2.3 Approximation semi-classique
2.3.3 Prendre en compte les sym´etries
En outre, pour pouvoir ´etablir des comparaisons avec des r´esultats exp´erimentaux, j’ai
´et´e amen´e `a prendre en compte l’effet des sym´etries, quelles soient discr`etes (parit´e) ou
continues (invariance par rotation). Dans le premier cas, la th´eorie des groupes fournit
directement les modifications `a apporter pour calculer les corrections en ~ aux formules de
traces restreintes `a des ´etats appartenant `a une des repr´esentations du groupe de sym´etrie.
Le cas de l’invariance par rotation est plus d´elicat puisque, pour une valeur donn´ee du
mo-ment angulaire, il faut prendre en compte l’effet des termes centrifuges dans le hamiltonien,
typiquement ~
2L
2/2/r
2, qui dans la limite ~ → 0 ne modifient pas la dynamique classique,
mais contribuent aux corrections. L`a encore, j’ai montr´e comment prendre proprement en
compte ces termes, en particulier leur singularit´e en r = 0, pour calculer leur contribution
aux corrections au premier ordre en ~ aux formules de trace. Les comparaisons num´eriques,
dans le cas de l’hydrog`ene en champ magn´etique, ont montr´e l’excellent accord entre ces
pr´edictions et les calculs quantiques exacts.
Semiclassical analysis of real atomic spectra beyond Gutzwiller’s approximation
Benoît Grémaud*Laboratoire Kastler Brossel, Université Pierre et Marie Curie, Case 74, 4, place Jussieu, 75252 Paris Cedex 05, France
共Received 24 January 2005; published 13 October 2005兲
Real atomic systems, like the hydrogen atom in a magnetic field or the helium atom, whose classical dynamics are chaotic, generally present both discrete and continuous symmetries. In this paper, we explain how these properties must be taken into account in order to obtain the proper共i.e., symmetry projected兲 ប expansion of semiclassical expressions like the Gutzwiller trace formula. In the case of the hydrogen atom in a magnetic field, we shed light on the excellent agreement between present theory and exact quantum results. DOI:10.1103/PhysRevE.72.046208 PACS number共s兲: 05.45.Mt, 03.65.Sq, 32.30.⫺r
In the studies of the quantum properties of systems whose classical counterparts depict chaotic behavior, semiclassical formulas are essential links between the two worlds, empha-sized by Gutzwiller’s work 关1兴. More specifically, starting from Feynman’s path formulation of quantum mechanics, he has been able to express the quantum density of states as a sum over all共isolated兲 periodic orbits of the classical dynam-ics. This formula, and extensions of it, have been widely used to understand and obtain properties of the energy levels of many classically chaotic systems, among which is the hy-drogen atom in a magnetic field 关2,3兴, the helium atom 关4–6兴, or billiards 关7–10兴.
At the same time, because the trace formula共and its varia-tions兲 as derived by Gutzwiller only contained the leading term of the asymptotic expansion of the quantum level den-sity, the systematic expansion of the semiclassical propagator in powers of ប has been the purpose of several studies 关9–12兴, but which focused on billiards, for which both clas-sical and quantum properties are easier to calculate.
