HYDROGEN ATOM IN STRONG MAGNETIC FIELDS : REGULAR AND IRREGULAR MOTIONS

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HYDROGEN ATOM IN STRONG MAGNETIC FIELDS : REGULAR AND IRREGULAR MOTIONS

M. Robnik

To cite this version:

M. Robnik. HYDROGEN ATOM IN STRONG MAGNETIC FIELDS : REGULAR AND IRREGULAR MOTIONS. Journal de Physique Colloques, 1982, 43 (C2), pp.C2-45-C2-61.

�10.1051/jphyscol:1982205�. �jpa-00221814�

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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'ex- pliquer 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 exist- ence 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 op- tical 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 temper- ature 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 consist- ent 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 be- lieving 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-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982205

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quency, becomes comparable o r even g r e a t e r than t h e spacings between t h e unperturbed atomiclevels. The magnetic i n t e r a c t i o n i s competing w i t h t h e e l e c t r o s t a t i c i n t e r a c - t i o n o f comparable strength, r e f l e c t i n g a l s o t h e approximate e q u a l i t y o f t h e Landau and atomic r a d i i . That p a r t o f the spectrum, where these c o n d i t i o n s a r e met, has been c a l l e d t h e " s t r o n g f i e l d m i x i n g regime" [8] and i s b e l i e v e d t o have some q u i t e general f e a t u r e s common t o a l l s i t u a t i o n s , where two e q u a l l y s t r o n g forces of d i f - f e r e n t symmetries a r e a c t i n g r 9 J . For the hydrogen atom t h e c r i t i c a l magnetic f i e l d Bo i s d e f i n e d by t h e 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 t h e Landau energy fiw/2, so t h a t B0=a3 e 2 / r o = 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 r a d i u s ) . For hydrogen-1 i ke i o n s w i t h t h e nuclear charge Ze t h e l i k e w i s e d e f i n e d 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 n a t u r a l u n i t o f t h e magnetic f i e l d and i n t h e f o l l o w i n g t h e d i - mensionless s t r e n g t h y := B/Bo w i l l be used. F o r hydrogen atoms i n s t r o n g cosmic magnetic f i e l d s , a l l l e v e l s , i n c l u d i n g t h e ground l e v e l , can l i e w i t h i n t h e s t r o n g f i e l d m i x i n g regime. Moreover, i n case o f neutron s t a r s , t h i s i s m a r g i n a l l y t r u e even f o r one-electron i r o n ions.

Now t h e n a i v e q u e s t i o n may be asked: Why i s t h i s s p e c t r a l problem so d i f f i c u l t ? We know t h e p o t e n t i a l , we have t h e Schrodinger equation, so i n p r i n c i p l e we should be a b l e t o c a l c u l a t e t h e energy l e v e l s . However, t h e problem i s n o t separable, n o t even approximately, so t h a t t h e wonderful w o r l d o f t h e p e r t u r b a t i o n t h e o r i e s 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 t h e l o w e r l e v e l s o f t h e spec- trum have been obtained using various numerical methods, so t h a t a l o t i s known now on t h e hydrogen spectrum i n s t r o n g magnetic f i e l d s . (The reader should c o n s u l t the review by Garstang

[lo],

h i s and t h e c o n t r i b u t i o n s by McDowell and by Wunner t o t h e proceedings o f t h i s conference.)

The b a s i c method i s t o represent t h e Hamilton o p e r a t o r i n some complete basis.

The b a s i s should be simple t o handle, and should p r o v i d e a r a p i d convergence o f t h e eigenvalues, so t h a t one can c o n t r o l t h e e r r o r i n s p i t e o f t h e necessary t r u n c a t i o n . When t h e hydrogen b a s i s i s chosen, t h e f o l l o w i n g d i f f i c u l t y a r i s e s : The basis i s n o t complete i n t h e L * ( R ~ ) space o f i n t e g r a b l e f u n c t i o n s , and t h e r e f o r e s u i t a b l e o n l y f o r c a l c u l a t i n g t h e low l e v e l s . The b a s i s c o u l d be completed by adding t h e continuum hydrogenic f u n c t i o n s , 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- t a i n e d by C l a r k e t a1 114, 15, 161. (See a l s o h i s c o n t r i b u t i o n t o t h i s meeting.) F o l l o w i n g a suggestion by Edmonds 1171 they use a basis o f Sturmian r a d i a l functions, a l l o w i n g p a r t i a l r e p r e s e n t a t i o n o f t h e continuous spectrum. A r a p i d convergence of even t h e h i g h l y e x c i t e d l e v e l s i n i n t e r m e d i a t e magnetic f i e l d s ( t y p i c a l l y 50kG) has been achieved. But t h e method i s e q u a l l y w e l l s u i t e d i n cases o f a s t r o p h y s i c a l i n t e r e s t .

The s t r o n g f i e l d m i x i n g regime e x i s t s o f course a l s o a t l a b o r a t o r y f i e l d strenghts ( a p r e f e r r e d value being y = 2.10-5), b u t then o n l y f o r h i g h l y e x c i t e d l e v e l s , which are now observable thanks t o t h e new experimental techniques (see an extensive r e - view by Gay [18]). To be s p e c i f i c , t h e r e are several d i f f e r e n t regimes, d e f i n e d by t h e 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 t h e diamagnetic p e r t u r b a t i o n term HD o f t h e Hamiltonian i s much s m a l l e r than t h e spacing between t h e unperturbed Coulomb l e v e l s 2Ry/n3; ( i i ) i n t e r - n - m i x i n g when t h e y are comparable;

( i i i ) s t r o n g f i e l d m i x i n g regime already defined; ( i v ) t h e Landau regime, when 2Ry/n3 i s much s m a l l e r than t h e Landau spacing4o. While t h e n a t u r e and t h e o r i g i n of t h e e q u a l l y spaced quasi-Landau resonances observed i n t h e s t r o n g f i e l d m i x i n g r e - gime discovered by Garton and T o m k i n s [I91 can be explained s e m i c l a s s i c a l l y [8,9, 20-231, a l o t o f t h e s t r u c t u r e i n t h e i n t e r - n - m i x i n g regime must be explained, e.g.

t h e observed precursors o f t h e quasi-Landau spectrum r e p o r t e d by Delande and Gay1241.

