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HYDROGEN ATOM IN STRONG MAGNETICFIELDS : REGULAR AND IRREGULAR MOTIONS

M. Robnik

To cite this version:M. Robnik. HYDROGEN ATOM IN STRONG MAGNETIC FIELDS : REGULAR ANDIRREGULAR MOTIONS. Journal de Physique Colloques, 1982, 43 (C2), pp.C2-45-C2-61.�10.1051/jphyscol:1982205�. �jpa-00221814�

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JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°lla Tome 43, novembre 1982 page C2^45

HYDROGEN ATOM IN STRONG MAGNETIC FIELDS : REGULAR AND IRREGULAR

MOTIONS

M. Robnik

Institut fur Astrophysik, Univevsitixt Bonn, Auf dem Ettgel 71, D-S300 Bonn, F.R.G.

Résumé.- La transition stochastique dans le système hamiltonien classique associé à 1'atome d'hydrogène dans un champ magnétique intense est mise en évidence au moyen d'applications de Poincaré. Le résultat est discuté dans le contexte de la théorie générale de la destruction des tores. La découverte d'une troisième intégrale du mouvement lorsque l'énergie est inférieure à l'énergie critique peut permettre d'expliquer l'existence d'anticroisements exponentiellement petits des niveaux d'énergie lorsque le mouvement est régulier. Au-dessus de l'énergie critique, le mouvement stochastique est responsable de la structure irréguliêre du spectre dans le régime d'inter-n-mixing. La régularité du spectre dans le régime des résonances quasi-Landau est associée à l'existence d'invariants adiafaatiques au-dessus de la limite d'ionisation.

Abstract.- The stochastic transition in the classical Hamilton System describing the hydrogen atom in a strong magnetic field is observed by means of Poincaré mappings, and discussed in the framework of the gênerai theory of tori destruction. The existence of the third intégral in the regular région below the critical energy can ex-plain the exponentially small séparations at avoided crossings of the energy levels, while the stochastic motion above the critical energy accounts for the irreqular structure of the spectrum in the inter-n-mixing régime. The regularity of the quasi-Landau résonances corresponds to the existence of the adiabatic invariants above the escape energy.

1. Introduction.- There are many reasons that so much attention is being paid to the problem of the hydrogen atom in strong magnetic fields. For example, in astrophysics we have firm evidence that strongly magnetic white dwarfs can have magnetic fields up to 5.108G [1,2]. This has been inferred from polarization measurements in the optical continuum initiated by Kemp et al. [3]. On the polar caps of the neutron stars the field can be as high as 5.1013G. This value is in agreement with the observed pulsar spin-down rates, assuming that they are isolated rotating neutron stars with known moment of inertia £4J. Neutron stars in binary systems can accrete material flowing from its normal companion star. The material falling down onto the polar cap of the neutron star gains the gravitational potential energy of the order of 10% of its rest mass. This will be ultimately dissipated and radiated away. The high temperature of several 108K then means that we see the star in X-rays [5], and gives us a chance of a direct measurement of the surface magnetic field. Indeed, Trlimper et al [6} discovered an electron cyclotron line in the X-ray spectrum of the binary X-ray source Her X-l, from which they inferred B=5.1012G for the magnetic field. Other lines, e.g. FeXXVI, have been observed in X-ray binaries [7]. It should be noted that the observed magnetic fields in white dwarfs and in neutron stars are consistent with the predicted values, when these objects are assumed to be collapsed normal stars. Their high magnetic field is simply explained as a compressed initial magnetic field of some 10G, which occurs in some main-sequence stars. We are thus safe in believing that the magnetic fields in astrophysical objects can have strengths by a factor of 107 in excess of those available in laboratory.

A quantitative theoretical understanding of the physical and chemical properties of matter in strong magnetic fields is thus of great importance in astrophysics.

Note that in the astrophysical situations sketched above the magnetic field is so strong that the spacing between the Landau levels hoiju being the cyclotron fre-

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JOURNAL DE PHYSIQUE

quency, becomes comparable o r even greater than the spacings between the unperturbed atomiclevels. The magnetic i n t e r a c t i o n i s competing w i t h the e l e c t r o s t a t i c in te rac - t i o n o f comparable strength, r e f l e c t i n g a lso the approximate e q u a l i t y o f the Landau and atomic r a d i i . That p a r t o f the spectrum, where these condi t ions are met, has been c a l l e d the "s t rong f i e l d mix ing regime" [8] and i s be l ieved t o have some q u i t e general features common t o a l l s i tua t ions , where two equal ly s t rong forces of d i f - f e r e n t symmetries are a c t i n g r9J. For the hydrogen atom the c r i t i c a l magnetic f i e l d Bo i s def ined by the e q u a l i t y o f the i o n i z a t i o n energy i n vacuum o f 1Ry and o f the Landau energy fiw/2, so t h a t B0=a3 e2 / ro = 2.35.1096 (a=the f i n e s t r u c t u r e constant, ro=e2/mc2=the c l a s s i c a l e l e c t r o n radius) . For hydrogen-1 i ke ions w i t h the nuclear charge Ze the l i k e w i s e def ined c r i t i c a l magnetic f i e l d i s B0Z2. For FeXXVI i t equals 1.59.1012G, Bo i s a na tu ra l u n i t o f the magnetic f i e l d and i n the f o l l o w i n g the d i - mensionless s t reng th y := B/Bo w i l l be used. For hydrogen atoms i n s t rong cosmic magnetic f i e l d s , a l l l eve ls , i n c l u d i n g the ground leve l , can l i e w i t h i n the s t rong f i e l d mix ing regime. Moreover, i n case o f neutron s tars , t h i s i s marg ina l l y t r u e even f o r one-electron i r o n ions.

Now the naive quest ion may be asked: Why i s t h i s spect ra l problem so d i f f i c u l t ? We know the p o t e n t i a l , we have the Schrodinger equation, so i n p r i n c i p l e we should be able t o ca lcu la te the energy leve ls . However, the problem i s no t separable, no t even approximately, so t h a t the wonderful wor ld o f the per tu rba t ion theor ies appears i n e f f i c i e n t . Nevertheless, q u i t e accurate r e s u l t s on the lower l e v e l s o f the spectrum have been obtained using various numerical methods, so t h a t a l o t i s known now on the hydrogen spectrum i n s t rong magnetic f i e l d s . (The reader should consul t the review by Garstang [lo], h i s and the con t r ibu t ions by McDowell and by Wunner t o t h e proceedings o f t h i s conference.)

The basic method i s t o represent the Hamilton operator i n some complete basis. The basis should be simple t o handle, and should prov ide a r a p i d convergence o f t h e eigenvalues, so t h a t one can con t ro l the e r r o r i n s p i t e o f the necessary t runcat ion. When the hydrogen basis i s chosen, the f o l l o w i n g d i f f i c u l t y ar ises: The basis i s n o t complete i n the L * ( R ~ ) space o f i n tegrab le funct ions, and there fo re s u i t a b l e on ly f o r c a l c u l a t i n g the low l e v e l s . The basis could be completed by adding the continuum hydrogenic funct ions, b u t t h i s leads t o the p r a c t i c a l divergence d i f f i c u l t i e s i n c a l c u l a t i n g the m a t r i x elements. A s o l u t i o n t o t h i s problem has been r e c e n t l y ob- ta ined by C la rk e t a1 114, 15, 161. (See a lso h i s c o n t r i b u t i o n t o t h i s meeting.) Fol lowing a suggest ion by Edmonds 1171 they use a basis o f Sturmian r a d i a l functions, a l low ing p a r t i a l representat ion o f the continuous spectrum. A r a p i d convergence of even the h i g h l y exc i ted l e v e l s i n in termediate magnetic f i e l d s ( t y p i c a l l y 50kG) has been achieved. But the method i s equa l l y we l l su i ted i n cases o f ast rophysica l i n t e r e s t .