In a recent paper关13兴, general equations for efficient com-putation of ប corrections in semiclassical formulas for a cha-otic system with smooth dynamics were presented, together with explicit calculations for the hydrogen atom in a mag-netic field. However, only the two-dimensional case was considered, because for the three-dimensional共3D兲 case, dis-crete symmetries and centrifugal terms had to be taken into account. Actually, this situation occurs in almost all real atomic systems depicting a chaotic behavior共molecules, two electron atoms…兲, for which experimental data involve lev-els having well defined parity, total angular momentum, and, if relevant, exchange between particles. In particular, semi-classical estimations of experimental signals like photoion-ization cross sections are calculated with closed orbits with vanishing total angular momentum, whereas they usually in-volve P共L=1兲 quantum states, whose positions in energy are shifted with respect to S共L=0兲 states. Furthermore, in recent years, the development of the harmonic inversion method makes it possible to extract the relevant quantities共position of peaks, complex amplitudes兲 from both theoretical and ex-perimental data with a much higher accuracy than with the conventional Fourier transform关14兴. In particular, it becomes possible to measure the deviation of the exact quantum
re-sults from the semiclassical leading order predictions. Thus a detailed semiclassical analysis of experimental results, be-yond the leading order in ប, requires the understanding and the calculation of corrections due to both the discrete sym-metries and centrifugal terms. In addition, we would like to stress that even if the present analysis is made with the den-sity of states, it can also be made with the quantum Green function, which leads to expressions and numerical compu-tations of the first order ប corrections for physical quantities like the photoionization cross section关15,16兴, which could either be compared to available experimental data关17,18兴, or become a starting point for refined experimental tests of the quantum-classical correspondence in the chaotic regime.
បcorrections and discrete symmetries have already been discussed, but only for billiards 关9,10,12兴, whereas in the case of systems with smooth dynamics a detailed study is still lacking. Also, centrifugal terms and/or rotational sym-metries have been considered by many authors, but either in the case of integrable systems关19,20兴, or for values of the angular momentum comparable to the action of classical or-bits 关1,21,22兴. From this point of view, the present study, which focuses on fixed values of the quantum angular mo-mentum and the effect of the centrifugal terms on ប correc-tions for systems with smooth chaotic dynamics, goes be-yond the preceding considerations. More precisely, in this paper, we explain how to take into account both discrete symmetries and centrifugal terms in order to obtain a full semiclassical description of the first order ប corrections for the 3D hydrogen atom in a magnetic field.
At first, in the case of a chaotic system, whose Hamil-tonian H = p2/ 2 + V共q兲 is invariant under a group S of dis-crete transformations, the leading order of semiclassical approximation for the trace of the Green function G共E兲 = 1 /共E−H兲, restricted to the mth irreducible representation is given by关23兴 gm sc共E兲 =dm iប
兺
l Tl 兩Kl兩兺
n m共ln兲g共l,n兲共0兲共E兲 共1兲 with g共l,n兲共0兲共E兲 = 1 兩det共Aln − 1兲兩1/2exp冋
i បnSl− inl 2册
, 共2兲 where the l sum is taken over all primitive共isolated兲 orbits which become periodic through the symmetry operationl*Electronic address: Benoit.Gremaud@spectro.jussieu.fr
PHYSICAL REVIEW E 72, 046208共2005兲
关i.e., final position 共respectively, velocity兲 is mapped back to initial position共respectively, velocity兲 byl兴.m共ln兲 is the character oflnin the mth irreducible representation of di-mension dm. Slis the action of the orbit l ,lis the Maslov index, Tlis the “period,” Alnrepresents the Poincaré surface-of-section map linearized around the orbit, and Klis the sub-group ofS leaving each point of the orbit l invariant. Adding first order ប corrections, the preceding equation共1兲 becomes
gm sc共E兲 =dm iប
兺
l Tl 兩Kl兩兺
n m共ln兲g共0兲共l,n兲共E兲共
1 + iបC共l,n兲tr兲
. 共3兲 Cl,ntr can be derived by a detailed analysis of the stationary phase approximations starting from the Feynman path inte-gral, following the same steps as in Refs.关10,13兴 and reads as follows:
Cltr,n= ClT,n→E+ 1
nTl
冕
0nTldt0Cl,n共t0兲, 共4兲 where ClT,n→Earises from the time to energy domain transfor-mation. Cl,n共t0兲 共see Ref. 关13兴 for the expressions兲 involves the classical Green functionsGl,n共t,t⬘兲, i.e., the solutions of the equations controlling the linear stability around the clas-sical trajectory qlcl,n共t兲:
冉
−d 2 dt21 − 2V qq关qlcl,n共t兲兴冊
Gl,n共t,t⬘兲 = 1␦共t − t⬘兲. 共5兲 The fact that the orbits are periodic after the symmetry trans-formation ln determines the boundary conditions that the classical Green functionsGl,n共t,t⬘兲 must fulfill, namely,冦
l−nGl,n共nTl,t⬘兲 = Gl,n共0,t⬘兲Pt0Gl,n共0,t⬘兲 = 0 ∀ t⬘苸 关0, nTl兴, Qt0l−nG˙l,n共nTl,t⬘兲 = Qt0G˙l,n共0,t⬘兲
冧
共6兲 wherePt0is the projector along the “periodic” orbit at the position depicted by time t0andQt0= 1 − Pt0. Of course, for
l= 1, one recovers the boundary conditions given in Ref. 关13兴. Finally, all technical steps of Ref. 关13兴 leading to effi-cient computation ofGl,n共t,t⬘兲 and ប corrections, that is, so-lutions of sets of first order differential equations, can easily be adapted to take into account these modified boundary con-ditions.
