Helium atom. Chapter Classical dynamics of collinear helium. e e. He ++ CHAPTER 41. HELIUM ATOM 734

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1 CHAPTER 41. HELIUM ATOM 734 e e Chapte 41 Helium atom Figue 41.1: Coodinates fo the helium thee body poblem in the plane. Figue 41.: Collinea helium, with the two electons on opposite sides of the nucleus. e θ 1 ++ He He e S But, Boh potested, nobody will believe me unless I can explain evey atom and evey molecule. Ruthefod was quick to eply, Boh, you explain hydogen and you explain helium and eveybody will believe the est. John Achibald Wheele (1986) (G. Tanne) ofa much has been said about 1-dimensional maps, game of pinball and othe cuious but athe idealized dynamical systems. If you have become impatient and stated wondeing what good ae the methods leaned so fa in solving eal physical poblems, we have good news fo you. We will show in this chapte that the concepts of symbolic dynamics, unstable peiodic obits, and cycle expansions ae essential tools to undestand and calculate classical and quantum mechanical popeties of nothing less than the helium, a deaded theebody Coulomb poblem. This sounds almost like one step too much at a time; we all know how ich and complicated the dynamics of the thee-body poblem is can we eally jump fom thee static disks diectly to thee chaged paticles moving unde the influence of thei mutually attacting o epelling foces? It tuns out, we can, but we have to do it with cae. The full poblem is indeed not accessible in all its detail, but we ae able to analyze a somewhat simple subsystem collinea helium. This system plays an impotant ole in the classical dynamics of the full thee-body poblem and its quantum spectum. The main wok in educing the quantum mechanics of helium to a semiclassical teatment of collinea helium lies in undestanding why we ae allowed to do so. We will not woy about this too much in the beginning; afte all, 80 yeas and many failed attempts sepaate Heisenbeg, Boh and othes in the 190ties fom the insights we have today on the ole chaos plays fo helium and its quantum spectum. We have intoduced collinea helium and leaned how to integate its 733 tajectoies in sect. B.. Hee we will find peiodic obits and detemine the elevant eigenvalues of the Jacobian matix in sect We will explain in sect why a quantization of the collinea dynamics in helium will enable us to find pats of the full helium spectum; we then set up the semiclassical spectal deteminant and evaluate its cycle expansion. A full quantum justification of this teatment of helium is biefly discussed in sect Classical dynamics of collinea helium Recapitulating biefly what we leaned in sect. B.: the collinea helium system consists of two electons of mass m e and chage e moving on a line with espect to a fixed positively chaged nucleus of chage+e, as in figue 41.. The Hamiltonian can be bought to a non dimensionalized fom H= p 1 + p 1 + = 1. (41.1) The case of negative enegies chosen hee is the most inteesting one fo us. It exhibits chaos, unstable peiodic obits and is esponsible fo the bound states and esonances of the quantum poblem teated in sect Thee is anothe classical quantity impotant fo a semiclassical teatment of quantum mechanics, and which will also featue pominently in the discussion in the next section; this is the classical action (37.15) which scales with enegy as S (E)= dq(e) p(e)= e m 1/ e S, (41.) ( E) 1/ with S being the action obtained fom (41.1) fo E= 1, and coodinates q= ( 1, ), p=(p 1, p ). Fo the Hamiltonian (41.1), the peiod of a cycle and its action ae elated by (37.17), T p = 1 S p. Afte a Kustaanheimo Stiefel tansfomation 1 = Q 1, =Q, p 1= P 1 Q 1, p = P Q, (41.3)

2 CHAPTER 41. HELIUM ATOM 735 CHAPTER 41. HELIUM ATOM 736 a) 10 b) Figue 41.3: (a) A typical tajectoy in the 1 plane; the tajectoy entes hee along the 1 axis and escapes to infinity along the axis; (b) Poincaé map ( =0) fo collinea helium. Stong chaos pevails fo small 1 nea the nucleus. 6 p Figue 41.4: The cycle 011 in the fundamental domain 1 (full line) and in the full domain (dashed line) and epaametization of time by dτ=dt/ 1, the equations of motion take fom (B.15) execise 41.1 Ṗ 1 = Q 1 P 8 Q 1+ Q R 4 ; Q 1 = 1 4 P 1Q (41.4) 1 Ṗ = Q P 1 8 Q 1 1+ Q 1 ; Q = 1 4 P Q 1. R 4 1 Individual electon nucleus collisions at 1 = Q 1 = 0 o =Q = 0 no longe pose a poblem to a numeical integation outine. The equations (B.15) ae singula only at the tiple collision R 1 = 0, i.e., when both electons hit the nucleus at the same time. The new coodinates and the Hamiltonian (B.14) ae vey useful when calculating tajectoies fo collinea helium; they ae, howeve, less intuitive as a visualization of the thee-body dynamics. We will theefoe efe to the old coodinates 1, when discussing the dynamics and the peiodic obits. 41. Chaos, symbolic dynamics and peiodic obits Let us have a close look at the dynamics in collinea helium. The electons ae attacted by the nucleus. Duing an electon nucleus collision momentum is tansfeed between the inne and oute electon. The inne electon has a maximal sceening effect on the chage of the nucleus, diminishing the attactive foce on the oute electon. This electon electon inteaction is negligible if the oute electon is fa fom the nucleus at a collision and the oveall dynamics is egula like in the 1-dimensional Keple poblem. Things change dastically if both electons appoach the nucleus nealy simultaneously. The momentum tansfe between the electons depends now sensitively on how the paticles appoach the oigin. Intuitively, these nealy missed tiple collisions ende the dynamics chaotic. A typical tajectoy is plotted in figue 41.3 (a) whee we used 1 and as the elevant axis. The dynamics can also be visualized in a Poincaé suface of section, see figue 41.3 (b). We plot hee the coodinate and momentum of the oute electon