The s t rong f i e l d mix ing regime e x i s t s o f course a lso a t l abora to ry f i e l d strenghts (a p re fe r red value being y = 2.10-5), b u t then on ly f o r h i g h l y exc i ted l e v e l s , which are now observable thanks t o the new experimental techniques (see an extensive re - view by Gay [18]). To be s p e c i f i c , the re are several d i f f e r e n t regimes, def ined by the f o l l o w i n g a p r i o r i q u a l i t a t i v e c r i t e r i a : ( i ) inter-1-mixing, when the diamagnetic per tu rba t ion term HD o f the Hamil tonian i s much smal ler than the spacing between the unperturbed Coulomb l e v e l s 2Ry/n3; ( i i ) in ter -n-mix ing when they are comparable; ( i i i ) s t rong f i e l d m ix ing regime already defined; ( i v ) the Landau regime, when 2Ry/n3 i s much smal ler than the Landau spacing4o. While the nature and the o r i g i n of the equal ly spaced quasi-Landau resonances observed i n the s t rong f i e l d mix ing re - gime discovered by Garton and Tomkins [I91 can be explained semic lass ica l l y [8,9, 20-231, a l o t o f the s t r u c t u r e i n t h e in ter -n-mix ing regime must be explained, e.g. the observed precursors o f the quasi-Landau spectrum repor ted by Delande and Gay1241. Undoubtedly, the most remarkable s u b t l e t y o f the spectrum i s the existence o f close ant icross ings, which are revealed when the spectrum as a f u n c t i o n o f the magnetic f i e l d i s ca lcu la ted by h igh p r e c i s i o n numerical methods [25,26,16]. The ex is tence alone o f avoided crossings i s no t s u f f i c i e n t t o draw any conclusions, s ince the lev - e l repuls ions a re t y p i c a l o f almost a l l systems, even o f ergodic systems w i thou t any i n t e g r a l a t a l l . Berry 1271 gave the important example o f the S ina i b i l l i a r d r i g o r - ous ly known t o be ergodic and he ca lcu la ted t h e exact spectrum, showing many avoided

crossings. What matters i s the re fo re the f a c t t h a t the separat ion o f these near-degeneracies decreases exponent ia l ly w i t h the main quantum number n. This l e d t o the conjecture t h a t a hidden dynamical symmetry o f the Hamil tonian ex is ts , approximate b u t more and more accurate w i t h increasing n. Bearing i n mind t h a t degeneracies are t y p i c a l of separable systems, there was a hope t h a t the Schrodinger equation c a n 3 separated, a1 though approximately, f o r the h i g h l y exc i ted leve ls . Since the general understanding o f the correspondence between the f i n e s t r u c t u r e o f the spectra and t h e g lobal features o f the Hamil tonian systems i s s t i l l incomplete, i t seems appro- p r i a t e t o study the c l a s s i c a l system [28,21]: the hidden dynamical symmetry must show up as an add i t i ona l i n t e g r a l o f motion. The r e s u l t o f the numerical experiment, i n which the Poincar6 mappings were inspected, was no t as dramatic as the conjec- tures: The hydrogen atom i n a magnetic f i e l d i s a system o f t h e generic c lass. It disp lays a l l features o f an in tegrab le system f o r energies below some c r i t i c a l energy, becoming s tochas t i c very r a p i d l y upon increasing the energy above the c r i t i c a l energy. The present paper i s concerned mainly w i t h the consequences o f t h i s fact. I n any case, t h e g e n e r i c i t y o f the hydrogen atom i n a magnetic f i e l d i s a f o r t u n a t e circumstance, s ince many features are expected t o occur a lso i n more complex atoms.

The sequence o f the sect ions i s as fo l lows: a b r i e f review o f the c l a s s i c a l and quantum mechanical regu la r and i r r e g u l a r motion; presentat ion o f the Poincar6 mappings; p o s s i b i l i t i e s t o p r e d i c t the c r i t i c a l energy; approximate i n t e g r a l o f motion below the c r i t i c a l energy; correspondence between the c l a s s i c a l and the quantum system.

2. A b r i e f review o f the regu la r and i r r e g u l a r motion

Most o f the mater ia l o f t h i s sec t ion (and much more) can be found i n the recent reviews of Berry 1291 and by Zaslavsky [30]. Other reviews might be appreciated by i n t e r e s t e d readers [31,32,33J. Those f a m i l i a r w i t h the subject can s k i p t h i s sec t ion w i thou t any l o s s o f con t inu i t y .

The c l a s s i c a l Kepler problem i s a good text-book example o f an in tegrab le system. Owing t o i t s supersymmetry there are many more i n t e g r a l s o f motion than needed f o r i n t e g r a b i l i t y . By in tegrab le we mean t h a t there are a t l e a s t N (N = dimension o f the con f igu ra t ion space g lobal , a n a l y t i c a l , sing1 e-val ued and independent i n t e g r a l s o f motion, being i n in?o lu t ion , i .e . such t h a t a l l Poisson brackets vanish. By the im- p o r t a n t theorem s ta ted e x p l i c i t l y by Arnold [34] i n t e g r a b i l i t y impl ies t h a t each i n - va r ian t sur face i n phase space ( o f dimension 2N) must have the topology o f the N- dimensional to rus (sphere w i t h one handle). The phase space o f an in tegrab le system i s thus f i l l e d everywhere w i t h i n v a r i a n t t o r i . I n absence o f per turbat ions the motion i s conf ined t o an i n v a r i a n t torus, which j u s t i f i e s the a t t r i b u t e "s tab le system". I t i s then poss ib le t o use the t o p o l o g i c a l l y most na tu ra l canon ica l l y conjugate var ia- b les, namely t h e act ion-angle var iab les. The act ions Ij, 1 ( j 5 N, are def ined as

1 the i n t e g r a l s I := zJ 6.d: along i r r e d u c i b l e c i r c u i t s on the torus (these are j

loops, which cannot be s?hrunk t o a p o i n t ) . Theylabel a given torus, wh i le the pos i - t i o n o f a phase p o i n t on the to rus i s s p e c i f i e d by the angles o., 1 5 j 5 N, canoni- c a l l y conjugate t o t h e act ions I j . The angle 8 . hanges by Zn adong t h e closed ir- reduc ib le cyc le C j . The Hamil ton, an reads tl = A(+), so t h a t the angles are c y c l i c va r iab les by the const ruct ion. The equations o f motion

f = - a H aH + = 0 , @ = - =: ~ ( 7 ) = constant, a7

can be immediately in tegra ted t o y i e l d : Ij = constant, 0 - - w - t + constant, showing t h a t the motion on an i n v a r i a n t to rus i s quasiper iod ic . t o r i s i s c a l l e d r a t i o n a l (a l so resonant) o r i r r a t i o n a l according t o whether the frequencies w j = aH/aI. are r a t i o n a l l y connected o r not. I n the resonant case the motion can be p e r i o d i c ?see Fig. 1).

I n c i d e n t a l l y , f o r the Kepler problem a1 1 t o r i are resonant, s ince a1 1 frequencies depend on ly on the value o f the t o t a l energy, and are thus equal, as i s we l l known.

What happens t o t o r i o f an in tegrab le system under a small pe r tu rba t ion has been a long standing and most important quest ion o f c l a s s i c a l mechanics. Some people l i k e Landau thought t h a t a l l systems are in tegrable, b u t t h a t we are unable t o f i n d the i n t e g r a l s . The opposite extreme was favoured by Fermj , who be1 ieved t h a t in tegrab le

C2-4s J O U P d A L D E PHYSIQUE

systems a re exceptions (which i s co r rec t ) , and become ergodicl when s l i g h t l y per tu r - bed (which i s i n c o r r e c t ) .

lThere are several equiva lent d e f i n i t i o n s o f an ergodic system [35-37): (a) each i n - v a r i a n t s e t has e i t h e r measure zero o r one; the measure i s the normalized L i o u v i l l e measure, and the p roper ty i s c a l l e d met r i ca l t r a n s i t i v i t y ; ( b ) the L i o u v i l l e measure i s t h e on ly i n v a r i a n t abso lu te ly continuous measure; ( c ) the microcanonical d i s t r i - b u t i o n i s the on ly i n v a r i a n t d i s t r i b u t i o n ; (d) the t ime average o f each in tegrab le f u n c t i o n i s almost everywhere equal t o the phase average; i t can be most c l e a r l y seen from (c ) t h a t an ergodic Hamilton system can have no i n t e g r a l s o f motion except o f the Hamil tonian i t s e l f (Jeans Theorem!). Roughly speaking, i n an ergodic Hamilton system almost every p o i n t o f the 2N-1-dim energy surface w i l l v i s i t each neighbourhood o f each p o i n t on t h i s surface as t ime goes on.