As a numerical example, we have considered the 2D hy-drogen atom in a magnetic field, at scaled energy⑀= −0.1关2兴. More precisely, we have computed the trace of the quantum Green function, using roughly 8000 states belonging to the
EEErepresentation关24兴 of the group D4, corresponding to effective 1 / ប values ranging from 0 to 124共see Ref. 关13兴 for further details兲. In that case, the periodic orbit 1234 关25,26兴 共see inset of the top of Fig. 1 for the trajectory in semipara-bolic coordinates兲, being 共globally兲 invariant under a rotation of angle/ 2, gives rise to contributions in the semiclassical approximation of the trace at all multiples of S1234/ 4. In the same way, the periodic orbit 1243共see middle inset of Fig. 1兲 being invariant under a rotation of angle, contributions are
present at all multiples of S1243/ 2. For both these orbits, Table I displays the comparison of the present theoretical calculation and the numerical coefficientClHI,n
, extracted from the exact quantum Green function, using harmonic inversion 关13,14兴. As one can notice, the agreement is excellent for the amplitudes and rather good for the phases, which is the usual behavior of harmonic inversion. Furthermore, the same agreement has also been found for the other representations, thus emphasizing the present approach for the calculation of the first order ប corrections when taking into account discrete symmetries.
Contrary to the preceding, calculating first order ប correc-tions due to centrifugal terms is more complicated and is best explained in the case of the 3D hydrogen atom in a magnetic
FIG. 1.共Color online 兲 First order ប correction to the semiclas-sical approximation of the trace of the quantum Green function for the hydrogen atom in a magnetic field for different values of the magnetic number M , M = 1 / 2 corresponding to the 2D case关13兴. Crosses depict the values extracted from the exact quantum func-tion using harmonic inversion, whereas the solid line corresponds to the classical results given by Eq.共10兲. For the three different peri-odic orbits, whose trajectories in the共u,v兲 plane are plotted 共the nucleus being depicted by the black dot兲, the agreement is excellent, thus emphasizing the validity of Eqs.共9兲 and 共10兲.
TABLE I. Numerical comparison between the theoretical ប cor-rectionsCltr
for the trace of the quantum Green function, restricted to the EEE representation, of the 2D hydrogen atom in a magnetic field and the numerical coefficientsClHIextracted from exact quan-tum function using harmonic inversion. The agreement is excellent for the amplitudes and rather good on the phases, thus emphasizing the validity of the present approach.