wheneve the inne paticle hits the nucleus, i.e., 1 o = 0. As the unstuctued gay egion of the Poincaé section fo small 1 illustates, the dynamics is chaotic wheneve the oute electon is close to the oigin duing a collision. Convesely, egula motions dominate wheneve the oute electon is fa fom the nucleus. As one of the electons escapes fo almost any stating condition, the system is unbounded: one electon (say electon 1) can escape, with an abitay amount of kinetic enegy taken by the fugative. The emaining electon is tapped in a Keple ellipse with total enegy in the ange [ 1, ]. Thee is no enegy baie which would sepaate the bound fom the unbound egions of the phase space. Fom geneal kinematic aguments one deduces that the oute electon will not etun when p 1 > 0, at p = 0, the tuning point of the inne electon. Only if the two electons appoach the nucleus almost symmetically along the line 1 =, and pass close to the tiple collision can the momentum tansfe between the electons be lage enough to kick one of the paticles out completely. In othe wods, the electon escape oiginates fom the nea tiple collisions. The collinea helium dynamics has some impotant popeties which we now list Reflection symmety The Hamiltonian (B.6) is invaiant with espect to electon electon exchange; this symmety coesponds to the mio symmety of the potential along the line 1 =, figue As a consequence, we can estict ouselves to the dynamics in the fundamental domain 1 and teat a cossing of the diagonal 1 = as a had wall eflection. The dynamics in the full domain can then be econstucted by unfolding the tajectoy though back-eflections. As explained in chapte 5, the dynamics in the fundamental domain is the key to the factoization of spectal deteminants, to be implemented hee in (41.15). Note also the similaity between the fundamental domain of the collinea potential figue 41.4, and the fundamental domain figue 15.1 (b) in the 3 disk system, a simple poblem with the same binay symbolic dynamics.

3 CHAPTER 41. HELIUM ATOM 737 CHAPTER 41. HELIUM ATOM 738 in depth: sect. 5.6, p Symbolic dynamics We have aleady made the claim that the tiple collisions ende the collinea helium fully chaotic. We have no poof of the assetion, but the analysis of the symbolic dynamics lends futhe cedence to the claim The potential in (41.1) foms a idge along the line 1 =. One can show that a tajectoy passing the idge must go though at least one two-body collision 1 = 0 o = 0 befoe coming back to the diagonal 1 =. This suggests a binay symbolic dynamics coesponding to the dynamics in the fundamental domain 1 ; the symbolic dynamics is linked to the Poincaé map = 0 and the symbols 0 and 1 ae defined as : if the tajectoy is not eflected fom the line 1 = between two collisions with the nucleus = 0; 1: if a tajectoy is eflected fom the line 1 = between two collisions with the nucleus = 0. Empiically, the symbolic dynamics is complete fo a Poincaé map in the fundamental domain, i.e., thee exists a one-to-one coespondence between binay symbol sequences and collinea tajectoies in the fundamental domain, with exception of the 0 cycle Peiodic obits The existence of a binay symbolic dynamics makes it easy to count the numbe of peiodic obits in the fundamental domain, as in sect Howeve, mee existence of these cycles does not suffice to calculate semiclassical spectal deteminants. We need to detemine thei phase space tajectoies and calculate thei peiods, topological indices and stabilities. A estiction of the peiodic obit seach to a suitable Poincaé suface of section, e.g. = 0 o 1 =, leaves us in geneal with a -dimensional seach. Methods to find peiodic obits in multidimensional spaces have been descibed in chapte 16. They depend sensitively on good stating guesses. A systematic seach fo all obits can be achieved only afte combining multi-dimensional Newton methods with intepolation algoithms based on the binay symbolic dynamics phase space patitioning. All cycles up to symbol length 16 (some 8000 pime cycles) have been computed by such methods, with some examples shown in figue All numeical evidence indicates that the dynamics of collinea helium is hypebolic, and that all peiodic obits ae unstable. Figue 41.5: Some of the shotest cycles in collinea helium. The classical collinea electon motion is bounded by the potential baie 1= / 1 / + 1/( 1+ ) and the condition i 0. The obits ae shown in the full 1 domain, the itineaies efes to the dynamics in the 1 fundamental domain. The last figue, the 14-cycle , is an example of a typical cycle with no symmety Note that the fixed point 0 cycle is not in this list. The 0 cycle would coespond to the situation whee the oute electon sits at est infinitely fa fom the nucleus while the inne electon bounces back and foth into the nucleus. The obit is the limiting case of an electon escaping to infinity with zeo kinetic enegy. The obit is in the egula (i.e., sepaable) limit of the dynamics and is thus maginally stable. The existence of this obit is also elated to intemittent behavio geneating the quasi egula dynamics fo lage 1 that we have aleady noted in figue 41.3 (b). Seach algoithm fo an abitay peiodic obit is quite cumbesome to pogam. Thee is, howeve, a class of peiodic obits, obits with symmeties, which can be easily found by a one-paamete seach. The only symmety left fo the dynamics in the fundamental domain is time evesal symmety; a time evesal symmetic peiodic obit is an obit whose tajectoy in phase space is mapped onto itself when changing (p 1, p ) ( p 1, p ), by evesing the diection of the momentum of the obit. Such an obit must be a libation o self-etacing cycle, an obit that uns back and foth along the same path in the ( 1, ) plane. The cycles 1, 01 and 001 in figue 41.5 ae examples of self-etacing cycles. Luckily, the shotest cycles that we desie most adently have this symmety.

