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QUASI-LANDAU RESONANCES : ANALYTIC TREATMENT OF THE HYDROGENIC SPECTRUM
IN THE TWO-DIMENSIONAL MODEL AND
RELATION TO OTHER STRONGFIELD PROBLEMS
J. Gallas, R. O’Connell
To cite this version:
J. Gallas, R. O’Connell. QUASI-LANDAU RESONANCES : ANALYTIC TREATMENT OF THE
HYDROGENIC SPECTRUM IN THE TWO-DIMENSIONAL MODEL AND RELATION TO
OTHER STRONGFIELD PROBLEMS. Journal de Physique Colloques, 1982, 43 (C2), pp.C2-435-
C2-437. �10.1051/jphyscol:1982234�. �jpa-00221846�
JOURNAL DE PHYSIQUE
Colloque C2, supplément au n°ll, Tome 4S3 novembve 1982 page C2-435
QUASI-LANDAU RESONANCES : ANALYTIC TREATMENT OF THE HYDROGENIC SPECTRUM IN THE TWO-DIMENSIONAL MODEL AND RELATION TO OTHER STRONG- FIELD PROBLEMS
J.A.C. G a l l a s and R.F. O'Connell*
Max-Planck Institut fur Quantenoptik, D-8046 Garohing bei Mtinehen, West Germany
and
Departamento de Fisioa, Universidade Federal de Santa Catarina, 88000 Florianopolis, SC, Brasil
*Department of Physios and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70805, U.S.A.
Résumé: Le modèle WKB à 2 dimensions est à l ' o r i g i n e de plusieurs approches numériques du spectre quasi-Landau de l'atome d'hydrogène. On montre i c i que les résultats peuvent être obtenus sous forme analytique en termes d'intégrales e l l i p t i q u e s . Celles-ci peuvent être calculées aisément ce qui permet d'obtenir les prédictions numériques en bon accord avec l'expérience. On montre aussi que plusieurs autres problèmes en champs intenses sont des cas particuliers d'un potentiel plus général pour lequel les niveaux d'énergie et leur espace- ment peuvent être exprimés en termes d'intégrales e l l i p t i q u e s .
Abstract: The two-dimensional WKB model has been the basis for several investigations of the quasi-Landau hydrogenic spectrum. Whereas other authors have used numerical integration, we show that the results can be obtained analytically in terms of elliptic integrals. The later are easily generated by even programmable pocket calculators, from which numerical results - which are in good agreement with experiments - are easily obtained. A further advantage of using elliptic integrals is that several strong-field problems can be shown to be special cases of a general potential whose energy and spacing is expressible in terms of them.
Discussion. In 1969 Garton and Tomkins |1| published a paper the result of which was to unleash an avalanche of research in the general area of what is now referred to as quasi-Landau resonances. They studied the absorption spectrum of barium in a magnetic field of 24 kG, for n values as high as 75, and found broad resonances spaced by approximately 1.5 "fico near E = 0 (to=eB/Mc is the cyclotron frequency where M is the electron mass). These quasi-Landau resonances were observed only in the a spectrum (dm= ±1j) and not in the IT spectrum {hm= 0 ) . Thus, since the a lines
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982234
C2-436 JOURNAL DE PHYSIQUE
come from s t a t e s which are e s s e n t i a l l y l o c a l i z e d i n the x-y plane (as d i s t i n c t from t h e IIl i n e s which come from a s t a t e w i t h a node i n t h e x-y plane), i t i s n a t u r a l t o n e g l e c t t h e motion i n t h e z d i r e c t i o n and t r e a t the problem as
two-
-dimensional. This was the procedure used by both Edmonds 121 and Starace 131, who reduced t h e problem t o motion i n a p o t e n t i a l V ( p ) where p 2 = x2 + y 2 . Thus i t was then n a t u r a l t o s o l v e t h e problem using WKB techniques. Whereas previous i n v e s t i g a t o r s have solved t h e WKB i n t e g r a l s u s i n g numerical techniques, we have r e c e n t l y shown 141 t h a t r e s u l t s can be obtained a n a l y t i c a l l y i n terms o f e l l i p t i c i n t e g r a l s , n o t o n l y f o r the energy spacings b u t a l s o f o r t h e energy l e v e l s f o r zero and non-zero values o f m. The r e s u l t s so obtained a r e i n very good agreement with, f o r example, t h e experimental r e s u l t s o f Delande and Gay 151 and w i t h t h e numerical 1 y generated r e s u l t s o f McDowell 16
1 .