As a su rp r i se came then a p a r t i a l answer due t o Kolmogorov, Arnold and Moser (KAM Theorem): A l l t o r i o f the unperturbed in tegrab le system H o ( I ) outs ide the resonant gaps surv ive a per tu rba t ion Eo+H = Ho + EV. The resonant gaps are centered around the r a t i o n a l t o r i , and t h e i r w id th i s determined by the i n e q u a l i t y 1353:

where K(E) i s some constant vanishing when E + 0, being the same f o r a l l t o r i , wh i le k j a re i n t e g e r numbers.The volume o f each gap i s small w i t h E, and summing up the volumes o f a l l gaps we s t i l l reach the same conclusion, s ince the ser ies i s conver- gent ( a >,,N-1). This i s the bas is f o r the condensed versions o f the KAM Theorem, such as: most t o r i survive", o r " a l l s u f f i c i e n t l y i r r a t i o n a l t o r i survive", a l t h o w they may be s l i g h t l y d is to r ted . However, t h i s theorem says noth ing about the motion w i t h i n the resonant gaps. No general p red ic t ions are known, b u t there are examples f o r the pers is tence o f even the r a t i o n a l t o r i , the f i r s t one having been demonstrated by Henon and Hei les [38]. But usua l l y t o r i w i t h i n the resonant gaps a re destructed, r e s u l t i n g i n a h ierarchy o f smal ler and smaller t o r i imbedded i n the chao t i c region.

sos Of integrr b2e sy~t-

He+ H=&+rV

@ mayw'fIca*ibn dJ OL eItpttc &tand:yl WtSu'tC k J w c h y

It i s extremely d i f f i c u l t t o p r e d i c t what i s a c t u a l l y going t o happen when a per tu rba t ion i s appl ied. One important concept f o r such an analys is i s the Poincar6 mapping o f a Surface of Sect ion. It i s a mapping o f some surface 2 i n phase space onto i t s e l f , generated by the phase f low. I f the energy i s the on ly i n t e g r a l o f mo- t i o n we know, SOS should be (2N-2)-dimensional. When each o r b i t o f given energy passes through the SOS, then the Poincar6 mapping w i l l g i ve f u l l in format ion on the s t a b i l i t y o f motion. For example, when N=2, the i r r a t i o n a l i n v a r i a n t t o r i w i l l appear i n the SOS as i n v a r i a n t curves, wh i le the resonant t o r i w i l l appear as a curve con- s i s t i n g o f p e r i o d i c po ints . Simple p e r i o d i c o r b i t s correspond t o f i x e d po in ts , wh i le a chain o f n p e r i o d i c po in ts i s c a l l e d n-cycle, and i s a f i x e d p o i n t o f the n - th i t e r a t e o f the Poincar'e mapping (see Fig. 1). One impor tant proper ty o f the Poincar6 mappings i s t h a t they a re area preserving.

The s t a b i l i t y analys is o f p e r i o d i c o r b i t s i s then reduced t o the s t a b i l i t y i n - v e s t i g a t i o n o f the Poincat-6 mapping o r o f i t s i t e r a t e s , i . e . the s t a b i l i t y o f f i x e d po in ts o f some area preserv ing mapping T. A f i x e d p o i n t xo o f an area preserv ing mapping T i s s a i d t o be s tab le i f f o r each neighbourhood U o f xo the re e x i s t s a sub- neighbourhood V 1 U such t h a t a l l i t e r a t e s T ~ ( v ) l i e i n U. P u t t i n g the coordinate o r i g i n t o xo and l i n e a r i z i n g T, we f i n d t h a t i t s l i n e a r p a r t L i s a 2x2 unimodular m a t r i x w i t h constant r e a l c o e f f i c i e n t s . I t s eigenvalues are thus e i t h e r complex con- jugates A, ?on the u n i t c i r c l e , o r rec ip roca ls A, l / A on the r e a l ax is . They are determined by A 2 - x Tr (L) - 1 = 0. I n the former case L describes e l l i p t i c ro ta t ion , the d i s c r e t e motion being confined t o e l l i p s e s ; xo i s c a l l e d an e l l i p t i c f i x e d p o i n t o f T. I n the l a t t e r case we have hyperbol ic r o t a t i o n , the motion being confined t o hyperbolae on the same o r in terchanging branches, depending on whether the eigenvalues are p o s i t i v e o r negative, respec t i ve ly . A hyperbol ic f i x e d p o i n t of T i s un- s tab le, and nearby o r b i t s separate exponent ia l ly . An e l l i p t i c f i x e d p o i n t i s s tab le (and nearby o r b i t s separate l i n e a r l y ) except i n cases o f low-order resonances, 1.e. i n cases t h a t the r o t a t i o n angle o f L i s 2a/m, m=1,2,3,4, where t h e l i n e a r s t a b i l i t y analys is i s n o t s u f f i c i e n t (Moser Twist Theorem [35]; T i s assumed t o be of c lass C6;) .

The s tab le e l l i p t i c is lands surrounding the e l l i p t i c f i x e d po in ts w i t h i n the otherwise chaot ic resonant gaps are sketched i n F ig. 1. How they come about upon a small pe r tu rba t ion o f an in tegrab le system i s explained by the Poincar6-Bi rkhof f theorem: when an area preserv ing mapping has a simple closed i n v a r i a n t curve consis t - i n g o f f i x e d points , then due t o the KAM Theorem and the area preserv ing property, an even number o f these f i x e d po in ts i s shown t o surv ive a small per turbat ion. H a l f o f them are e l l i p t i c , and t h e others are hyperbol ic . The almost, s e l f - s i m i l a r i n - f i n i t e h ierarchy i s explained by the f a c t t h a t a l l considerat ions above are v a l i d for any i t e r a t e o f the PoincarP mapping.

I t remains t o understand how the chao t i c motion can a r i se . This i s connected w i t h the hyperbol ic po in ts . A l l what has been done so f a r i s the l i n e a r i z a t i o n around a hyperbol ic f i x e d po in t . We s h a l l concentrate on the asymptotes o f the hyperbolae and the motion on them. We c a l l them incoming o r outgoing strand, according t o whether L acts as a con t rac t ion o r expansion, respec t i ve ly (F ig. 2). Ac tua l l y , these l i n e a r p a r t s can be extended i n t o the nonl inear reg ion o f T 1403, and are then c a l l e d s t a b l e and unstable manifold, respect ive ly . Now, the most impor tant t h i n g i s how they meet each other. I f they j o i n smoothly, noth ing special happens. But such a smooth j o i n i n g i s exceptional, s ince they w i l l gener i ca l l y ( i .e. i n almost a11 cases) meet t ransversa l l y a t a p o i n t which i s n o t a f i x e d po in t . When they do so, they w i l l do i t i n f i n i t e l y many times, as a l i t t l e r e f l e c t i o n shows. Moreover, due t o the area preserv ing proper ty o f the mapping the amplitude o f the o s c i l l a t i o n s w i l l become l a r g e r and la rger , so t h a t an extremely complex motion ar ises, which has l i t t l e i n common w i t h the laminar in tegrab le behaviour. Such an o s c i l l a t o r y motion i s c a l l e d homocl inic o s c i l l a t i o n . I t i s a prototype o f chaot ic motion. Indeed, i t can be shown t h a t no smooth i n t e g r a l o f motion e x i s t s fo r homocl inic o s c i l l a t i o n (by imbedding the Smale horseshoe mapping 1401 ) . We see t h a t hyperbol i c i t y + t ransversal i t y imply chaot ic motion. Since both occur i n almost a l l cases upon a per turbat ion, i f n o t otherwise, we get a f e e l i n g t h a t the type o f motion w i t h i n the resonant gaps, as shown i n F ig. 1, i s o f the generic c lass.