Undoubtedly, t h e most remarkable s u b t l e t y o f the spectrum i s t h e existence o f close a n t i c r o s s i n g s , which a r e revealed when t h e spectrum as a f u n c t i o n o f t h e magnetic f i e l d i s c a l c u l a t e d by h i g h p r e c i s i o n numerical methods [25,26,16]. The e x i s t e n c e alone o f avoided crossings i s n o t s u f f i c i e n t t o draw any conclusions, s i n c e t h e l e v - e l r e p u l s i o n s a r e t y p i c a l o f almost a l l systems, even o f ergodic systems w i t h o u t any i n t e g r a l a t a l l . Berry 1271 gave t h e important example o f t h e S i n a i b i l l i a r d r i g o r - o u s l y known t o be e r g o d i c and he c a l c u l a t e d t h e exact spectrum, showing many avoided

crossings. What matters i s t h e r e f o r e t h e f a c t t h a t t h e s e p a r a t i o n o f these near-de- generacies decreases e x p o n e n t i a l l y w i t h t h e main quantum number n. This l e d t o t h e c o n j e c t u r e t h a t a hidden dynamical symmetry o f t h e Hamiltonian e x i s t s , approximate b u t more and more accurate w i t h i n c r e a s i n g n. Bearing i n mind t h a t degeneracies a r e t y p i c a l of separable systems, t h e r e was a hope t h a t t h e Schrodinger equation c a n 3 separated, a1 though approximately, f o r t h e h i g h l y e x c i t e d l e v e l s . Since t h e general understanding o f the correspondence between t h e f i n e s t r u c t u r e o f t h e spectra and t h e g l o b a l f e a t u r e s o f t h e Hamiltonian systems i s s t i l l incomplete, i t seems appro- p r i a t e t o study t h e c l a s s i c a l system [28,21]: t h e hidden dynamical symmetry must show up as an a d d i t i o n a l i n t e g r a l o f motion. The r e s u l t o f t h e numerical experiment, i n which t h e Poincar6 mappings were inspected, was n o t as dramatic as t h e 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 l a s s . It d i s p l a y s a l l f e a t u r e s o f an i n t e g r a b l e system f o r energies below some c r i t i c a l ener- gy, becoming s t o c h a s t i c very r a p i d l y upon i n c r e a s i n g t h e energy above t h e c r i t i c a l energy. The present paper i s concerned mainly w i t h t h e 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 t h e 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 i n c e many f e a t u r e s are expected t o occur a l s o i n more complex atoms.

The sequence o f t h e sections i s as f o l l o w s : a b r i e f review o f the c l a s s i c a l and quantum mechanical r e g u l a r and i r r e g u l a r motion; p r e s e n t a t i o n o f t h e Poincar6 mappings; p o s s i b i l i t i e s t o p r e d i c t t h e c r i t i c a l energy; approximate i n t e g r a l o f motion below t h e c r i t i c a l energy; correspondence between t h e c l a s s i c a l and t h e quan- tum system.

2. A b r i e f review o f t h e r e g u l a r and i r r e g u l a r motion

Most o f the m a t e r i a l o f t h i s s e c t i o n (and much more) can be found i n t h e r e c e n t 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 s u b j e c t can s k i p t h i s s e c t i o n w i t h o u t any l o s s o f c o n t i n u 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 i n t e g r a b l e system.

Owing t o i t s supersymmetry t h e r e a r e 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 i n t e g r a b l e we mean t h a t t h e r e a r e a t l e a s t N (N = dimension o f t h e c o n f i g u r a t i o n space g l o b a l

,

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 i n ? o l u t i o n , i . e . such t h a t a l l Poisson brackets vanish. By t h e im- p o r t a n t theorem s t a t e d 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 i m p l i e s t h a t each i n - v a r i a n t s u r f a c e i n phase space ( o f dimension 2N) must have t h e topology o f t h e N- dimensional t o r u s (sphere w i t h one handle). The phase space o f an i n t e g r a b l e 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 p e r t u r b a t i o n s t h e motion i s confined t o an i n v a r i a n t torus, which j u s t i f i e s t h e a t t r i b u t e " s t a b l e system". I t i s then p o s s i b l e t o use t h e t o p o l o g i c a l l y most n a t u r a l c a n o n i c a l l y conjugate v a r i a - b l e s , namely t h e action-angle v a r i a b l e s . The a c t i o n s Ij, 1 ( j 5 N, a r e d e f i n e d as t h e i n t e g r a l s I : =

z J

1 6.d: along i r r e d u c i b l e c i r c u i t s on t h e t o r u s (these are

j

loops, which cannot be s?hrunk t o a p o i n t ) . Theylabel a given t o r u s , w h i l e t h e p o s i - t i o n o f a phase p o i n t on t h e t o r u s i s s p e c i f i e d by t h e angles o., 1 5 j 5 N, canoni- c a l l y conjugate t o t h e a c t i o n s I j . The angle 8 . hanges by Zn adong t h e closed ir- r e d u c i b l e c y c l e C j

.

The Hamil ton, an reads tl = A(+), so t h a t t h e angles are c y c l i c v a r i a b l e s by t h e c o n s t r u c t i o n . The equations o f motion

f = - a H _{= 0}

,

@ =

-

aH ^=: + ~ ( 7 ) = constant,

a7

can be immediately i n t e g r a t e d t o y i e l d : I j = constant, 0 -

-

w - t

+

constant, showing t h a t t h e motion on an i n v a r i a n t t o r u s i s q u a s i p e r i o d i c . t o r i s i s c a l l e d r a t i o n a l

( a l s o resonant) o r i r r a t i o n a l according t o whether t h e frequencies w j = aH/aI. are r a t i o n a l l y connected o r not. I n t h e resonant case t h e motion can be p e r i o d i c ?see F i g . 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 i n c e a1 1 frequencies depend o n l y on t h e value o f t h e t o t a l energy, and a r e thus equal, as i s w e l l known.

