) 2697.

[lo] P. Schmelcher, Phys. Rev. B 48 (1992) 14 642. [ 111 W.R.S. Garton and ES. Tomkins, Astrophys. J. 158 ( 1969) 839. [ 121 J.C. Gay, D. Delande and F. Biraben, J. Phys. B 13 (1980) L720. [ 131 N.P. Economou, R.R. Freeman and P.F. Liao, Phys. Rev. A 18 (1978) 2506. [ 1410. Akimoto and H. Hasegawa, J. Phys. Sot. Japan 22 (1967) 181. [ 151 D.C. Rogers, J. Singleton, R.J. Nicholas, C.T. Foxton and K. Woodbridge, Phys. Rev. B 34 ( 1986) 4002. [ 161 M.M. Dignam and J.E. Sipe, Phys. Rev. B 45 ( 1992) 68 19. [ 171 L.P. Gor’kov and I.E. Dzyaloshinskii, Sov. Phys. JETP 26 (1968) 449;

L.A. Burkova, I.E. Dzyaloshinskii, G.F. Drukarev and B.S. Monozon, Sov. Phys. JETP 44 (1976) 276. [ 181 F. Penent, D. Delande and J.C. Gay, Phys. Rev. A 37 (1988) 4707. [ 191 G. Raithel, M. Fauth and H. Walther, Phys. Rev. A 44 ( 1991) 1898. [ 201 M.J. Gourlay, T. Uzer and D. Farrelly, Phys. Rev. A 47 ( 1993) 3 113. [21] A. Deprit, in: ThebigbangandGeorge Lemaitre, ed. A. Berger (Reidel, Dordrecht, 1984) pp. 151-180. [ 22 ] H. Goldstein, Classical mechanics (Addison-Wesley, Reading, 198 1) p. 380. [23] A. Deprit and B.R. Miller, Ann. NY Acad. Sci. 536 (1988) 101. [24] J.A. GriBiths and D. Farrelly, Phys. Rev. A 45 (1992) R2678. [25] J.A. Milligan and D. Farrelly, Phys. Rev. A 47 (1993) 3137. [26] K. Rzazewski and B. Pirau, Phys. Rev. A 47 ( 1993) R2468. [ 27 ] J. Zakrzewski, D. Delande, J.C. Gay and K. Rzazewski, Phys. Rev, A 47 ( 1993 ) R 16 12. [28] P. Kappertz and M. Nauenberg, Phys. Rev. A 47 ( 1993) 4749.