We can now understand the nature o f the extreme o f completely unstable systems, as opposed t o the s tab le in tegrab le system: They must be f u l l o f hyperbol ic p e r i o d i c o r b i t s . Indeed, an ergodic system can be and must be f u l l o f p e r i o d i c hyperbol ic

JOURNAL DE PHYSIQUE

o r b i t s . The i r measure i s zero, so t h a t t h i s does no t c o n t r a d i c t the proper ty (a). I n a c e r t a i n sense the p e r i o d i c o r b i t s , being a l l unstable, span the f low o f even an ergodic system. This f a c t i s n o t less s u r p r i s i n g than the approximation o f r e a l numbers by the r a t i o n a l s .

However, a generic Hamilton system i s n e i t h e r in tegrab le nor ergodic, both ex- tremes being exceptional. (We consider on ly systems w i t h a few degrees o f freedom. I t should be noted t h a t the ergodic systems are less exceptional than the in tegrab le ones, s ince they are s t r u c t u r a l l y s tab le: a small pe r tu rba t ion does n o t change the two fundamental proper t ies, hyperbol i c i t y and t ransversal i ty. ) To be s p e c i f i c , a t y p i c a l system disp lays a l l features o f an in tegrab le system a t low energies. A t some c r i t i c a l energy a r a t h e r sharp t r a n s i t i o n , the so-ca l led s tochas t i c t r a n s i t i o n occurs: the i n v a r i a n t t o r i a re observed t o disappear r a p i d l y as the energy i s increased, and the motion becomes i r r e g u l a r . When the motion i s numer ica l ly in tegra ted the best way t o observe the t r a n s i t i o n f rom t h e regu la r t o the i r r e g u l a r motion i s by means o f Poincar'e mappings. But i t i s very d i f f i c u l t t o p r e d i c t the c r i t i c a l energy. We s h a l l r e t u r n t o t h i s quest ion i n a l a t e r sect ion.

WorCA pi^& & exceptiond

Figure 2: When the stable and unstable manifolds o f an hyperbolic point H meet t r a n s versally a t a homoclinic point Po, then there i s i n f i n i t y o f homoclinic points PI,PZ, ... No smooth integral o f motion e x i s t s i n homoclinic oscil- lat ion. I t I S a prototype o f irregular motion.

While a l o t i s known on the c l a s s i c a l i r r e g u l a r motion, the study o f i t s quantum mechanical aspects i s now i n progress [29,30, and references therein]. The f a c t t h a t the Schrodinger equation i s known and i n p r i n c i p l e so lvable f o r any system helps as l i t t l e as the analogous f a c t o f the Hamilton-Jacobi equation i n the c l a s s i c a l case d id . What we need are more e x p l i c i t quan t i za t ion condi t ions, and t h i s i s the reason f o r a considerable r e v i t a l i z a t i o n o f the semiclassical mechanics. There a re two major problems t o be solved: f i r s t l y , the t ime evo lu t ion o f nonstat ionary wavefunc- t i ons , which i s ignored here; secondly, the p roper t ies o f the energy spectra. The l a t t e r problem has no analogy i n c l a s s i c a l mechanics.

Before I s t a r t rev iewing some general b u t fragmentary r e s u l t s , I should l i k e t o emphasize the f a c t t h a t nobody so f a r has c l e a r l y formulated what i s the quantum analogy o f the KAM Theorem, which must e x i s t , and should be a natura l bas is f o r the study o f s t o c h a s t i c i t y i n quantum mechanics. I be l ieve t h a t i t can be expressed i n terms o f i n t e g r a l s o f motion, i .e . i n terms o f commutators.

I n the f o l l o w i n g the terms "chaot ic" and " in tegrab le quantum system" a re meant t o imply t h a t they are i r r e g u l a r o r in tegrab le i n the c lass ica l l i m i t , respec t i ve ly . I n the r a r e cases t h a t a system i s in tegrab le we have the we l l def ined semic lass ica l quan t i za t ion condi t ions, i n i t i a t e d by Einste in , and r i g o r o u s l y developed by Maslov. The method i s known under the names: t o r i quant izat ion, topologica l quant izat ion, Maslov quant izat ion, EBK-quantization, . . . It i s j u s t the quan t i za t ion o f the actions,

Ij = ( rn .+c i j /4 )b J , (2)

where m j = 0,1,2,. . . , whi le a j i s the number o f caust ics encountered i n conf igurat ion

space upon t ravers ing the irreducible cyc le C j , and i s c a l l e d Maslov index. The spectrum i s then given by E$ = H(I3).

Un l i ke the c l a s s i c a l case, the quantum spectra l problem poses g rea t d i f f i c u l t i e s even i n the in tegrab le system: the exact quant izat ion can y i e l d an e x p l i c i t s o l u t i o n t y p i c a l l y when the system belongs t o the very specia l class o f separable systems. O f course t h i s touches the pure ly c lass ica l problem o f when i s an in tegrab le system a ls , separable. (Kaln ins and M i l l e r [41] have r e c e n t l y s tud ied the separat ion problem i n nonorthogonal systems.) To my knowledge there i s n o t even a s i n g l e example o f a E- - separable b u t in tegrab le system, whose exact spectrum i s known. So we do n o t know the gener ic f i n e s t r u c t u r e o f t h e i r spectra: are the degeneracies t y p i c a l j u s t as i n the s e p a r a b l ~ s t e m s , o r i s the exponent ia l ly small separat ion o f l e v e l s a t c lose a n t i crossings c h a r a c t e r i s t i c f o r them, as Berry i s suggesting [29].

So f a r no quan t i za t ion condi t ions as e x p l i c i t as (2 ) a re known f o r i r r e g u l a r systems. I t i s n o t unexpected t h a t a search f o r them i s d i rec ted towards the usage o f Feynman path i n t e g r a l s (o rd inary and phase space path i n t e g r a l s ) , s ince these present the most i n t i m a t e l i n k between the c l a s s i c a l and quantum dynamics. The approach has been 1 argely developed by Gutzwi 1 l e r [42].

To see how i t works, observe t h a t the Kernel K(qa, qb, t ) f o r the Schrodinger equation w i t h eigenvalues Ej can be expanded i n terms o f the eigenfunct ions $ j ( q ) ,

so t h a t i t s Four ie r t ransform (Green func t ion ) equals,

where 6(E-E .) are the Di rac d e l t a funct ions. P u t t i n g qa=qb, and i n t e g r a t i n g over qa, we f i n d t h a i the response d(E) i s equal t o the sum o f the d e l t a spikes:

Hence, knowing the kernel (3) we know the spectrum (5) . But Feynman has shown t h a t the kernel K(qa,qb,t) can be exac t l y given by the sum o f a l l paths going from qa a t t ime t = 0 t o qb a t t ime t,

~ ( q ~ , g ~ , t ) = J e i s [ a ~ b l / ~ ~ q ( t ) ( 6 )

where S[a,bJ = f ~ ( ~ , q , t ) d t i s t h e a c t i o n o f a path. I n the semic lass ica l l i m i t P + 0 two f a c t s are important. Since the con t r ibu t ions t o (6) w i l l d i sp lay r a p i d os- c i l l a t i o n s on ly the s t a t i o n a r y ones w i l l n o t cancel out, whence: ( i ) f o r f i x e d e-nd- po in ts a,b the s t a t i o n a r y pathsare exac t l y the c l a s s i c a l paths; ( i i ) the i d e n t i f i c a - t i o n qa=qb demands t h a t on ly those c l a s s i c a l paths, which are recur ren t i n conf igu- r a t i o n space should be taken. But since they should a lso be s ta t ionary w i t h respect t o v a r i a t i o n s o f the endpoints qa, qb, due t o i n t e g r a t i o n over qa i n passing from(4) t o (5), we reach the conclusion tha t : i n the semiclassical l i m i t on ly the closed ( i . e. pe r iod ic ) c l a s s i c a l paths con t r ibu te t o the response d(E). I n t h i s sense, the semic lass ica l spectrum i s spanned by the p e r i o d i c o r b i t s .