4 CHAPTER 41. HELIUM ATOM 739 CHAPTER 41. HELIUM ATOM 740 Why is this obsevation helpful? A self-etacing cycle must stat pependicula to the bounday of the fundamental domain, that is, on eithe of the axis = 0 o 1 =, o on the potential bounday = 1. By shooting off tajectoies pependicula to the boundaies and monitoing the obits etuning to the bounday with the ight symbol length we will find time evesal symmetic cycles by vaying the stating point on the bounday as the only paamete. But how can we tell whethe a given cycle is self-etacing o not? All the elevant infomation is contained in the itineaies; a cycle is self-etacing if its itineay is invaiant unde time evesal symmety (i.e., ead backwads) and a suitable numbe of cyclic pemutations. All binay stings up to length 5 fulfill this condition. The symbolic dynamics contains even moe infomation; we can tell at which bounday the total eflection occus. One finds that an obit stats out pependicula to the diagonal 1 = if the itineay is time evesal invaiant and has an odd numbe of 1 s; an example is the cycle 001 in figue 41.5; to the axis = 0 if the itineay is time evesal invaiant and has an even numbe of symbols; an example is the cycle 0011 in figue 41.5; to the potential bounday if the itineay is time evesal invaiant and has an odd numbe of symbols; an example is the cycle 011 in figue All cycles up to symbol length 5 ae time evesal invaiant, the fist two non-time evesal symmetic cycles ae cycles and in figue Thei detemination would equie a two-paamete seach. The two cycles ae mapped onto each othe by time evesal symmety, i.e., they have the same tace in the 1 plane, but they tace out distinct cycles in the full phase space. We ae eady to integate tajectoies fo classical collinea helium with the help of the equations of motions (B.15) and to find all cycles up to length 5. Thee execise 41.5 is only one thing not yet in place; we need the govening equations fo the matix elements of the Jacobian matix along a tajectoy in ode to calculate stability indices. We will povide the main equations in the next section, with the details of the deivation elegated to the appendix C Local coodinates, Jacobian matix In this section, we will deive the equations of motion fo the Jacobian matix along a collinea helium tajectoy. The Jacobian matix is 4-dimensional; the two tivial eigenvectos coesponding to the consevation of enegy and displacements along a tajectoy can, howeve, be pojected out by suitable othogonal coodinates tansfomations, see appendix C. We will give the tansfomation to local coodinates explicitly, hee fo the egulaized coodinates (B.13), and state the esulting equations of motion fo the educed [ ] Jacobian matix. The vecto locally paallel to the tajectoy is pointing in the diection of the phase space velocity (8.3) v m = ẋ m (t)=ω mn H x n = (H P1, H P, H Q1, H Q ), = H with H Qi Q i, and H Pi = H P i, i=1,. The vecto pependicula to a tajectoy x(t)=(q 1 (t), Q (t), P 1 (t), P (t)) and to the enegy manifold is given by the gadient of the Hamiltonian (B.14) γ= H= (H Q1, H Q, H P1, H P ). H H By symmety v m γ m =ω mn x n x m = 0, so the two vectos ae othogonal. Next, we conside the othogonal matix O = (γ 1,γ,γ/R, v) (41.5) H P /R H Q H Q1 /R H P1 H = P1 /R H Q1 H Q /R H P H Q /R H P H P1 /R H Q1 H Q1 /R H P1 H P /R H Q with R= H = (HQ 1 + HQ + HP 1 + HP ), which povides a tansfomation to local phase space coodinates centeed on the tajectoy x(t) along the two vectos (γ, v). The vectosγ 1, ae phase space vectos pependicula to the tajectoy and execise 41.6 to the enegy manifold in the 4-dimensional phase space of collinea helium. The Jacobian matix (4.5) otated to the local coodinate system by O then has the fom m 11 m 1 0 m m= 1 m , M= OT mo 1 The lineaized motion pependicula to the tajectoy on the enegy manifold is descibed by the [ ] matix m; the tivial diections coespond to unit eigenvalues on the diagonal in the 3d and 4th column and ow. The equations of motion fo the educed Jacobian matix m ae given by ṁ=l(t)m(t), (41.6) with m(0)=1. The matix l depends on the tajectoy in phase space and has the fom l 11 l 1 0 l l= 1 l , 0

5 CHAPTER 41. HELIUM ATOM 741 CHAPTER 41. HELIUM ATOM 74 Table 41.1: Action S p (in units of π), Lyapunov exponent Λ p /T p fo the motion in the collinea plane, winding numbeσ p fo the motion pependicula to the collinea plane, and the topological index m p fo all fundamental domain cycles up to topological length 6. p S p /π ln Λ p σ p m p whee the elevant matix elements l i j ae given by l 11 = 1 R [H Q 1Q (H Q H P1 + H Q1 H P ) (41.7) +(H Q1 H P1 H Q H P )(H Q1Q 1 H QQ H P1P 1 + H PP )] l 1 = H Q1Q (H Q1 H Q H P1 H P ) +(HQ 1 + HP )(H QQ + H P1P 1 )+(HQ + HP 1 )(H Q1Q 1 + H PP ) 1 l 1 = R [(H Q 1P + H QP 1 )(H Q H P1 + H Q1 H P8 ) (HP 1 + HP )(H Q1Q 1 + H QQ ) (HQ 1 + HQ )(H P1P 1 + H PP )] l = l 11. Hee H QiQ j, H PiP j, i, j=1, ae the second patial deivatives of H with espect to the coodinates Q i, P i, evaluated at the phase space coodinate of the classical tajectoy Getting eady Now eveything is in place: the egulaized equations of motion can be implemented in a Runge Kutta o any othe integation scheme to calculate tajectoies. We have a symbolic dynamics and know how many cycles thee ae and how to find them (at least up to symbol length 5). We know how to compute the Jacobian matix whose eigenvalues ente the semiclassical spectal deteminant (38.1). By (37.17) the action S p is popotional to the peiod of the obit, S p = T p. Thee is, howeve, still a slight complication. Collinea helium is an invaiant 4-dimensional subspace of the full helium phase space. If we estict the dynamics to angula momentum equal zeo, we ae left with 6 phase space coodinates. That is not a poblem when computing peiodic