Code Cltr 兩ClHI兩 Rel. error argClHI
1 41234 −0.094 430 0.09445 ⬇2⫻10−4 0.9996⫻ 1 21234 −0.361 689 0.3611 ⬇2⫻10−3 0.996⫻ 3 41234 −0.400 555 0.3992 ⬇3⫻10−3 1.005⫻ 1 21243 0.049 399 0.0493 ⬇8⫻10−4 −0.075⫻ 046208-2
field. The regularized Hamiltonian in semiparabolic coordi-nates, for fixed value M of the projection of the angular momentum along the field axis, is given by关2兴
H= −ប 2 2
再
2 u2+ 2 v2+冉
1 4−兩M兩2冊冋
1 u2+ 1 v2册冎
−⑀共u2+ v2兲 +1 8u 2 v2共u2+ v2兲 = H0+ប 2 2冉
兩M兩2−1 4冊
U共u,v兲. 共7兲H0is then the Hamiltonian of the 2D hydrogen atom in a magnetic field. If U共u,v兲 was regular, then the additional first order ប correction for the orbit l would simply be
−1 2
冉
兩M兩2−14
冊冕
0Tldt U„ul共t兲,vl共t兲…. 共8兲 One must mention that in this case, the Langer transforma-tion关27兴 of the coordinates 共u,v兲→(exp共−x兲,exp共−y兲) gives rise to a Hamiltonian which does not separate into kinetic and potential energies and for which no expressions for ប corrections are available.
On the other hand, the fact that U共u,v兲 is singular im-poses boundary conditions on both classical and quantum dynamics. The classical trajectories have to make共smooth兲 bounces near u = 0 and v = 0 and for vanishing values of ប, we expect the trajectories of H to be those of H0, but mapped onto the reduced phase space共u⬎0,v⬎0兲, i.e., making hard bounces on the共u,v兲 axis. From the quantum point of view, depending on the parity of M, only wave functions belonging to given representations of D4are allowed. Thus first order ប corrections due to the singular part of the potential U, are given by the preceding considerations on the symmetries, whereas remaining corrections are given by Eq.共8兲, where U has to be replaced by a smooth counterpart, namely,
U˜ = lim ⑀→0+ 1 2
冉
1 共u + i⑀兲2+ 1 共u − i⑀兲2+ 1 共v + i⑀兲2+ 1 共v − i⑀兲2冊
. 共9兲 Actually, one can show that the preceding equation gives the right answers for ប expansion of the propagator of the free particle共up to ប3兲 and the harmonic oscillator 共up to ប2兲, for which analytical expressions for classical trajectories, classi-cal Green functions, and quantum propagators exist共higher orders have not been checked yet兲. However, even if a de-tailed analysis of the derivation of the trace formula in pres-ence of centrifugal terms seems to show that the precedingapproach works in general cases, rigorous proof of Eq.共9兲 is lacking.
Nevertheless, in the case of the 3D hydrogen atom in a magnetic field, we have compared the first order ប correc-tions, for different periodic orbits and for different values of the magnetic number M, with the present prediction, namely,
Cltr共M兲 = Cl
tr共2D兲 −1
8共4兩M兩2− 1兲
冕
0TldtU˜ „ul共t兲,vl共t兲…. 共10兲 The results are displayed in Fig. 1 for M = 0,1,2 and for three different orbits, namely 1234, 1243, and 12343, whose tra-jectories in the共u,v兲 plane are plotted. The solid line is the theoretical result given by Eq.共10兲, whereas the crosses are the values extracted from the trace of the exact quantum Green function, using harmonic inversion共for scaled energy
⑀= −0.1, roughly 8000 effective 1 / ប values ranging from 0 to 124兲. As one can notice the agreement is excellent, thus giv-ing strong support for the validity of Eqs.共9兲 and 共10兲. Fur-thermore, the simplicity of the replacement U˜ may serve as a
guideline for a rigorous treatment of the ប corrections arising from the centrifugal terms. In particular, the calculation of higher orders involves products of the derivatives of these centrifugal terms and those of the potential V0, giving rise to nontrivial mixing between centrifugal and standard ប correc-tions.
In conclusion, we have presented a semiclassical analysis, beyond the usual Gutzwiller approximation, including first order ប corrections, of the quantum properties of real chaotic systems. More specifically, we have explained the additional corrections arising when taking into account both discrete symmetries and centrifugal terms. In the case of the共3D兲 hydrogen in a magnetic field, the agreement between the theory and the numerical data extracted from exact quantum results is excellent, emphasizing the validity of the analysis, especially of Eqs.共9兲 and 共10兲.