What happens t o t o r i o f an i n t e g r a b l e system under a small p e r t u r b a t i o n has been a l o n g standing and most important question 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 i n t e g r a b l e , b u t t h a t we a r e unable t o f i n d t h e i n t e g r a l s . The opposite extreme was favoured by Fermj

,

who be1 i e v e d t h a t i n t e g r a b l e

C2-4s J O U P d A L D E P H Y S I Q U E

systems a r e exceptions (which i s c o r r e c t ) , and become ergodicl when s l i g h t l y p e r t u r - bed (which i s i n c o r r e c t ) .

lThere are several e q u i v a l e n t 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 t h e normalized L i o u v i l l e measure, and the p r o p e r t y i s c a l l e d m e t r i c a 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 o n l y i n v a r i a n t a b s o l u t e l y continuous measure; ( c ) t h e microcanonical d i s t r i - b u t i o n i s t h e o n l y i n v a r i a n t d i s t r i b u t i o n ; ( d ) t h e t i m e average o f each i n t e g r a b l e 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 t h e Hamiltonian 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 t h e 2N-1-dim energy surface w i l l v i s i t each neighbour- hood o f each p o i n t on t h i s surface as time goes on.

As a s u r p r i s e 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 i n t e g r a b l e system H o ( I ) o u t s i d e t h e resonant gaps s u r v i v e a p e r t u r b a t i o n Eo+H = Ho

+

EV. The resonant gaps a r e centered around the r a t i o n a l t o r i , and t h e i r w i d t h i s determined by t h e 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 , w h i l e k j a r e 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 t h e volumes o f a l l gaps we s t i l l reach t h e same conclusion, s i n c e t h e s e r i e s i s conver- gent ( a >,,N-1). T h i s i s t h e b a s i s 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 i s t o r t e d . However, t h i s theorem says n o t h i n g about the motion w i t h i n the resonant gaps. No general p r e d i c t i o n s a r e known, b u t t h e r e a r e examples f o r t h e p e r s i s t e n c e o f even t h e r a t i o n a l t o r i , t h e f i r s t one having been demonstrated by Henon and H e i l e s [38]. But u s u a l l y t o r i w i t h i n t h e resonant gaps a r e destructed, r e s u l t i n g i n a h i e r a r c h y o f s m a l l e r and smaller t o r i imbedded i n the c h a o t i c region.

sos Of

integrr b2e s y ~ t -

He+ H=&+rV

@

mayw'fIca*ibn dJ

OL

eItpttc &tand:yl WtSu'tC

k J w c h y

I t 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 p e r t u r b a t i o n i s applied. One important concept f o r such an a n a l y s i s i s t h e Poincar6 mapping o f a Surface

of

S e c t i o n . It i s a mapping o f some surface

2

i n phase space onto i t s e l f , generated by t h e phase f l o w . I f t h e energy i s t h e o n l y 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 t h e SOS, then t h e Poincar6 mapping w i l l g i v e f u l l i n f o r m a t i o n on t h e s t a b i l i t y o f motion. For example, when N=2, t h e 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 t h e SOS as i n v a r i a n t curves, w h i l e t h e 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 p o i n t s . Simple p e r i o d i c o r b i t s correspond t o f i x e d p o i n t s , w h i l e a chain o f n p e r i o d i c p o i n t s 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 - t h i t e r a t e o f t h e Poincar'e mapping (see Fig. 1). One i m p o r t a n t p r o p e r t y o f t h e Poincar6 mappings i s t h a t they a r e area preserving.

The s t a b i l i t y a n a l y s i s o f p e r i o d i c o r b i t s i s then reduced t o t h e s t a b i l i t y i n - v e s t i g a t i o n o f t h e Poincat-6 mapping o r o f i t s i t e r a t e s , i . e . t h e s t a b i l i t y o f f i x e d p o i n t s o f some area p r e s e r v i n g mapping T. A f i x e d p o i n t xo o f an area p r e s e r v i n g mapping T i s s a i d t o be s t a b l e i f f o r each neighbourhood U o f xo t h e r e 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 t h e u n i t c i r c l e , o r r e c i p r o c a l s A, l / A on t h e r e a l a x i s . They are determined by A 2 - x T r ( L )

-

1 = 0. I n t h e former case L describes e l l i p t i c r o t a t i o n , t h e 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 t h e l a t t e r case we have h y p e r b o l i c r o t a t i o n , the motion being confined t o hyperbolae on t h e same o r i n t e r c h a n g i n g branches, depending on whether the eigen- values are p o s i t i v e o r negative, r e s p e c t i v e l y . A h y p e r b o l i c f i x e d p o i n t of T i s un- s t a b l e , and nearby o r b i t s separate e x p o n e n t i a l l y . An e l l i p t i c f i x e d p o i n t i s s t a b l e (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 t h e 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 a n a l y s i s i s n o t s u f f i c i e n t (Moser T w i s t Theorem [35]; T i s assumed t o be of c l a s s

C 6 ; ) .

The s t a b l e e l l i p t i c i s l a n d s surrounding the e l l i p t i c f i x e d p o i n t s w i t h i n t h e otherwise c h a o t i c resonant gaps a r e sketched i n F i g . 1. How they come about upon a small p e r t u r b a t i o n o f an i n t e g r a b l e system i s explained by t h e P o i n c a r 6 - B i r k h o f f theorem: when an area p r e s e r v i n g mapping has a simple closed i n v a r i a n t curve c o n s i s t - i n g o f f i x e d p o i n t s , then due t o t h e KAM Theorem and t h e area p r e s e r v i n g property, an even number o f these f i x e d p o i n t s i s shown t o s u r v i v e a small p e r t u r b a t i o n . H a l f o f them are e l l i p t i c , and t h e o t h e r s are h y p e r b o l i c . The almost, s e l f - s i m i l a r i n - f i n i t e h i e r a r c h y i s explained by t h e f a c t t h a t a l l considerations above a r e v a l i d for any i t e r a t e o f t h e PoincarP mapping.