This seems paradoxical as the p e r i o d i c o r b i t s w i l l have as a r u l e measure zero i n phase space, no mat ter whether the system i s regu la r o r i r r e g u l a r . Zaslavsky [30] considers almost p e r i o d i c o r b i t s instead.

I n an in tegrab le system the per iod ic o r b i t s l i e on resonant t o r i , and these are p rec ise ly those allowed t o be destructed according t o the KAM Theorem. We see t h a t des t ruc t ion o f t o r i connected w i t h the onset o f chao t i c motion w i l l have important consequences f o r the (semic lass ica l ) spectrum. However, n o t a l l resonant t o r i are gener i ca l l y destroyed a t the same t ime upon a small per turbat ion: the des t ruc t ion begins f o r h igh energy t o r i , appearing a t lower and lower energies as the perturba-

JOURNAL DE PHYSIQUE

tion gets stronger. A warning i s in order concerning the semiclassical quantization resulting in the

response (5) : the quantization of the action of a single periodic classical o rb i t can give only accidentally a correct energy eigenvalue. Berry (291 points out that closed orbi ts and energy levels correspond to each other only collectively.

Notwithstanding some major disagreements [29,30] as concerns the quantization of irregular systems, there i s a generally accepted f ac t tha t the i r energy spectra are s t a t i s t i c a l i n nature. This comes not only from the methodologically s t a t i s t i ca l approach to the response ( 5 ) , as was the case in the early s t a t i s t i ca l theory of spectra of complex nuclei, developed by Wigner, Porter and Dyson. A jus t i f ica t ion fo r the s t a t i s t i ca l description i s believed to be found in the f ac t , t ha t t h i s i s the only meaningful description. Since the works by Percival (431 and Pomphrey 1441 , enough observational material has been gathered to be convinced that energy levels corresponding to the irregular motion are unstable with respect to small variations of some (family) parameters of the Hamiltonian. Since a l l real physical systems are actually ensembles with some scat ter in external conditions, such as magnetic f i e ld , the observed spectra of irregular systems will be rather smooth, reminiscent more of the continuous than of the discrete spectra. Zaslavsky [30] links th i s ins tabi l i - ty of levels directly to the ins tabi l i ty of the classical motion, via the K-entropy. (Roughly speaking, th is i s equal to the phase average of the largest Lyapunov exponent. I t measures the r a t e of increase in time of the coarsened phase volume, i .e . the ra te of entropy increase.) On these grounds the response formula (5) should be understood more as a distr ibution of energy levels rather than a quantization con- dit ion for individual levels.

I t i s then appropriate to define the probability density P(S) tha t the separation of two neighbouring levels i s equal to S (normalized to the mean separation of lev- e l s ) . For integrable systems there i s a generally accepted resul t that P(S)=exp(-S), reflecting the possible existence of level crossings. In th i s case the levels are clustered. In the irregular case, there i s agreement that P(S) i s given by some pow- e r law, P(S) = const. x sb. Berry 1293 suggests b = 1, while Zaslavsky argues tha t b = const./h, h being the Kolmogorov K-entropy of the corresponding classical system in the energy shell around the levels considered. As P(S) vanishes with S, t h i s accounts for the f ac t that level repulsions are typical of the irregular systems. Therefore, no clustering occurs and the spectra appear more random.

3. The classical Hamiltonian and the stochastic transit ion

In the following I shall consider the "naked problem" of the hydrogen atom in a magnetic f i e ld , which remains a f t e r dropping a l l e f fec ts except the diamagnetic, which i s essential for the transit ion from the regular to the irregular motion. Thus, the t r iv i a l paramagnetic term i s omitted; the nucleus i s assumed to have in f in i t e mass, so tha t no coupling between the electronic s ta tes and the CM-motion i s present; the CM i s assumed to be a t res t in a homogeneous magnetic f i e l d , accounting for the absence of the motional Stark ef fec t . Then the classical electron motion i s governed by the dimensionless Hamiltonian

1 + 2 1 2 2 1 H = p p + - y P - - 8 r 3 (7

2 where p 2 := x + y2. The unit of length i s equal to the Bohr radius, the energy i s measured in atomic units (har t ree) , and y i s the already defined dimensionlessmagnetic f i e ld strength, y = B/Bo. The only continuous sym~eir ies of (7) are time translation and rotation around the z-axis. The energy E = H(p,r) and the angular momentum ( i .e . , i t s z-component) L = xpy-ypx are two obvious integralsof motion. Since N = 3, a third integral i s needed fo r integrabil i ty.

Accounting for the invariance of L we can reduce ( 7 ) by using the cylindrical coordinates:

0

The three-dimensional parameter space ( E , L , V ) in which the Hamilton flow i s analyzed can be reduced by one by a simple stretching of variables: new coordinates = a x old coordinates, new momenta = ~x old momenta. A simple calculation shows that i t i s im-

possib le t o e l im ina te two parameters, t h e remaining p o s s i b i l i t i e s being: (A) a = const. y2l3, 0 = c ~ n s t . y - l / ~ , when L = 0, y f 0, w i t h the Hamil tonian

(B ) a = const. L-', = const. L, when L 0, w i t h the Hamil tonian

where a l l constants are independent o f L and y. We have taken them equal t o u n i t y . Here the same symbols are used f o r the new var iab les p , g , z,pz. The case (A) has been ex tens ive ly considered i n the works by Edmonds and Pu l len PI]. I s h a l l concen- t r a t e on case ( B ) o f nonvanishing angular momentum, where we have the f o l l o w i n g ex- p l i c i t s c a l i n g property: I f f o r L = 1 some fea tu re o f the f l o w (e.g. a b i f u r c a t i o n o f a p e r i o d i c o b i t ) o curs a t energy Ef(y), then a t any L + 0 the same fea tu re ap- h 5 oears a t E = L- EG(vL 1. I t i s thus s u f f i c i e n t t o exolore the svstem f o r L = 1. I \ . I

2 2 2 -1/2 The equ ipo ten t ia l l i n e s o f the p o t e n t i a l L'(p,z) := 1/& + A 0' -(o + z ) (see (8) and (10)) a r e p l o t t e d i n F ig . 3. I t s minimal value Emin = min U(P ,z) as a f u n c t i o n o f y i s given by

2 E . ( y ) = (2-3ho)/2ho mi n 3 (11

where A. := h(y/2), and t h e f u n c t i o n A(x) i s def ined as a p o s i t i v e r o o t o f t h e equa- t i o n x2X4 = l - A , SO t h a t hc[O,l]for non-negative y. A t y = 1 one has Emin=-0.3943046. The p o t e n t i a l l i n e s open a t the escape energy Eesc(y) = y/2, above which the e lect ron can escape the Coulomb p o t e n t i a l we l l , wh i le i t i s s t i l l t r a n s v e r s a l l y bounded by the magnetic f i e l d . Note t h a t Eesc i s no t equal t o the quantum mechanical i o n i z a t i o n energy Ei = y(L+1)/2, which i s h igher by e x a c t l y t h e Landau energy y/2, and does n o t obey ?!e sca l ing law. I f the e l e c t r o n escapes i n the d i r e c t i o n o f the magnetic m l d , then i t s k i n e t i c energy must be asympto t i ca l l y equal t o o r l a r g e r than the quantum mechanical zero p o i n t energy y/2. For t h i s reason the s ta tes w i t h energies between Ee and Eion are t r u e bo nd, l o c a l i z e d s tates, and n o t resonances.