Finally, since we know how to compute the ប corrections, it would be very interesting to work the other way around, that is, to perform the semiclassical quantization, thus getting បcorrections in the semiclassical estimations of the quantum quantities, like the eigenenergies. Of course, this represents a more considerable amount of work, since the Cltr,ncoefficients must be computed for all relevant orbits and then included in standard semiclassical quantization schemes, like the cycle expansion关5,11,28兴.
The author thanks D. Delande for his kind support during this work. Laboratoire Kastler Brossel is Laboratoire de l’Université Pierre et Marie Curie et de l’Ecole Normale Supérieure, unité mixte de recherche 8552 du CNRS.
关1兴 M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics 共Springer, New York, 1990兲.
关2兴 H. Friedrich and D. Wintgen, Phys. Rep. 183, 37 共1989兲. 关3兴 D. Delande, in Chaos and Quantum Physics, edited by M.-J.
Giannoni, A. Voros, and J. Zinn-Justin, Les Houches Summer School, Session LII共North-Holland, Amsterdam, 1991兲. 关4兴 G. S. Ezra, K. Richter, G. Tanner, and D. Wintgen, J. Phys. B
24, L413共1991兲.
SEMICLASSICAL ANALYSIS OF REAL ATOMIC… PHYSICAL REVIEW E 72, 046208共2005兲
关5兴 P. Gaspard and S. A. Rice, Phys. Rev. A 48, 54 共1993兲. 关6兴 B. Grémaud and P. Gaspard, J. Phys. B 31, 1671 共1998兲. 关7兴 L. A. Bunimovitch, Commun. Math. Phys. 65, 295 共1979兲. 关8兴 P. Cvitanović and B. Eckhardt, Phys. Rev. Lett. 63, 823
共1989兲.
关9兴 P. Gaspard and D. Alonso, Phys. Rev. A 47, R3468 共1993兲. 关10兴 P. Gaspard, D. Alonso, and I. Burghardt, Adv. Chem. Phys.
90, 105共1995兲.
关11兴 G. Vattay and P. E. Rosenqvist, Phys. Rev. Lett. 76, 335 共1996兲.
关12兴 K. Weibert, J. Main, and G. Wunner, Eur. Phys. J. D 19, 379 共2002兲.
关13兴 B. Grémaud, Phys. Rev. E 65, 056207 共2002兲. 关14兴 J. Main, Phys. Rep. 316, 233 共1999兲.
关15兴 E. P. Bogomolny, Zh. Eksp. Teor. Fiz. 96, 487 共1989兲 关Sov. Phys. JETP 69, 275共1989兲兴.
关16兴 J. Gao and J. B. Delos, Phys. Rev. A 46, 1455 共1992兲. 关17兴 A. Holle, J. Main, G. Wiebusch, H. Rottke, and K. H. Welge,
Phys. Rev. Lett. 61, 161共1988兲.
关18兴 J. Main, G. Wiebusch, K. Welge, J. Shaw, and J. B. Delos, Phys. Rev. A 49, 847共1994兲.
关19兴 L. S. Schulmann, Techniques and Applications of Path
Inte-gration共Wiley, New York, 1981兲.
关20兴 K. C. Khandekar and S. V. Lawande, J. Math. Phys. 16, 384 共1975兲.
关21兴 S. C. Creagh and R. G. Littlejohn, Phys. Rev. A 44, 836 共1991兲.
关22兴 S. C. Creagh and R. G. Littlejohn, J. Phys. A 25, 1643 共1992兲. 关23兴 J. M. Robbins, Phys. Rev. A 40, 2128 共1989兲.
关24兴 C. C. Martens, R. L. Waterland, and W. P. Reinhardt, J. Chem. Phys. 90, 2328共1989兲.
关25兴 B. Eckhardt and D. Wintgen, J. Phys. B 23, 355 共1990兲. 关26兴 K. T. Hansen, Phys. Rev. E 51, 1838 共1995兲. 关27兴 R. E. Langer, Phys. Rev. 51, 669 共1937兲.
关28兴 P. Cvitanović and B. Eckhardt, Phys. Rev. Lett. 63, 823 共1989兲.