I n t h e present b r i e f communication our emphasis w i l l be on why we f e e l i t i s advantageous t o work i n terms o f e l l i p t i c i n t e g r a l s , of which t h e r e are o n l y t h r e e kinds. F i r s t o f a l l , they a r e e a s i l y generated by even programmable pocket c a l c u l a t o r s , from which numerical r e s u l i s a r e e a s i l y obtained. As a r e s u l t i t very much f a c i l i t a t e s comparison w i t h work of o t h e r authors. I n a d d i t i o n , we see e x p l i - c i t e l y t h a t the quasi-Landau problem f i t s i n t o a general framework o f s t r o n g - f i e l d problems. This was emphasized by one o f us several years ago 171 and elaborated on a t more l e n g t h i n a review paper presented a t t h e present Colloque 181. To be
s p e c i f i c , we mention f i r s t o f a l l t h e c a l c u l a t i o n of t h e energy spectrum o f e l e c t r o n s o u t s i d e a f r e e surface o f l i q u i d helium. These e l e c t r o n s a r e trapped i n an image p o t e n t i a l which i s e s s e n t i a l l y one-dimensional Coulombic and f o r d i a g n o s t i c purposes an e l e c t r i c f i e l d i s a l s o g e n e r a l l y superimposed. This combined f i e l d problem i s another example o f a s t r o n g - f i e l d problem whose s o l u t i o n can be expressed i n terms o f e l l i p t i c i n t e g r a l s 17,81 as i n t h e case o f t h e quasi-landau resonances 141. A f u r t h e r example i s t h e Stark problem, which was solved t o very good accuracy using WKB techniques 191 and the r e s u l t s obtained
-
i n c l u d i n g t h e case o f a u t o i o n i z i n g s t a t e s-
were a l l expressed i n terms o f e l l i p t i c i n t e g r a l s . F i n a l l y , we mention t h e c a l c u l a t i o n o f t h e spectrum o f Charmonium] l o ] .
I n essence, t h e p o t e n t i a l s f o r a l l t h e problems a r e s p e c i a l cases o f a general p o t e n t i a l whose energy spectrum and spacing i s e x p r e s s i b l e i n terms o f e l l i p t i c i n t e g r a l s . Elsewhere, we w i l l present i n more d e t a i l a general framework which can be used f o r t h e c o n s i d e r a t i o n o f a l a r g e c l a s s of s t r o n g - f i e l d problems.Acknowledgments. We would l i k e t o thank Dr. J.-C. Gay and Prof. M.R.C. McDowell f o r discussions and communications o f t h e i r r e s u l t s . JACG i s supported a l s o by the Deutscher Akademischer Austauschdienst/Bonn-Bad Godesberg. RFO'C i s supported by the Department o f Energy, D i v i s i o n o f M a t e r i a l Sciences, c o n t r a c t n r .
DE-AS05-79ER10459.
References.
I. Garton W.R.S. and Tomkins F.S., Astrophys. J. - 158(1969) 839.
2. Edmonds A.R., J.Phys.(Paris) 31-C4(1970)71.
3.
Starace A.F., J.Phys. B
-6(1973) 585.
4. Gallas J.A.C. and O'Connell R.F., J.Phys. B - 15(1982)L75 and L309; ibid. to be published.
5. Delande D. and Gay J.-C., Phys. Lett. - 82A(1981) 399 and ref. therein.
6. McDowell M.R.C., these Proceedings.
7. 0' Connell R.F., Phys.Rev. A x(1978) 1984; Phys.Lett.
-60A(1977)
481.8.