obits, they ae oblivious to the othe dimensions. Howeve, the Jacobian matix does pick up exta contibutions. When we calculate the Jacobian matix fo the full poblem, we must also allow fo displacements out of the collinea plane, so the full Jacobian matix fo dynamics fo L = 0 angula momentum is 6 dimensional. Fotunately, the lineaized dynamics in and off the collinea helium subspace decouple, and the Jacobian matix can be witten in tems of two distinct [ ] matices, with tivial eigen-diections poviding the emaining two dimensions. The submatix elated to displacements off the linea configuation chaacteizes the lineaized dynamics in the additional degee of feedom, theθ-coodinate in figue It tuns out that the lineaized dynamics in the Θ coodinate is stable, coesponding to a bending type motion of the two electons. We will need the Floquet exponents fo all degees of feedom in evaluating the semiclassical spectal deteminant in sect The numeical values of the actions, Floquet exponents, stability angles, and topological indices fo the shotest cycles ae listed in table These numbes, needed fo the semiclassical quantization implemented in the next section, an also be helpful in checking you own calculations Semiclassical quantization of collinea helium Befoe we get down to a seious calculation of the helium quantum enegy levels let us have a bief look at the oveall stuctue of the spectum. This will give us a peliminay feel fo which pats of the helium spectum ae accessible with the help of ou collinea model and which ae not. In ode to keep the discussion as simple as possible and to concentate on the semiclassical aspects of ou calculations we offe hee only a ough oveview. Fo a guide to moe detailed accounts see emak 41.4.

6 CHAPTER 41. HELIUM ATOM 743 CHAPTER 41. HELIUM ATOM Stuctue of helium spectum 0 We stat by ecalling Boh s fomula fo the spectum of hydogen like oneelecton atoms. The eigenenegies fom a Rydbeg seies -0.5 N=5 N=6 N=7 N=8 E N = e4 m e Z N, (41.8) whee Ze is the chage of the nucleus and m e is the mass of the electon. Though the est of this chapte we adopt the atomic units e=m e = =1. -1 N=3 N=4 The simplest model fo the helium spectum is obtained by teating the two electons as independent paticles moving in the potential of the nucleus neglecting the electon electon inteaction. Both electons ae then bound in hydogen like states; the inne electon will see a chage Z=, sceening at the same time the nucleus, the oute electon will move in a Coulomb potential with effective chage Z 1=1. In this way obtain a fist estimate fo the total enegy E N,n = N 1 n with n> N. (41.9) This double Rydbeg fomula contains aleady most of the infomation we need to undestand the basic stuctue of the spectum. The (coect) ionizations thesholds E N = ae obtained in the limit n, yielding the gound and excited N states of the helium ion He +. We will theefoe efe to N as the pincipal quantum numbe. We also see that all states E N,n with N lie above the fist ionization theshold fo N = 1. As soon as we switch on electon-electon inteaction these states ae no longe bound states; they tun into esonant states which decay into a bound state of the helium ion and a fee oute electon. This might not come as a big supise if we have the classical analysis of the pevious section in mind: we aleady found that one of the classical electons will almost always escape afte some finite time. Moe emakable is the fact that the fist, N= 1 seies consists of tue bound states fo all n, an effect which can only be undestood by quantum aguments. The hydogen-like quantum enegies (41.8) ae highly degeneate; states with diffeent angula momentum but the same pincipal quantum numbe N shae the same enegy. We ecall fom basic quantum mechanics of hydogen atom that the possible angula momenta fo a given N span l=0, 1... N 1. How does that affect the helium case? Total angula momentum L fo the helium thee-body poblem is conseved. The collinea helium is a subspace of the classical phase space fo L=0; we thus expect that we can only quantize helium states coesponding to the total angula momentum zeo, a subspectum of the full helium spectum. Going back to ou cude estimate (41.9) we may now attibute angula momenta to the two independent electons, l 1 and l say. In ode to obtain total angula momentum L=0 we need l 1 = l = l and l z1 = l z, that is, thee ae N diffeent states coesponding to L=0 fo fixed quantum numbes N, n. That means that we expect N diffeent Rydbeg seies conveging to each ionization theshold E N = /N. This is indeed the case and the N diffeent seies can be identified also in the exact helium quantum spectum, see figue The Figue 41.6: The exact quantum helium spectum fo L = 0. The enegy levels denoted by bas have been obtained fom full 3-dimensional quantum calculations [41.3]. E [au] degeneacies between the diffeent N Rydbeg seies coesponding to the same pincipal quantum numbe N, ae emoved by the electon-electon inteaction. We thus aleady have a athe good idea of the coase stuctue of the spectum In the next step, we may even speculate which pats of the L=0 spectum can be epoduced by the semiclassical quantization of collinea helium. In the collinea helium, both classical electons move back and foth along a common axis though the nucleus, so each has zeo angula momentum. We theefoe expect that collinea helium descibes the Rydbeg seies with l=l 1 = l = 0. These seies ae the enegetically lowest states fo fixed (N, n), coesponding to the Rydbeg seies on the outemost left side of the spectum in figue We will see in the next section that this is indeed the case and that the collinea model holds down to the N = 1 bound state seies, including even the gound state of helium! We will also find a semiclassical quantum numbe coesponding to the angula momentum l and show that the collinea model descibes states fo modeate angula momentum l as long as l N.. emak 41.4 N=1 N= N=5 N=6 N=7 N=8