For smaTF x we have i ( x ) z 1-xY, w h i l e ~ ( x ) z l/G as x >> 1. Therefore, 1 1 2 4

Emin = - 2(1 - y + d (y ) ) , wh i le (Eesc - E . ) -. as y'-. La te r we s h a l l min use the r e l a t i v e energy g(E), def ined by

F i g u r e 3: T h e e q u i p o t e n t i a l lines o f U(p , z ) w i t h y=L=l . T h e m i n i m a l e n e r g y i s e q u a l t o Emin=-0.3943046 a n d the es- c a p e e n e r g y Eesc=0.5

JOUKNAL DE PHYSIQUE

The dynamics of the systemhas been examined by investigating the Poincari! mappings on the SOS in the plane (p,pp) of the phase space. The bounding curve of the allowed region i s determined through pl = 2E - pi - ~ U ( P ,z = 0) = 0, and the motion on th i s invariant curve corresponds to the bouncing of the electron along the gradient l i ne z = 0 in Fig. 3, i .e . t o plane orbi ts in the 3-dimensional picture. A Poincari! mapping was examined by taking some typical in i t i a l conditions, and by plott ing the suff ic ient ly large number of i t s i t e ra t e s to see whether the phase point i s confined to some invariant curve, In Figure 4 we show SOS a t ten different energies for y = 1. A t low energies (Figs. 4a-b) the motion appears integrable, since SOS i s f i l l e d with invariant curves, which are nested around an e l l i p t i c fixed point. This corresponds to the simple periodic bouncing between the potential l ines of Fig. 3, but now approximately along the z-axis and exactly vertical a t z = 0 (because pp = 0 a t the fixed point). A t higher energy (Fig. 4c) another fixed point occurs, while a t s t i l l higher energy (Fig. 4d) bifurcations resulting in new s table periodic orbi ts have occured. Here the hyperbolic points can be also observed, and in vicinity of them there are f i r s t signs of unstable motion. By increasing energy th i s chain of hyperbolic points i s seen to become a broad belt of stochastic motion (Fig. 4e). There are several definit ions of the stochastic t rans i t ion , a l l being almost equivalent due to the f ac t that the transit ion i s f a i r ly sharp. (The definit ion in terms of the K- entropy, which becomes positive, seems to be the best, since i t refers t o a global property of the flow 1301. S t i l l other definit ion refers to the destruction of the " l a s t large scale torus".) I have adopted the cr i ter ion, that the c r i t i ca l energy i s reached when the stochastic motion becomes observable on the largest scale. According t o th i s definition the cr i t ica l energy Ecrit(y=l) l i e s between -0.04 and - 0.03.

:ri;Di 0

1 -I r f ' a 1 9 1

'f ". 'DO 1;DI a

I ? L

= - - ; '?

t -I a 1 , L a I T 2

I

\ _I -mo --- -u 9 s o .

C'; m r 5 .

F i g u r e 4 ( a - j ) : T h e Pozncari . s u r f a c e s o f s e c t i o n f o r y=L=l a t t e n d i f f e r e n t e n e r g i e s , E=-0.3 (a ) , -0 .1 (b ) , -0 .05 ( e l , - 0 . 0 4 (d ) , -0 .03 (e ) , -0 .01 ( f ) , O . ( g ) , 0.1 ( h ) , 0 .3 ( z ) ,O. 5 (j) . T h e i r r e g u l a r m o t i o n b e c o m e s " m a c r o s c o p i c a l l y v i s i b l e " b e t w e e n E=-0.04 and -0.03, s o t h a t Ecrit -0.035. A t h i g h e r e n e r g i e s , when t h e " l a s t " l a r g e s c a l e z n v a r i a n t t o r u s i s d e s t r u c t e d , the m o t l o n becomes c o m p l e t e l y stochastic.

Upon increasing the energy above the c r i t i c a l energy the i r r e g u l a r reg ion becomes l a r g e r and covers the whole energy sur face when t h e energy approaches the escape energy.

How can we understand the ex is tence o f i n v a r i a n t t o r i and the emergence o f the i r r e g u l a r motion on the other hand ? When the system i s viewed as a perturbed Kepler system ( w i t h a r a t h e r l a r g e per turbat ion, s ince y = 1) the KAM Theorem al lows a l l t o r i t o be destroyed, because a l l t o r i o f the Kepler problem are resonant t o r i . How- ever, we can look a t the system as being in tegrab le a t the bottom o f the p o t e n t i a l we1 1, where the p o t e n t i a l can be approximated by a two-dimensional harmonic o s c i l - l a t o r which i s resonant on ly a t some d i s c r e t e values o f y, f o r instance y = 0, 7&, ... The energy i t s e l f can be t r e a t e d as a per tu rba t ion parameter, and the KAM Theorem can be applied, p r e d i c t i n g now the existence o f t o r i f i l l i n g l a r g e regions of the energy surface, the excluded volume being small w i t h (E - Emin). Upon increas- i n g the energy the resonant gaps become wider u n t i l the chaot ic motion w i t h i n the gaps becomes macroscopical ly v i s i b l e . When the l a s t large-scale i n v a r i a n t torus i s destructed, the motion i s i r r e g u l a r on the whole energy surface.

I n Sect ion 2 i t was shown t h a t the hyperbol ic f i x e d po in ts are "germs" o f the chao t i c motion ( h y p e r b o l i c i t y + t r a n s v e r s a l i t y = homocl inic o s c i l l a t i o n ) . To expla in t h e l a t t e r i t i s necessary t o show t h e way how the system can become f u l l o f hyper- b o l i c f i x e d po in ts . I n t h e i r b e a u t i f u l work Greene, MacKay, V i v a l d i and Feigenbaum 1461 discover a universa l behaviour o f fami l i e s o f area preserv ing mappings, q u i t e analogous t o the p roper t ies o f one-dimensional d i s s i p a t i v e mappings. The generic case i s as fo l l ows : as a parameter o f an area preserv ing mapping i s va r ied a s tab le, e l l i p t i c f i x e d p o i n t ( o r p e r i o d i c p o i n t ) may go unstable and a t the same t ime s tab le p e r i o d i c o r b i t s o f tw ice the o r i g i n a l pe r iod are born ou t o f i t . With the parameter p vary ing the process can be repeated r e s u l t i n g i n a sequence o f e r i o d doubl ing b i - fu rca t ions , occur ing a t po,p ly . . . ,p jy . . . The seql;ence converges t!wards some value p*, and i s asympto t i ca l l y geometric as j + -, w i t h the universa l r a t e

as j + m, where the " resca l ing f a c t o r " 6 = 8.721097200 ... The explanat ion f o r t h i s un iversa l behaviour, independent o f the p a r t i c u l a r map, i s found i n the existence o f a un iversa l map ac t ing as an a t t r a c t o r i n the f u n c t i o n space, so t h a t as a parameter o f a s p e c i f i c fami l y o f maps i s varied, the i t e r a t e s o f the maps w i l l a t l e a s t l o - c a l l y converge t o the universa l mapping. This un iversa l map has o ther resca l ing proper t ies , such as universa l convergence r a t e s o f the pos i t i ons o f the b i f u r c a t e d f ixed po in ts . I n any case, the map i s f u l l o f hyperbol ic o r b i t s and does indeed exp la in i r r e g u l a r motion, which need n o t be ergodic, however. The d iscovery has p r a c t i c a l imp l i ca t ions , s ince the convergence i s very rap id: I f we observe, o r estimate, two per iod doubl ing b i f u r c a t i o n s o f the same sequence appearing a t the parameter values po and p l , then we can p r e d i c t the l i m i t p ,

beyond which the motion becomes i r r e g u l a r . (Note t h a t the l i m i t i n g behaviour (13) i s i n v a r i a n t w i t h respect t o the reparametr izat ion.) I n our case the parameter i s the energy and t h i s o f f e r s a p o s s i b i l i t y t o p r e d i c t the c r i t i c a l energy, discussed i n the next sect ion.