I t remains t o understand how t h e c h a o t i c motion can a r i s e . T h i s i s connected w i t h t h e h y p e r b o l i c p o i n t s . A l l what has been done so f a r i s t h e l i n e a r i z a t i o n around a h y p e r b o l i c f i x e d p o i n t . We s h a l l concentrate on t h e asymptotes o f the hyperbolae and t h e motion on them. We c a l l them incoming o r outgoing strand, according t o whether L a c t s as a c o n t r a c t i o n o r expansion, r e s p e c t i v e l y ( F i g . 2). A c t u a l l y , these l i n e a r p a r t s can be extended i n t o t h e n o n l i n e a r r e g i o n o f T 1403, and a r e then c a l l e d s t a b l e and unstable manifold, r e s p e c t i v e l y . Now, t h e most i m p o r t a n t t h i n g i s how they meet each other. I f they j o i n smoothly, n o t h i n g special happens. But such a smooth j o i n i n g i s exceptional, since they w i l l g e n e r i c a l l y ( i .e. i n almost a11 cases) meet t r a n s v e r s a l l y a t a p o i n t which i s n o t a f i x e d p o i n t . When they do so, t h e y 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 t h e area p r e s e r v i n g p r o p e r t y o f t h e mapping t h e amplitude o f t h e o s c i l l a t i o n s w i l l become l a r g e r and l a r g e r , so t h a t an extremely complex motion a r i s e s , which has l i t t l e i n common w i t h t h e laminar i n t e g r a b l e behaviour. Such an o s c i l l a t o r y motion i s c a l l e d homoclinic o s c i l l a t i o n . I t i s a p r o t o t y p e o f c h a o t i c 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 f o r homoclinic o s c i l l a t i o n (by imbedding t h e Smale horseshoe mapping 1401 ) . We see t h a t hyperbol i c i t y

+

t r a n s v e r s a l i t y imply c h a o t i c motion. Since both occur i n almost a l l cases upon a p e r t u r b a t i o n , i f n o t otherwise, we g e t a f e e l i n g t h a t t h e type o f motion w i t h i n t h e resonant gaps, as shown i n F i g . 1, i s o f t h e generic c l a s s .

We can now understand t h e nature o f t h e extreme o f completely unstable systems, as opposed t o t h e s t a b l e i n t e g r a b l e system: They must be f u l l o f h y p e r b o l i c 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 h y p e r b o l i c

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o r b i t s . T h e i r measure i s zero, so t h a t t h i s does n o t c o n t r a d i c t the p r o p e r t y (a). I n a c e r t a i n sense t h e p e r i o d i c o r b i t s , being a l l unstable, span t h e f l o w o f even an e r g o d i c system. This f a c t i s n o t l e s s s u r p r i s i n g than t h e approximation o f r e a l num- bers by t h e r a t i o n a l s .

However, a generic Hamilton system i s n e i t h e r i n t e g r a b l e nor ergodic, b o t h ex- tremes being exceptional. (We consider o n l y systems w i t h a few degrees o f freedom.

I t should be noted t h a t t h e ergodic systems a r e l e s s exceptional than t h e i n t e g r a b l e ones, s i n c e they a r e s t r u c t u r a l l y s t a b l e : a small p e r t u r b a t i o n does n o t change t h e two fundamental p r o p e r t i e s , hyperbol i c i t y and t r a n s v e r s a l i ty. ) To be s p e c i f i c , a t y p i c a l system d i s p l a y s a l l features o f an i n t e g r a b l e 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 , t h e s o - c a l l e d s t o c h a s t i c t r a n s i t i o n oc- curs: t h e i n v a r i a n t t o r i a r e observed t o disappear r a p i d l y as t h e energy i s increased, and t h e motion becomes i r r e g u l a r . When t h e motion i s n u m e r i c a l l y i n t e g r a t e d t h e best way t o observe t h e t r a n s i t i o n f r o m t h e r e g u l a r t o t h e 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 q u e s t i o n i n a l a t e r s e c t i o n .

WorCA pi^& &

exceptiond

Figure 2: When t h e s t a b l e and unstable manifolds o f an hyperbolic point H meet t r a n s v e r s a l l y a t a homoclinic point Po, then t h e r e i s i n f i n i t y o f homoclinic points PI,PZ,

...

No smooth i n t e g r a l o f motion e x i s t s i n homoclinic o s c i l - l a t i o n . I t I S a prototype o f i r r e g u l a r motion.

While a l o t i s known on t h e c l a s s i c a l i r r e g u l a r motion, t h e 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 t h e Schrodinger equation i s known and i n p r i n c i p l e s o l v a b l e f o r any system helps as l i t t l e as t h e analogous f a c t o f t h e Hamilton-Jacobi equation i n the c l a s s i c a l case d i d . What we need a r e more e x p l i c i t q u a n t i z a t i o n c o n d i t i o n s , and t h i s i s t h e 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 r e two major problems t o be solved: f i r s t l y , t h e time e v o l u t i o n o f n o n s t a t i o n a r y wavefunc- t i o n s , which i s ignored here; secondly, t h e p r o p e r t i e s 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 r e v i e w i n g some general b u t fragmentary r e s u l t s , I should l i k e t o emphasize t h e 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 t h e quantum analogy o f t h e KAM Theorem, which must e x i s t , and should be a n a t u r a l b a s i s f o r t h e study o f s t o c h a s t i c i t y i n quantum mechanics. I b e l i e v e 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 t h e f o l l o w i n g t h e terms " c h a o t i c " and " i n t e g r a b l e quantum system" a r e meant t o imply t h a t they a r e i r r e g u l a r o r i n t e g r a b l e i n t h e c l a s s i c a l l i m i t , r e s p e c t i v e l y . I n t h e r a r e cases t h a t a system i s i n t e g r a b l e we have t h e w e l l d e f i n e d s e m i c l a s s i c a l q u a n t i z a t i o n c o n d i t i o n s , i n i t i a t e d by E i n s t e i n , and r i g o r o u s l y developed by Maslov.