7 CHAPTER 41. HELIUM ATOM 745 CHAPTER 41. HELIUM ATOM Semiclassical spectal deteminant fo collinea helium Nothing but lassitude can stop us now fom calculating ou fist semiclassical eigenvalues. The only thing left to do is to set up the spectal deteminant in tems of the peiodic obits of collinea helium and to wite out the fist few tems of its cycle expansion with the help of the binay symbolic dynamics. The semiclassical spectal deteminant (38.1) has been witten as poduct ove all cycles of the classical systems. The enegy dependence in collinea helium entes the classical dynamics only though simple scaling tansfomations descibed in sect. B..1 which makes it possible to wite the semiclassical spectal deteminant in the fom det (Ĥ E) sc = exp 1 e i(ss p mpπ ) ( det (1 M p ))1/ det (1 M, (41.10) p ) 1/ p =1 with the enegy dependence absobed into the vaiable s= e me E, obtained by using the scaling elation (41.) fo the action. As explained in sect. 41.3, the fact that the [4 4] Jacobian matix decouples into two [ ] submatices coesponding to the dynamics in the collinea space and pependicula to it makes it possible to wite the denominato in tems of a poduct of two deteminants. Stable and unstable degees of feedom ente the tace fomula in diffeent ways, eflected by the absence of the modulus sign and the minus sign in font of det (1 M ). The topological index m p coesponds to the unstable dynamics in the collinea plane. Note that the facto e iπ N(E) pesent in (38.1) is absent in (41.10). Collinea helium is an open system, i.e., the eigenenegies ae esonances coesponding to the complex zeos of the semiclassical spectal deteminant and the mean enegy staicase N(E) not defined. In ode to obtain a spectal deteminant as an infinite poduct of the fom (38.18) we may poceed as in (.8) by expanding the deteminants in (41.10) in tems of the eigenvalues of the coesponding Jacobian matices. The matix epesenting displacements pependicula to the collinea space has eigenvalues of the fom exp(±πiσ), eflecting stable lineaized dynamics. σ is the full winding numbe along the obit in the stable degee of feedom, multiplicative unde multiple epetitions of this obit.the eigenvalues coesponding to the unstable dynamics along the collinea axis ae paied as{λ, 1/Λ} with Λ > 1 and eal. As in (.8) and (38.18) we may thus wite [ det (1 M ) det (1 M ) ] 1/ = [ (1 Λ )(1 Λ ) (1 e πiσ )(1 e πiσ ) ] 1/ 1 = Λ 1/ Λ k e i(l+1/)σ. k,l=0 (41.11) The ± sign coesponds to the hypebolic/invese hypebolic peiodic obits with positive/negative eigenvalues Λ. Using the elation (41.1) we see that the sum ove in (41.10) is the expansion of the logaithm, so the semiclassical spectal deteminant can be ewitten as a poduct ove dynamical zeta functions, as in (.8): det (Ĥ E) sc = ζk,m 1 = k=0 m=0 whee the cycle weights ae given by (1 t (k,m) k=0 m=0 p p ), (41.1) t (k,m) 1 π p mp p = Λ 1/ ei(ss 4π(l+1/)σ p) Λk, (41.13) and m p is the topological index fo the motion in the collinea plane which equals twice the topological length of the cycle. The two independent diections pependicula to the collinea axis lead to a twofold degeneacy in this degee of feedom which accounts fo an additional facto in font of the winding numbeσ. The values fo the actions, winding numbes and stability indices of the shotest cycles in collinea helium ae listed in table The intege indices l and k play vey diffeent oles in the semiclassical spectal deteminant (41.1). A lineaized appoximation of the flow along a cycle coesponds to a hamonic appoximation of the potential in the vicinity of the tajectoy. Stable motion coesponds to a hamonic oscillato potential, unstable motion to an inveted hamonic oscillato. The index l which contibutes as a phase to the cycle weights in the dynamical zeta functions can theefoe be intepeted as a hamonic oscillato quantum numbe; it coesponds to vibational modes in theθcoodinate and can in ou simplified pictue developed in sect be elated to the quantum numbe l=l 1 = l epesenting the single paticle angula momenta. Evey distinct l value coesponds to a full spectum which we obtain fom the zeos of the semiclassical spectal deteminant 1/ζ l keepingl fixed. The hamonic oscillato appoximation will eventually beak down with inceasing off-line excitations and thus inceasing l. The index k coesponds to excitations along the unstable diection and can be identified with local esonances of the inveted hamonic oscillato centeed on the given obit. The cycle contibutions t (k,m) p decease exponentially with inceasing k. Highe k tems in an expansion of the deteminant give coections which become impotant only fo lage negative imaginay s values. As we ae inteested only in the leading zeos of (41.1), i.e., the zeos closest to the eal enegy axis, it is sufficient to take only the k=0 tems into account. Next, let us have a look at the discete symmeties discussed in sect Collinea helium has a C symmety as it is invaiant unde eflection acoss the 1 = line coesponding to the electon-electon exchange symmety. As explained in example 5.9 and sect. 5.5, we may use this symmety to factoize the semiclassical spectal deteminant. The spectum coesponding to the states symmetic o antisymmetic with espect to eflection can be obtained by witing the dynamical zeta functions in the symmety factoized fom 1/ζ (l) = (1 t a ) s a (1 t s ). (41.14)