The s tochas t i c t r a n s i t i o n has been observed by inspec t ion o f the Poincari? mappings fo r o ther magnetic f i e l d strengths, and the r e s u l t i s shown i n F igure 5, where the c r i t i c a l energy i s p l o t t e d as a func t ion o f the magnetic f i e l d . The r e l a t i v e c r i t i - ca l energy (see (12)) g ( ~ c r i t ( y ~ 3 ) ) =: . u ( y ~ 3 ) , which depends only on yL3, i s p l o t t e d i n F igure 6. Here i t i s seen t h a t the i r r e g u l a r reg ion becomes smal ler i n the two in tegrab le 1 i m i t i n g cases ( y = 0, y = -), wh i le i t extends down t o almost Emin a t yL3 = 2.7.

The quest ion o f the l i m i t L + 0 connects the cases ( A ) and (B) (see (9,10)), and i s a t the same t ime r e l a t e d t o the l i m i t o f vanishing magnetic f i e l d . It i s reason- ab le t o assume t h a t some topo log ica l fea tu re o f the f l o w obeys a power law

JOURNAL DE PHYSIQUE

I ESCAPE REGION E > E, I

F i g u r e 6 : The r e l a t i v e c r i t i c a l e n e r g y p = g ( ~ ~ ~ ~ ) = p ( y ~ ~ ) , s e p a r a t i n g t h e r e g u l a r and i r r e g u l a r r e g i o n . The l a t t e r o c c u p z e s t h e l a r g e s t f r a c t i o n of t h e phase s p a c e around y ~ 3 =" 2.7.

- 2 as y + 0. The rescaling property of the energy then implies Ef = L ( y ~ 3 ) a , fo r any L#O. I f the l imit L -t 0 should exis t , then a > 2/3, or a = 2/3. In the former case the feature would approach the escape energy, while in t h a t t e r case i t becomes 5/i independent of L , and scales with the magnetic f i e ld as y . As regards the cr i t ica l energy E c r i t , 5dmonds and Pullen f211 have shown that in case t = 0 i t i s given by E c r i t = -0.5 y j3. The l imi t as L + 0 therefore exis ts , and does not nish, whence a c r i t = 213 as well. From this i t follows (Le - ECrjt) = O.5(y + y2YQ) as y - 0, with L = 1. Since the observed slope i s 0.87 ?see Figure 5 ) , the asymptotic r gion has not yet been reached completely. However, the theoretical slope 2/3 + y1/'/3 a t y = 0.1 equals 0.82, which i s not f a r from the observed value 0.87.

In the reference [28] the escape orbi ts have also been observed. Some fac ts are worth stressing: The motion near the plane z = 0 i s very irregular. Among some typical randomly chosen i n i t i a l conditions no trapped orbi ts were seen. I t i s conjec- tured that the measure of orbi ts which are trapped in a f i n i t e part of the phase space with energy above Ee i s negligibly small. However, suff ic ient ly f a r from the plane z = 0 the Coulomb po%ential becomes weaker, the motion becomes ordered and the adiabatic invariant $pp dp becomes more and more exact. In order to escape, the electron must have the appropriate value of the adiabatic invariant. If not, i t will be reflected and will return to the plane z = 0, where the value of the adiabatic invariant can change due to the chaotic motion. After several reflections the electron will ultimately escape in direction of the magnetic f i e ld . This i s i l lus t ra ted in Figure 7.

Figure 7(a-b) : The intersection points of an escape orbi t (E=0.6, y=L=l) w i t h the plane p =O (a). The same orbi t projected to the plane z=O(b).

P

4. Some poss ib i l i t ies to calculate the c r i t i ca l energy

The attempts to predict the cr i t ica l energy, i . e . t o reproduce the Figure 6 , by using the approximate analytical methods have been inconclusive so far . For instance, the considerations based on the curvature R ( i s negative R a valid cr i ter ion for stochastici ty?)

C2-58 JOURNAL DE PHYSIQUE

in the Jacobi metric, deri ed from the potential U, lead t o the conclusion that R Y i s always positive when y: >1/8, which disagrees with the observed stochastic tran- s i t i on a t y=L=l. That should not surprise us however, since the "negative curvature cri terion" i s known to f a i l in some cases [45,47,48,28].

The application of the Chiri kov cri terion of overlapping resonances 1321 as well as of the renormalization approach ( to the case of two resonances and the destruction of the " l a s t invariant torus" separating them) of Escande and Doveil [49] hinges upon the f ac t tha t i t i s not easy t o express the diamagnetic perturbation term by the Keplerian action-angle variab es in an expl ic i t way. Hence, the classical perturbation problem in the l imi t fi3 + 0 i s d i f f i c u l t t o t r e a t . Si i l a r l y , but fo r other reasons, one faces d i f f i cu l t i e s in the opposite extreme yLy + -. Here, the problem i s that the motions decouple.

The only case where the action-angle formalism i s easy to handle i s the neighbourhood of the c r i t i ca l point, i.e. near the bottom of the potential well. To the lowest approximation the Hamiltonian i s harmonic (being nonresonant except a t hO=h(yL3/2)=1,4/7 ,...), while the higher order terms in the power expansion of the potential can be treated as perturbations, and can be easily expressed in action- angle variables. A cri terion a l a Chirikov, saying that the stochastici ty will s e t in when the third-order and the fourth-order resonant terms have comparable ampli- tudes, results in

3 2 2 Ecr i t - Emin = (1 - h o ) / f l L X o (15)

which yields E c r i t = 0.05 when y = L = 1. The agreement with the observations de- scribed in the forego~ng section i s excellent. However, such an expansion i s e f f i - cient only when a few low-order terms represent a significant departure from an integrable system. This i s not t rue in the limiting cases v ~ 3 + 0,-, which renders the validity of eq. (15) to the neighbourhood of yL3=1. (As can be seen, the expression (15) does not have the correct asymptotic behaviour in these l imits.) I t i s not quite c lear whether the excellent agreement in t h i s region i s accidental, since the r i 5 - terion i s largely quali tat ive in nature. Nevertheless, i t shows tha t around yL %1 the expansion i s meaningful and could be made rigorous by a detailed study of the truncated Hamiltonian. I f we succeed t o construct a few s table periodic orbi ts and some of the i r period doubling bifurcations, then th i s would enable us to use the formula (13).

A f inal remark of th i s section concerns again the important case of the weak diamagnetic term, i .e . the case of the s l ight ly perturbed Kepler-system. The KAMTheo- rem admits the destruction of a l l t o r i , since a l l tor i of the Kepler problem are resonant, but we can rely on the structural s t a b i l i t y of the real physical systems, and believe, that only high energy tor i will be destroyed. To be more precise but s t i l l quali tat ive, we expect th i s to happe2 yhen the Coulomb pot t i a l and the diamagnetic potential are comparable, l/$% y p 18, whence r = 2<2'? This yields the correct scaling E r i t = -l/r = -0.5 y 1 3 , and (incidentally) also the correct numerical factor. ~ k e ?mportance i s not in rederiving a resul t of the previous section, but in showing that in the l imit of weak magnetic f i e ld , interesting fo r experimen- ta l investigations, the classical irregular region i s coincident with the inter-n- mixing regime.

5. An approximate integral of the motion below the cr i t ica l energy

The third integral of motion exis ts almost everywhere on the energy surface below the c r i t i ca l energy, except in th in , macroscopically invisible, gaps. I t can be exact there, as the KAM Theorem suggests, b u t i s certainly not analytic, and therefore d i f f i c u l t to construct. I t would be useful to find an approximate formal integral of motion, using the computer algebra to overcome the d i f f i cu l t i e s in formal expansions [50]. If three integrals E , L and F of the Hamil tonian ( 7 ) a re known, and are in in- volution, we can use the tor i quantization (2 ) .