The method i s known under t h e names: t o r i q u a n t i z a t i o n , t o p o l o g i c a l q u a n t i z a t i o n , Maslov q u a n t i z a t i o n , EBK-quantization,

. . .

I t i s j u s t t h e q u a n t i z a t i o n o f t h e actions,

Ij = ( r n . + c i j / 4 ) b J

,

( 2 )

where m j = 0,1,2,.

. . ,

w h i l e a j i s t h e number o f c a u s t i c s encountered i n c o n f i g u r a t i o n

space upon t r a v e r s i n g t h e

irreducible

c y c l e C j , and i s c a l l e d Maslov index. The spectrum i s then given by E$ = H(I3).

U n l i k e t h e c l a s s i c a l case, the quantum s p e c t r a l problem poses g r e a t d i f f i c u l t i e s even i n t h e i n t e g r a b l e system: the exact q u a n t i z a t i o n 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 t h e system belongs t o t h e very s p e c i a l class o f separable systems. O f course t h i s touches t h e p u r e l y c l a s s i c a l problem o f when i s an i n t e g r a b l e system a l s , separable. ( K a l n i n s and M i l l e r [41] have r e c e n t l y s t u d i e d t h e separation problem i n nonorthogonal systems.) To my knowledge t h e r e i s n o t even a s i n g l e example o f a

E- -

separable b u t i n t e g r a b l e system, whose exact spectrum i s known. So we do n o t know

t h e g e n e r i c f i n e s t r u c t u r e o f t h e i r spectra: a r e t h e degeneracies t y p i c a l j u s t as i n t h e s e p a r a b l ~ s t e m s , o r i s t h e e x p o n e n t i a l l y small separation o f l e v e l s a t c l o s e 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 q u a n t i z a t i o n c o n d i t i o n s as e x p l i c i t as ( 2 ) a r e known f o r i r r e g u l a r sys- tems. I t i s n o t unexpected t h a t a search f o r them i s d i r e c t e d towards t h e usage o f Feynman p a t h i n t e g r a l s ( o r d i n a r y and phase space path i n t e g r a l s ) , since these pre- sent the most i n t i m a t e l i n k between t h e c l a s s i c a l and quantum dynamics. The approach has been 1 a r g e l y developed by Gutzwi 1 le r [42].

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

so t h a t i t s F o u r i e r t r a n s f o r m (Green f u n c t i o n ) equals,

where 6(E-E .) a r e the D i r a c d e l t a f u n c t i o n s . 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 t h e response d(E) i s equal t o the sum o f t h e d e l t a spikes:

Hence, knowing t h e kernel ( 3 ) we know t h e spectrum ( 5 ) . But Feynman has shown t h a t t h e kernel K(qa,qb,t) can be e x a c t l y given by t h e sum o f a l l paths going from qa a t time t = 0 t o qb a t time 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 t h e s e m i c l a s s i c a l l i m i t

P

⁺ 0 two f a c t s a r e important. Since t h e c o n t r i b u t i o n s t o ( 6 ) w i l l d i s p l a y r a p i d os- c i l la t i o n s o n l y t h e 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- p o i n t s a,b t h e s t a t i o n a r y pathsare e x a c t l y the c l a s s i c a l paths; ( i i ) t h e i d e n t i f i c a - t i o n qa=qb demands t h a t o n l y those c l a s s i c a l paths, which a r e r e c u r r e n t i n configu- r a t i o n space should be taken. But since they should a l s o be s t a t i o n a r y w i t h r e s p e c t 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 t h e conclusion t h a t : i n t h e semiclassical l i m i t o n l y t h e closed ( i . e. p e r i o d i c ) c l a s s i c a l paths c o n t r i b u t e t o t h e response d(E). I n t h i s sense, t h e s e m i c l a s s i c a l spectrum i s spanned by t h e p e r i o d i c o r b i t s .

This seems paradoxical as t h e 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 m a t t e r whether t h e system i s r e g u l a 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 i n t e g r a b l e system t h e p e r i o d i c o r b i t s l i e on resonant t o r i , and these are p r e c i s e l y those allowed t o be destructed according t o the KAM Theorem. We see t h a t d e s t r u c t i o n o f t o r i connected w i t h t h e onset o f c h a o t i c motion w i l l have important consequences f o r the ( s e m i c l a s s i c a l ) spectrum. However, n o t a l l resonant t o r i are g e n e r i c a l l y destroyed a t the same time upon a small p e r t u r b a t i o n : the d e s t r u c t i o n begins f o r h i g h energy t o r i , appearing a t lower and lower energies as t h e 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 c l a s s i c a l o r b i t can give only accidentally a correct energy eigenvalue. Berry (291 points out t h a t closed o r b i t s and energy levels correspond t o each other only collectively.

Notwithstanding some major disagreements [29,30] as concerns the quantization of irregular systems, there i s a generally accepted f a c t t h a t t h e 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 c a l approach t o the response ( 5 ) , as was the case in the early s t a t i s t i c a l theory of spectra of complex nuclei, developed by Wigner, Porter and Dyson. A j u s t i f i c a t i o n f o r the s t a t i s t i c a l description i s believed t o be found in the f a c t , t h a t t h i s i s t h e only meaningful description. Since the works by Percival (431 and Pomphrey 1441

,

enough observational material has been gathered t o be convinced t h a t energy levels corresponding t o the i r r e g u l a r motion a r e unstable with respect t o small variations of some (family) parameters of the Hamiltonian. Since a l l real physical systems a r e actually ensembles with some s c a t t e r in external conditions, such as magnetic f i e l d , the observed spectra of i r r e g u l a r systems will be rather smooth, reminiscent more of the continuous than of the d i s c r e t e spectra. Zaslavsky [30] links t h i s i n s t a b i l i - t y of levels d i r e c t l y t o the i n s t a b i l i t y of the classical motion, via the K-entropy.