8 CHAPTER 41. HELIUM ATOM 747 CHAPTER 41. HELIUM ATOM 748 Hee, the fist poduct is taken ove all asymmetic pime cycles, i.e., cycles that ae not self-dual unde the C symmety. Such cycles come in pais, as two equivalent obits ae mapped into each othe by the symmety tansfomation. The second poduct uns ove all self-dual cycles; these obits coss the axis 1 = twice at a ight angle. The self-dual cycles close in the fundamental domain 1 aleady at half the peiod compaed to the obit in the full domain, and the cycle weights t s in (41.14) ae the weights of fundamental domain cycles. The C symmety now leads to the factoization of (41.14) 1/ζ = ζ+ζ 1 1, with 1/ζ (l) + = 1/ζ (l) = (1 t a ) (1 t s ), a s (1 t a ) (1+t s ), (41.15) a s setting k = 0 in what follows. The symmetic subspace esonances ae given by the zeos of 1/ζ + (l), antisymmetic esonances by the zeos of 1/ζ (l), with the two dynamical zeta functions defined as poducts ove obits in the fundamental domain. The symmety popeties of an obit can be ead off diectly fom its symbol sequence, as explained in sect An obit with an odd numbe of 1 s in the itineay is self-dual unde the C symmety and entes the spectal deteminant in (41.15) with a negative o a positive sign, depending on the symmety subspace unde consideation Cycle expansion esults So fa we have established a factoized fom of the semiclassical spectal deteminant and have theeby picked up two good quantum numbes; the quantum numbe m has been identified with an excitation of the bending vibations, the exchange symmety quantum numbe ±1 coesponds to states being symmetic o antisymmetic with espect to the electon-electon exchange. We may now stat witing down the binay cycle expansion (3.8) and detemine the zeos of spectal deteminant. Thee is, howeve, still anothe poblem: thee is no cycle 0 in the collinea helium. The symbol sequence 0 coesponds to the limit of an oute electon fixed with zeo kinetic enegy at 1 =, the inne electon bouncing back and foth into the singulaity at the oigin. This intoduces intemittency in ou system, a poblem discussed in chapte 9. We note that the behavio of cycles going fa out in the channel 1 o is vey diffeent fom those staying in the nea coe egion. A cycle expansion using the binay alphabet epoduces states whee both electons ae localized in the nea coe egions: these ae the lowest states in each Rydbeg seies. The states conveging to the vaious ionization thesholds E N = /N coespond to eigenfunctions whee the wave function of the oute electon is stetched fa out into the ionization channel 1,. To include those states, we have to deal with the dynamics in the limit of lage 1,. This tuns out to be equivalent to switching to a symbolic dynamics with an infinite alphabet. With this obsevation in mind, we may wite the cycle expansion emak 41.5 (...) fo a binay alphabet without the 0 cycle as 1/ζ l (s)= 1 t (l) 1 t(l) 01 [t(l) t(l) 011 t(l) 01 t(l) 1 ] [t (l) t(l) 0011 t(l) 001 t(l) 1 + t(l) 0111 t(l) 011 t(l) 1 ]... (41.16) The weights t (l) p ae given in (41.1), with contibutions of obits and composite obits of the same total symbol length collected within squae backets. The cycle expansion depends only on the classical actions, stability indices and winding numbes, given fo obits up to length 6 in table To get eacquainted with the cycle expansion fomula (41.16), conside a tuncation of the seies afte the fist tem 1/ζ (l) (s) 1 t 1. The quantization condition 1/ζ (l) (s)=0 leads to (S 1 /π) E m,n = [m+ 1 + (N+ 1 )σ m, N= 0, 1,,..., (41.17) 1], with S 1 /π=1.890 fo the action andσ 1 = fo the winding numbe, see table 41.1, the 1 cycle in the fundamental domain. This cycle can be descibed as the asymmetic stetch obit, see figue The additional quantum numbe N in (41.17) coesponds to the pincipal quantum numbe defined in sect The states descibed by the quantization condition (41.17) ae those centeed closest to the nucleus and coespond theefoe to the lowest states in each Rydbeg seies (fo a fixed m and N values), in figue The simple fomula (41.17) gives aleady a athe good estimate fo the gound state of helium! Results obtained fom (41.17) ae tabulated in table 41., see the 3d column unde j=1 and the compaison with the full quantum calculations. In ode to obtain highe excited quantum states, we need to include moe obits in the cycle expansion (41.16), coveing moe of the phase space dynamics futhe away fom the cente. Taking longe and longe cycles into account, we indeed eveal moe and moe states in each N-seies fo fixed m. This is illustated by the data listed in table 41. fo symmetic states obtained fom tuncations of the cycle expansion of 1/ζ +. execise 41.7 Results of the same quality ae obtained fo antisymmetic states by calculating the zeos of 1/ζ (l). Repeating the calculation withl=1 o highe in (41.15) eveals states in the Rydbeg seies which ae to the ight of the enegetically lowest seies in figue Résumé We have coveed a lot of gound stating with consideations of the classical popeties of a thee-body Coulomb poblem, and ending with the semiclassical hehelium - 7dec004 ChaosBook.og vesion15.7, Ap 8 015

9 CHAPTER 41. HELIUM ATOM 749 CHAPTER 41. HELIUM ATOM 750 Table 41.: Collinea helium, eal pat of the symmetic subspace esonances obtained by a cycle expansion (41.16) up to cycle length j. The exact quantum enegies [41.3] ae in the last column. The states ae labeled by thei pincipal quantum numbes. A dash as an enty indicates a missing zeo at that level of appoximation. N n j=1 j=4 j=8 j=1 j=16 E qm lium spectum. We saw that the thee-body poblem esticted to the dynamics on a collinea appeas to be fully chaotic; this implies that taditional semiclassical methods such as WKBquantization will not wok and that we needed the full peiodic obit theoy to obtain leads to the semiclassical spectum of helium. As a piece of unexpected luck the symbolic dynamics is simple, and the semiclassical quantization of the collinea dynamics yields an impotant pat of the helium spectum, including the gound state, to a easonable accuacy. A