In a,recent wogk Solov'ey [5~J+considgr~ tke time averages of the Poisson brack- e t s {H,L] and { H , A ) , where L = rxp, = pxL - r / r are the angular momentum and the Runge-Lenz vector. He finds two approximate integrals of motion Q and A ,

4 The i r mean v a r i a t i o n i n t ime i? o f the order y . (The Poisson brackets {H,Q] and { H A $ r e s t i l o the order y . ) When we in t roduce the s p l i t t i n g H=Ho + HD, HD=y p /8, andlus: the r e l a t i o n A ~ = ~ L ~ H ~ + 1, we see t h a t Q=-l/ZH,.

The t r i c k behind t h i s idea o f averaging i s connected w i t h the degeneracy o f the Kepler Hamiltonian. Because o f the degeneracy the i n v a r i a n t t o r i o f Ho are no t unique [343. I n other words, Kepler o r b i t s can be embedded i n t o several d i f f e r e n t systems o f nested t o r i . The p o i n t i s t o f i n d those t o r i on which the secular changes can be neglected. Indeed, LZ,Ho= -1/2Q and A are th ree independent i n t e g r a l s o f the Kepler problem, they are i n i n v o l u t i o n and thus def ine the i n v a r i a n t t o r i . The r e s u l t o f the averaging i s t h a t these t o r i are preserved t o the order y4 under the diamagnetic per tu rba t ion HD.

The semiclassical topolog 'ca l quan t i za t ion leads t o the usual r e s u l t f o r the Coulomb spectrum, Eo t -1/2n3, where n=1,2,. . . i s the main quantum number, wh i le the diamagnetic c o n t r i b u t i o n i s obtained by averaging HD over the torus l a b e l l e d by Lz, H,,A, and then expressed by the corresponding quantum numbers. Sol ov 'ev 's r e s u l t reads, 2 2

E = E + <HD> = - 2 2 + JLL (n2 + m + n hk) 0 2 l6 3 (17)

where Ak i s the quantized value o f the i n t e g r a l A, determined by

m 2 l 4 d e / n 2 ( l + A ) - T = 2n(k + 1/2).

1-5sin e s i n 6 (18)

Here m=L i s the quantized z-component o f the angular momentum, equal t o an in teger . The a r i a b l e e i n the in tegrand i s def ined as the angle between the Runge-Lenz vec- t o r "A and the z-axis. The in tegrand i s then the momentum canonica l ly conjugate t o 9. Solov'ev po in ts ou t the importance o f the pole i n eq. (18). It imp l ies t h a t t o r i belong t o two d i s t i n c t classes, according t o the s ign o f A. When pro jected down onto t h e con f igu ra t ion space, those w i t h A nega t i ve - l i e i n s i d e the cone 0 < e < eQ, c o t e0=2, wh i le others l i e i n the complementary cone. Hence, t o r i d i f f e r i n g i n the s ign o f A have nonoverlapping p ro jec t ions i n con f igu ra t ion space, thus the semiclas- s i c a l wavefunctions associated w i t h them are also nonoverlapping. Two l e v e l s corresponding t o such s tates a re the re fo re al lowed t o cross i n the semiclassical approximation. However, s ince the exact wavefunctions do no t vanish exac t l y beyond the caust ics, b u t decay exponent ia l ly instead, the corresponding mat r i x elements <$+IHgl$-> w i l l be exponent ia l ly small. These mat r i x elements f o r s ta tes w i t h i n an n-manifold w i t h constant m a re o f the order exp(-2n+m), a r e s u l t obtained by Solov'ev which i s i n b e a u t i f u l agreement w i t h the numerical ca lcu la t ions o f the s p l i t t i n g s a t the avoided crossings from the work by Delande and Gay 1261. (The s p l i t t i n g s are p ropor t iona l t o the mat r i x elements 1523.) The r e s u l t i s a n ice example i n favour of the conjecture by Berry [29] t h a t the exact spectra o f nonseparable in tegrab le sys- t e m s w i l l gener i ca l l y show exponent ia l ly small separat ions a t avoided crossings.

6. The correspondence between the c l a s s i c a l and the quantum system

Above the c r i t i c a l energy, i .e . i n the in ter -n-mix ing regime, the t o r i spanned by the i n t e g r a l s LZ,Ho and A w i l l begin t o break up, and when the l a s t large-scale torus i s destructed, the system wi 11 become completely s tochast ic . As a consequence, there w i l l be no caust ics and no l o c a l i z e d s tates, so t h a t the corresponding l e v e l s w i l l no longer be exponent ia l ly small separated, bu t w i l l d i sp lay avoided crossings t y p i - ca l of the ergodic systems. The spectrum w i l l be l a r g e l y i r r e g u l a r , which as been observed by Clark and Taylor n 6 1 t o appear a t t h e energy o f the order y2/' below the z e r o - f i e l d - i o n i z a t i o n l i m i t . The on ly " la rge scale torus" remaining i n t h i s ir- regu la r reg ion i s the boundary o f the Poincari! SOS (see F ig . 4). It corresponds t o plane o r b i t s i n z=O, i .e. along the gradient l i n e z=O o f F ig . 3, and i t s quant izat ion may exp la in the precursors o f quasi-Landau resonances repor ted by Delande and Gay [24]. Obviously above the escape energy such plane o r b i t s become even more important due t o the ad iaba t i c invar ian ts . The correspondence between t h e c l a s s i c a l regu la r and i r r e g u l a r motion and the s t r u c t u r e o f the quantum mechanical spectrum i s sham i n F igure8 f o r the case o f l abora to ry f i e l d s t rengths.

C2- 6 0 JOURNAL DE PHYSIQUE

T h e c l a s s i c a l m o t i o n

t r a p i d e s c a p e f r o m z=0 t o w a r d s a s y m p t o t i c a l l y

T h e q u a n t u m s p e c t r u m

LANDAU REGIME

REGULAR MOTION --,, , \ - \ -, s l o w , oscillating e s c a p e f r o m the p l a n e orbits z=0, d u e t o

ADIABATIC INVARIANTS

" -L - '1--'. 1- . --. '. ' \ ' s t r o n g f l e l d m i x i n g r e g i m e

EQUALLY SPACED QUASI-LANDAU RESONANCES ( 3 y/2)

ESCAPE REGION

, E EESC = ?,L/2 - - - - - - - - - - -

A t very s t rong magnetic f i e l d s , ? , > > I , the spectrum o f the hydrogen atom i s again Coulomb-like f o r s ta tes w i t h the same m, except f o r the ground state, which scales as l n 2 y. Apart from the ground s t a t e a l l l e v e l s are w i t h i n =; 1Ry below the ion iza t ion energy. Since the i o n i z a t i o n energy i s h igher than the escape energy by y/2, t h i s im- p l i e s t h a t a l l l e v e l s are above the i r r e g u l a r region, and t h a t the chaot ic features o f the spectrum w i l l even tua l l y disappear as ?, + m. However, i n whi te dwarfs y i s n o t h igher than 0.1, so t h a t there remains the p o s s i b i l i t y o f exp la in ing some con- t inuum- l ike features [1,2] o f t h e i r spectra by s t o c h a s t i c i t y involved i n the problem.

E I EION = y(m+l) /2

i n t e r - n - m i x i n g r e g i m e

USULL E I E E D CROSSINGS - - - - - - IRREGULAR FEATURES

IRREGULAR MOTION

d e s t r u c t i o n of tor i

- 2 / 3 E _C ECRIT = - 0 . 5 y

REGULAR MOTION

on i n v a r i a n t tor i

Acknowledgements

The author thanks the Sonderforschungsbereich Radioastronomie f o r f i n a n c i a l support. The valuable and s t i m u l a t i n g communications by J.C. Gay and C.W. C lark are g r a t e f u l l y acknowledged. Sincere thanks a re due t o Wi l l i am P. Reinhardt f o r h i s comments on the incompleteness o f the hydrogenic bas is (see the in t roduc t ion ) . The work has n o t been supported by any m i l i t a r y agency.

PRECURSORS OF QUASI-LANDAU RESONANCES

i n t e r - 1 - m i x i n g r e g i m e

EXPONENTIALLY SMALL SPLITTINGS

AT AVOIDED CROSSINGS

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