(Roughly speaking, t h i s i s equal t o the phase average of the l a r g e s t Lyapunov ex- ponent. I t measures t h e r a t e of increase in time of the coarsened phase volume, i . e . the r a t e of entropy increase.) On these grounds the response formula ( 5 ) should be understood more as a d i s t r i b u t i o n of energy levels rather than a quantization con- d i t i o n f o r individual levels.

I t i s then appropriate t o define the probability density P(S) t h a t the separation of two neighbouring l e v e l s i s equal t o S (normalized to the mean separation of lev- e l s ) . For integrable systems there i s a generally accepted r e s u l t t h a t P(S)=exp(-S), r e f l e c t i n g the possible existence of level crossings. In t h i s case the levels are clustered. In the i r r e g u l a r case, there i s agreement t h a t P(S) i s given by some pow- e r law, P(S) = const.

x sb.

Berry 1293 suggests b = 1, while Zaslavsky argues t h a t b = const./h, h being the Kolmogorov K-entropy of the corresponding c l a s s i c a l system in the energy shell around the levels considered. As P(S) vanishes with S, t h i s ac- counts f o r the f a c t t h a t level repulsions a r e typical of the irregular systems.

Therefore, no clustering occurs and the spectra appear more random.

3. The classical Hamiltonian and the stochastic t r a n s i t i o n

In the following I shall consider the "naked problem" of the hydrogen atom in a magnetic f i e l d , which remains a f t e r dropping a l l e f f e c t s except the diamagnetic, which i s essential f o r the t r a n s i t i o n from the regular t o the i r r e g u l a r motion. Thus, t h e t r i v i a l paramagnetic term i s omitted; the nucleus i s assumed t o have i n f i n i t e mass, so t h a t no coupling between the electronic s t a t e s and the CM-motion i s present;

the CM i s assumed t o be a t r e s t in a homogeneous magnetic f i e l d , accounting f o r the absence of the motional Stark e f f e c t . Then the classical electron motion i s governed by the dimensionless Hamiltonian

1 + 2 1 2 2 1

H = p p + - y ₈ P

- -

_r 3 (7

where p 2 :=

x

2

+

y2. The unit of length i s equal t o the Bohr radius, the energy i s measured in atomic units ( h a r t r e e ) , and y i s the already defined dimensionlessmagnetic f i e l d strength, y = B/Bo. The only continuous s y m ~ e i r i e s 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 a r e two obvious integralsof motion. Since

N

= 3, a t h i r d integral i s needed f o r i n t e g r a b i l i t y .

Accounting f o r the invariance of L we can reduce ( 7 ) by using the cylindrical co- ordinates:

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 t h a t i t i s im-

p o s s i b l e t o e l i m i n a t e 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 t h e Hamiltonian

( B ) a = const. L-', = const. L, when L 0, w i t h t h e Hamiltonian

where a l l constants a r e independent o f L and y. We have taken them equal t o u n i t y . Here t h e same symbols are used f o r t h e new v a r i a b l e s p

, g ,

z,pz. The case (A) has been e x t e n s i v e l y considered i n t h e works by Edmonds and P u l l e n

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 f e a t u r e o f t h e 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

h 5 +

0 t h e same f e a t u r e ap- oears a t E = L- EG(vL _I\. 1. I I t i s thus s u f f i c i e n t t o e x o l o r e t h e svstem f o r L = 1.

2 2 2 -1/2

The e q u i p o t e n t i a 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 i g . 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

E mi n . ( y ) = (2-3ho)/2ho 2 ³ (11

where A. := h(y/2), and t h e f u n c t i o n A(x) i s d e f i n e d 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 , S O 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 t h e escape energy Eesc(y) = y/2, above which t h e e l e c t r o n can escape t h e Coulomb p o t e n t i a l w e l l , w h i l e i t i s s t i l l t r a n s v e r s a l l y bounded by t h e magnetic f i e l d . Note t h a t Eesc i s n o t equal t o t h e quantum mechanical i o n i z a t i o n energy Ei = y(L+1)/2, which i s h i g h e r by e x a c t l y t h e Landau energy y/2, and does n o t obey ?!e s c a l i n g law. I f the e l e c t r o n escapes i n t h e 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 a s y m p t o t i c a l l y equal t o o r l a r g e r than t h e quantum mechanical zero p o i n t energy y/2. For t h i s reason t h e s t a t e s w i t h energies between Ee and Eion a r e t r u e bo nd, l o c a l i z e d s t a t e s , 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 ) ) , w h i l e (Eesc

-

E . )

-.

^{as y'-.} L a t e r we s h a l l min

use t h e r e l a t i v e energy g(E), d e f i n e d by

F i g u r e 3: T h e e q u i p o t e n t i a l l i n e s 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 t h i s invariant curve corresponds t o the bouncing of the electron along the gradient l i n e z = 0 in Fig. 3, i . e . t o plane o r b i t s in the 3-dimensional picture. A Poincari! map- ping was examined by taking some typical i n i t i a l conditions, and by p l o t t i n g the s u f f i c i e n t l y large number of i t s i t e r a t e s t o see whether the phase point i s confined to some invariant curve, In Figure 4 we show SOS a t ten d i f f e r e n t energies f o r 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 a r e nested around an e l l i p t i c fixed point. This corresponds t o the simple periodic bouncing between t h e potential l i n e s of Fig. 3, but now ap- proximately 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 t a b l e periodic o r b i t s have occured. Here the hyperbolic points can be also observed, and in v i c i n i t y of them there are f i r s t signs of unstable motion. By increasing energy t h i s chain of hyper- bolic points i s seen t o become a broad b e l t of stochastic motion (Fig. 4e). There a r e several d e f i n i t i o n s of the stochastic t r a n s i t i o n , a l l being almost equivalent due t o the f a c t t h a t the t r a n s i t i o n i s f a i r l y sharp. (The d e f i n i t i o n in terms of the K- entropy, which becomes positive, seems t o be the best, since i t refers t o a global property of the flow 1301. S t i l l other d e f i n i t i o n r e f e r s t o the destruction of the

" l a s t large scale t o r u s " . ) I have adopted the c r i t e r i o n , t h a t the c r i t i c a l energy i s reached when the stochastic motion becomes observable on the l a r g e s t scale. According t o t h i s definition the c r i t i c a 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 ". ^'

D

O

1;DI

^a I ? L

=^'? - - ;

t -I

a 1

,

^{L a I T 2}

I

\ _I

-mo

---

9 s o . ^-u

C';

m r 5 .