sceptic might say: Why bothe with all the semiclassical consideations? A staightfowad numeical quantum calculation achieves the same goal with bette pecision. While this is tue, the semiclassical analysis offes new insights into the stuctue of the spectum. We discoveed that the dynamics pependicula to the collinea plane was stable, giving ise to an additional (appoximate) quantum numbel. We thus undestood the oigin of the diffeent Rydbeg seies depicted in figue 41.6, a fact which is not at all obvious fom a numeical solution of the quantum poblem. Having tavesed the long oad fom the classical game of pinball all the way to a cedible helium spectum computation, we could declae victoy and fold down this entepise. Nevetheless, thee is still much to think about - what about such quintessentially quantum effects as diffaction, tunnelling,...? As we shall now see, the peiodic obit theoy has still much of inteest to offe. Commentay Remak 41.1 Souces. The full 3-dimensional Hamiltonian afte elimination of the cente of mass coodinates, and an account of the finite nucleus mass effects is given in ef. [41.]. The geneal two body collision egulaizing Kustaanheimo Stiefel tansfomation [41.5], a genealization of Levi-Civita s [41.13] Pauli matix two body collision egulaization fo motion in a plane, is due to Kustaanheimo [41.1] who ealized that the coect highe-dimensional genealization of the squae oot emoval tick (B.11), by intoducing a vecto Q with popety = Q, is the same as Diac s tick of getting linea equation fo spin 1/ femions by means of spinos. Vecto spaces equipped with a poduct and a known satisfy Q Q = Q define nomed algebas. They appea in vaious physical applications - as quatenions, octonions, spinos. The technique was oiginally developed in celestial mechanics [41.6] to obtain numeically stable solutions fo planetay motions. The basic idea was in place as ealy as 1931, when H. Hopf [41.14] used a KS tansfomation in ode to illustate a Hopf s invaiant. The KS tansfomation fo the collinea helium was intoduced in ef. [41.]. Remak 41. Complete binay symbolic dynamics. No stable peiodic obit and no exception to the binay symbolic dynamics of the collinea helium cycles have been found in numeical investigations. A poof that all cycles ae unstable, that they ae uniquely labeled by the binay symbolic dynamcis, and that this dynamics is complete is, howeve, still missing. The conjectued Makov patition of the phase space is given by the tiple collision manifold, i.e., by those tajectoies which stat in o end at the singula point 1 = = 0. See also ef. [41.]. Remak 41.3 Spin and paticle exchange symmety. In ou pesentation of collinea helium we have completely ignoed all dynamical effects due to the spin of the paticles involved, such as the electonic spin-obit coupling. Electons ae femions and that detemines the symmety popeties of the quantum states. The total wave function, including the spin degees of feedom, must be antisymmetic unde the electon-electon exchange tansfomation. That means that a quantum state symmetic in the position vaiables must have an antisymmetic spin wave function, i.e., the spins ae antipaallel and the total spin is zeo (singletstate). Antisymmetic states have symmetic spin wave function with total spin 1 (tipletstates). The theefold degeneacy of spin 1 states is lifted by the spin-obit coupling. Remak 41.4 Helium quantum numbes. The classification of the helium states in tems of single electon quantum numbes, sketched in sect , pevailed until the 1960 s; a gowing discepancy between expeimental esults and theoetical pedictions made it necessay to efine this pictue. In paticula, the diffeent Rydbeg seies shaing a given N-quantum numbe coespond, oughly speaking, to a quantization of the inte electonic angleθ, see figue 41.1, and can not be descibed in tems of single electon quantum numbes l 1, l. The fact that something is slightly wong with the single electon pictue laid out in sect is highlighted when consideing the collinea configuation whee both electons ae on the same side of the nucleus. As both electons again have angula momentum equal to zeo, the coesponding quantum states should also belong to single electon quantum numbes (l 1, l )=(0, 0). Howeve, the single electon pictue beaks down completely in the limit Θ = 0 whee electon-electon inteaction becomes the dominant effect. The quantum states coesponding to this classical configuation ae distinctively diffeent fom those obtained fom the collinea dynamics with electons on diffeent sides of the nucleus. The Rydbeg seies elated to the classical Θ = 0 dynamics ae on the outemost igth side in each N subspectum in figue 41.6, and contain the

10 EXERCISES 751 REFERENCES 75 enegetically highest states fo given N, n quantum numbes, see also emak A detailed account of the histoical development as well as a moden intepetation of the spectum can be found in ef. [41.1]. Remak 41.5 Beyond the unstable collinea helium subspace. The semiclassical quantization of the chaotic collinea helium subspace is discussed in efs. [41.7, 41.8, 41.9]. Classical and semiclassical consideations beyond what has been discussed in sect follow seveal othe diections, all outside the main of this book. A classical study of the dynamics of collinea helium whee both electons ae on the same side of the nucleus eveals that this configuation is fully stable both in the collinea plane and pependicula to it. The coesponding quantum states can be obtained with the help of an appoximate EBK-quantization which eveals helium esonances with extemely long lifetimes (quasi - bound states in the continuum). These states fom the enegetically highest Rydbeg seies fo a given pincipal quantum numbe N, see figue Details can be found in efs. [41.10, 41.11]. In