F i g u r e 4 ( a - j ) : T h e P o z n c a r i . 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 . 0 5 ( e l , - 0 . 0 4 ( d ) , - 0 . 0 3 ( e ) , - 0 . 0 1 ( 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 . 0 3 , 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 b e c o m e s c o m p l e t e l y stochastic.

Upon i n c r e a s i n g t h e energy above t h e c r i t i c a l energy t h e i r r e g u l a r r e g i o n becomes l a r g e r and covers t h e whole energy s u r f a c e when t h e energy approaches t h e escape energy.

How can we understand t h e e x i s t e n c e o f i n v a r i a n t t o r i and the emergence o f t h e i r r e g u l a r motion on t h e o t h e r hand ? When t h e 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 p e r t u r b a t i o n , s i n c e y = 1) t h e

KAM

Theorem allows a l l t o r i t o be destroyed, because a l l t o r i o f t h e Kepler problem are resonant t o r i . How- ever, we can l o o k a t t h e system as being i n t e g r a b l e a t the bottom o f the p o t e n t i a l we1 1, where t h e 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 o n l y 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 p e r t u r b a t i o n parameter, and t h e KAM Theorem can be applied, p r e d i c t i n g now t h e existence o f t o r i f i l l i n g l a r g e r e g i o n s of t h e energy surface, t h e excluded volume being small w i t h ( E

-

Emin). Upon increas- i n g t h e energy t h e resonant gaps become wider u n t i l t h e c h a o t i c motion w i t h i n t h e gaps becomes macroscopically v i s i b l e . When t h e l a s t l a r g e - s c a l e i n v a r i a n t t o r u s i s destructed, t h e motion i s i r r e g u l a r on t h e whole energy surface.

I n S e c t i o n 2 i t was shown t h a t t h e h y p e r b o l i c f i x e d p o i n t s a r e "germs" o f t h e c h a o 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 = homoclinic o s c i l l a t i o n ) . To e x p l a i n t h e l a t t e r i t i s necessary t o show t h e way how t h e system can become f u l l o f hyper- b o l i c f i x e d p o i n t s . 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 u n i v e r s a l behaviour o f f a m i l i e s o f area p r e s e r v i n g mappings, q u i t e analogous t o t h e p r o p e r t i e s o f one-dimensional d i s s i p a t i v e mappings. The generic case i s as f o l l o w s : as a parameter o f an area p r e s e r v i n g mapping i s v a r i e d a s t a b l e , 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 t h e same t i m e s t a b l e p e r i o d i c o r b i t s o f t w i c e t h e o r i g i n a l p e r i o d are born o u t o f i t . With t h e parameter p v a r y i n g t h e process can be repeated r e s u l t i n g i n a sequence o f e r i o d d o u b l i n g b i - f u r c a t i o n s , o c c u r i n g a t p o , p l y . .

.

, p j y . .

.

The seql;ence converges t!wards some value p*, and i s a s y m p t o t i c a l l y geometric as j +

-,

w i t h t h e u n i v e r s a l r a t e

as j + m, where t h e " r e s c a l i n g f a c t o r " 6 = 8.721097200

...

The e x p l a n a t i o n f o r t h i s u n i v e r s a l behaviour, independent o f t h e p a r t i c u l a r map, i s found i n t h e existence o f a u n i v e r s a l map a c t i n g as an a t t r a c t o r i n t h e 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 f a m i l y o f maps i s varied, t h e i t e r a t e s o f t h e maps w i l l a t l e a s t l o - c a l l y converge t o t h e u n i v e r s a l mapping. This u n i v e r s a l map has o t h e r r e s c a l i n g pro- p e r t i e s , such as u n i v e r s a l convergence r a t e s o f t h e p o s i t i o n s o f t h e b i f u r c a t e d fixed p o i n t s . I n any case, t h e map i s f u l l o f h y p e r b o l i c o r b i t s and does indeed e x p l a i n i r r e g u l a r motion, which need n o t be ergodic, however. The d i s c o v e r y has p r a c t i c a l i m p l i c a t i o n s , s i n c e the convergence i s very r a p i d : I f we observe, o r estimate, two p e r i o d doubling b i f u r c a t i o n s o f t h e same sequence appearing a t t h e 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 t h e motion becomes i r r e g u l a r . (Note t h a t t h e 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 t h e reparametrization.) I n our case t h e parameter i s t h e 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 t h e c r i t i c a l energy, discussed i n t h e n e x t s e c t i o n .

The s t o c h a s t i c t r a n s i t i o n has been observed by i n s p e c t i o n o f t h e Poincari? mappings f o r o t h e r magnetic f i e l d strengths, and t h e r e s u l t i s shown i n F i g u r e 5, where t h e c r i t i c a l energy i s p l o t t e d as a f u n c t i o n o f t h e magnetic f i e l d . The r e l a t i v e c r i t i - c a l energy (see ( 1 2 ) ) g ( ~ c r i t ( y ~ 3 ) ) =: . u ( y ~ 3 ) , which depends o n l y on yL3, i s p l o t t e d i n F i g u r e 6. Here i t i s seen t h a t t h e i r r e g u l a r r e g i o n becomes s m a l l e r i n t h e two i n t e g r a b l e 1 im i t i n g cases ( y = 0, y = -), w h i l e i t extends down t o almost Emin a t yL3 = 2.7.

The question o f t h e l i m i t L + 0 connects t h e cases ( A ) and (B) (see (9,10)), and i s a t t h e same t i m e r e l a t e d t o t h e l i m i t o f vanishing magnetic f i e l d . It i s reason- a b l e t o assume t h a t some t o p o l o g i c a l f e a t u r e o f t h e f l o w obeys a power law

JOURNAL DE PHYSIQUE

I

^{ESCAPE REGION E}

^>

^,E

I

F i g u r e 6 : T h e 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 . T h e 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 p h a s e 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 , f o r any L#O. I f the l i m i t L ^-t 0 should e x i s 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 l d as y

.

As regards the c r i t i c a l energy E c r i t , 5dmonds and Pullen f211 have shown t h a t i n case t = 0 i t i s given by E c r i t = -0.5 y j3. The l i m i t as L + 0 therefore e x i s t s , and does not nish, whence a c r i t = 213 as well. From t h i s 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 y e t 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 o r b i t s have also been observed. Some f a c t s a r e worth stressing: The motion near the plane

z

= 0 i s very i r r e g u l a r . Among some typi- cal randomly chosen i n i t i a l conditions no trapped o r b i t s were seen. I t i s conjec- tured t h a t the measure of o r b i t s which a r e trapped in a f i n i t e p a r t of the phase space with energy above Ee i s negligibly small. However, s u f f i c i e n t l y 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 t o escape, the elec- tron must have the appropriate value of the adiabatic invariant. I f not, i t will be reflected and will return t o the plane

z

= 0 , where the value of the adiabatic in- variant can change due t o the chaotic motion. After several reflections the electron will ultimately escape in direction of the magnetic f i e l d . This i s i l l u s t r a t e d in Figure 7.

Figure 7(a-b) : The i n t e r s e c t i o n points of an escape o r b i t (E=0.6, y=L=l) w i t h the plane p =O (a). The same o r b i t projected t o t h e plane z=O(b).

P

4. Some p o s s i b i l i t i e s t o calculate the c r i t i c a l energy

The attempts t o predict the c r i t i c a l energy, i . e . t o reproduce the Figure 6 , by using the approximate analytical methods have been inconclusive so f a r . For in- stance, the considerations based on the curvature

R

( i s negative R a valid c r i t e r i o n f o r stochastici ty?)

C2-58 JOURNAL DE PHYSIQUE

in the Jacobi metric, deri ed from the potential U, lead t o the conclusion t h a t R i s always positive when y:

Y

>1/8, which disagrees with the observed s t o c h a s t i c tran- s i t i o n a t y=L=l. That should not surprise us however, since the "negative curvature c r i t e r i o n " i s known t o f a i l in some cases [45,47,48,28].

The application of the Chiri kov c r i t e r i o n of overlapping resonances 1321 a s well as of the renormalization approach ( t o the case of two resonances and the destruc- tion of the " l a s t invariant torus" separating them) of Escande and Doveil [49] hinges upon the f a c t t h a t i t i s not easy t o express the diamagnetic perturbation term by the Keplerian action-angle variab e s in an e x p l i c i t way. Hence, the c l a s s i c a l per- turbation problem i n t h e l i m i 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 f o r other reasons, one faces d i f f i c u l t i e s in the opposite extreme yLy +

-.

Here, the problem i s t h a t the motions decouple.

The only case where the action-angle formalism i s easy t o handle i s the neigh- bourhood of the c r i t i c a 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 e a s i l y expressed in action- angle variables. A c r i t e r i o n a l a Chirikov, saying t h a t the s t o c h a s t i c i t y will s e t in when the third-order and the fourth-order resonant terms have comparable ampli- tudes, r e s u l t s in

3 2 2

Ecrit

-

Emin = (1

-

h o ) / f l L Xo (15)

which y i e l d s E c r i t = 0.05 when y = L = 1. The agreement with the observations de- scribed i n the f o r e g o ~ n g section i s excellent. However, such an expansion i s e f f i - c i e n t only when a few low-order terms represent a s i g n i f i c a n t departure from an in- tegrable system. This i s not t r u e i n the limiting cases v ~ 3 + 0,-, which renders the v a l i d i t y of eq. (15) t o the neighbourhood of yL3=1. (As can be seen, the expression (15) does not have the correct asymptotic behaviour in these l i m i t s . ) I t i s not quite c l e a r whether the excellent agreement in t h i s region i s accidental, since the r i

5 -

terion i s largely q u a l i t a t i v e in nature. Nevertheless, i t shows t h a 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 t a b l e periodic o r b i t s and some of t h e i r period doubling bifurcations, then t h i s would enable us t o use the formula (13).

A f i n a l remark of t h i s section concerns again the important case of the weak dia- magnetic term, i . e . t h e case of the s l i g h t l y perturbed Kepler-system. The KAMTheo- rem admits the destruction of a l l t o r i , since a l l t o r i of the Kepler problem a r e re- sonant, but we can r e l y on the structural s t a b i l i t y of the real physical systems, and believe, t h a t only high energy t o r i will be destroyed. To be more precise but s t i l l q u a l i t a t i v e , we expect t h i s t o happe2 yhen the Coulomb pot t i a l and the dia- magnetic potential a r e 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 ( i n c i d e n t a l l y ) also t h e correct nu- merical f a c t o r . ~ k e ?mportance i s not in rederiving a r e s u l t of the previous section, but in showing t h a t in the l i m i t of weak magnetic f i e l d , i n t e r e s t i n g f o r experimen- t a l investigations, the c l a s s i c a l i r r e g u l a r region i s coincident with the inter-n- mixing regime.

5. An approximate integral of the motion below the c r i t i c a l energy

The t h i r d integral of motion e x i s t s almost everywhere on t h e energy surface below the c r i t i c a l energy, except i n t h i n , macroscopically i n v i s i b l e , gaps. I t can be exact there, as the KAM Theorem suggests, b u t i s certainly not a n a l y t i c , 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 t o overcome the d i f f i c u l t i e s in formal expansions [50]. I f three i n t e g r a l s E , L and F of the Hamil tonian ( 7 ) a r e known, and a r e in in- volution, we can use the t o r i quantization ( 2 ) .

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