ode to obtain the Rydbeg seies stuctue of the spectum, i.e., the succession of states conveging to vaious ionization thesholds, we need to take into account the dynamics of obits which make lage excusions along the 1 o axis. In the chaotic collinea subspace these obits ae chaacteized by symbol sequences of fom (a0 n ) whee a stands fo an abitay binay symbol sequence and 0 n is a succession of n 0 s in a ow. A summation of the fom n=0 t a0 n, whee t p ae the cycle weights in (41.1), and cycle expansion of indeed yield all Rydbeg states up the vaious ionization thesholds, see ef. [41.4]. Fo a compehensive oveview on specta of two-electon atoms and semiclassical teatments ef. [41.1]. ough patition can be used to initiate dimensional Newton-Raphson method seaches fo helium cycles, execise Collinea helium cycles. The motion in the ( 1, ) plane is topologically simila to the pinball motion in a 3-disk system, except that the motion is in the Coulomb potential. Just as in the 3-disk system the dynamics is simplified if viewed in the fundamental domain, in this case the egion between 1 axis and the 1 = diagonal. Modify you integation outine so the tajectoy bounces off the diagonal as off a mio. Miaculously, the symbolic dynamics fo the suvivos again tuns out to be binay, with 0 symbol signifying a bounce off the 1 axis, and 1 symbol fo a bounce off the diagonal. Just as in the 3-disk game of pinball, we thus know what cycles need to be computed fo the cycle expansion (41.16). Guess some shot cycles by equiing that topologically they coespond to sequences of bounces eithe etuning to the same i axis o eflecting off the diagonal. Now eithe Use special symmeties of obits such as self-etacing to find all obits up to length 5 by a 1- dimensional Newton seach Collinea helium cycle stabilities. Compute the eigenvalues fo the cycles you found in execise 41.5, as descibed in sect You may eithe integate the educed matix using equations (41.6) togethe with the geneating function l given in local coodinates by (41.7) o integate the full 4 4 Jacobian matix, see sect. L.1. Integation in 4 dimensions should give eigenvalues of the fom (1, 1,Λ p, 1/Λ p ); The unit eigenvalues ae due to the usual peiodic obit invaiances; displacements along the obit as well as pependicula to the enegy manifold ae conseved; the latte one povides a check of the accuacy of you computation. Compae with table 41.1; you should get the actions and Lyapunov exponents ight, but topological indices and stability angles we take on faith Helium eigenenegies. Compute the lowest eigenenegies of singlet and tiplet states of helium by substituting cycle data into the cycle expansion (41.16) fo the symmetic and antisymmetic zeta functions (41.15). Pobably the quickest way is to plot the magnitude of the zeta function as function of eal enegy and look fo the minima. As the eigenenegies in geneal have a small imaginay pat, a contou plot such as figue 3.1, can yield infomed guesses. Bette way would be to find the zeos by Newton method, sect. 3.. How close ae you to the cycle expansion and quantum esults listed in table 41.? You can find moe quantum data in ef. [41.3]. Refeences [41.1] G. Tanne, J-M. Rost and K. Richte, Rev. Mod. Phys. 7, 497 (000). Execises Kustaanheimo Stiefel tansfomation. Check the Kustaanheimo Stiefel egulaization fo collinea helium; deive the Hamiltonian (B.14) and the collinea helium equations of motion (B.15) Helium in the plane. Stating with the helium Hamiltonian in the infinite nucleus mass appoximation m he =, and angula momentum L=0, show that the thee body poblem can be witten in tems of thee independent coodinates only, the electon-nucleus distances 1 and and the inte-electon angleθ, see figue B Helium tajectoies. Do some tial integations of the collinea helium equations of motion (B.15). Due to the enegy consevation, only thee of the phase space coodinates (Q 1, Q, P 1, P ) ae independent. Altenatively, you can integate in 4 dimensions and use the enegy consevation as a check on the quality of you integato. The dynamics can be visualized as a motion in the oiginal configuation space ( 1, ), i 0 quadant, o, bette still, by an appopiately chosen -dimensional Poincaé section, execise Most tajectoies will un away, do not be supised - the classical collinea helium is unbound. Ty to guess appoximately the shotest cycle of figue A Poincaé section fo collinea Helium. Constuct a Poincaé section of figue 41.3b that educes the helium flow to a map. Ty to delineate egions which coespond to finite symbol sequences, i.e. initial conditions that follow the same topological itineay in figue 41.3a space fo a finite numbe of bounces. Such [41.] K. Richte, G. Tanne, and D. Wintgen, Phys. Rev. A 48, 418 (1993). [41.3] Büges A., Wintgen D. and Rost J. M., J. Phys. B 8, 3163 (1995). [41.4] G. Tanne and D. Wintgen Phys. Rev. Lett (1995). [41.5] P. Kustaanheimo and E. Stiefel, J. Reine Angew. Math. 18, 04 (1965). [41.6] E.L. Steifel and G. Scheifele, Linea and egula celestial mechanics (Spinge, New Yok 1971). [41.7] G.S. Eza, K. Richte, G. Tanne and D. Wintgen, J. Phys. B 4, L413 (1991). [41.8] D. Wintgen, K. Richte and G. Tanne, CHAOS, 19 (199). [41.9] R. Blümel and W. P. Reinhadt, Diections in Chaos Vol 4, eds. D. H. Feng and J.-M. Yuan (Wold Scientific, Hongkong), 45 (199). [41.10] K. Richte and D. Wintgen, J. Phys. B 4, L565 (1991). exehelium - 16ap00 ChaosBook.og vesion15.7, Ap efshelium - 6ma00 ChaosBook.og vesion15.7, Ap 8 015

11 Refeences 753 [41.11] D. Wintgen and K. Richte, Comments At. Mol. Phys. 9, 61 (1994). [41.1] P. Kustaanheimo, Ann. Univ. Tuku, Se. AI., 73 (1964). [41.13] T. Levi-Civita, Opee mathematische (1956). [41.14] H. Hopf, Math. Ann. 104 (1931). efshelium - 6ma00 ChaosBook.og vesion15.7, Ap 8 015