One-electron ion in a quantizing magnetic field
aa r X i v : . [ phy s i c s . a t o m - ph ] F e b One-electron ion in a quantizing magnetic field ∗ Ivan V. Demidov † Res. & Engn. Corp. “Mekhanobr-Tekhnika”, 22th Line of Vasilievsky Island, 3, Saint Petersburg 199106, Russia
Alexander Y. Potekhin ‡ Ioffe Institute, Politekhnicheskaya 26, Saint Petersburg 194021, Russia
A charged particle in a magnetic field possesses discrete energy levels associated with particlerotation around the field lines. A bound complex of particles with a nonzero net charge possesses ananalogous levels associated with its center-of-mass motion and, in addition, the levels associated withinternal degrees of freedom, that is with relative motions of its constituent particles. The center-of-mass and internal motions are mutually dependent, which complicates theoretical studies of thebinding energies, radiative transitions and other properties of the complex ions moving in quantizingmagnetic fields. In this work, we present a detailed derivation of practical expressions for thenumerical treatment of such properties of the hydrogenlike ions moving in strong quantizing magneticfields, which follows and supplements the previous works of Bezchastnov et al. Second, we deriveasymptotic analytic expressions for the binding energies, oscillator strengths, and photoionizationcross sections of the moving hydrogenlike ions in the limit of an ultra-strong magnetic field.
Keywords: atomic processes — magnetic fields — radiation mechanisms: general — stars: neutron
Contents
I. Introduction II. Generalities
III. Transverse basis
IV. Solution of the Schr¨odinger equation
V. Interaction with radiation
VI. Approximate solutions for ultra-strongfields ∗ This article is based on I. V. Demidov’s master thesis [1]. † Electronic address: [email protected] ‡ Electronic address: [email protected]
A. Wave functions and eigenenergies 191. Exterior solution for bound states 192. Interior solution for even states 193. Binding energies of even states 204. Odd bound states 225. Continuum states 22B. Overlap integrals between even states 221. The case of tightly bound states 232. Overlaps of tightly bound states withother even states 233. The bound-free case 23C. Transverse geometric size 24D. Radiative transitions for circular polarization 25
VII. Conclusions Acknowledgments Appendices C ( N ,n + ) k J N at large N ν ) 30E. Proof of Equation (249) 31 References I. INTRODUCTION
Properties of atomic and molecular systems in externalmagnetic fields have been intensively studied for severaldecades (see, e.g., extensive reviews [2–4]). The majorityof the studies considered them to be at rest and assumedthe atomic nuclei to be infinitely massive (fixed in space).The model of infinitely massive nuclei can serve as a con-venient first approximation, but it is a gross simplifica-tion for astrophysical simulations, because thermal mo-tion of atoms and ions across magnetic field lines breaksthe axial symmetry.The theory of motion of a system of point charges ina constant magnetic field is reviewed in [2, 5]. A com-prehensive calculation of hydrogen-atom energy spectra,taking account of the effects of motion across the strongmagnetic fields, was carried out in Refs. [6, 7]. Calcu-lations of the rates of different types of radiative tran-sitions and the absorption coefficients in neutron staratmospheres were performed in a series of studies (e.g.,Refs. [8, 9], and references therein). Based on these data,a model of the hydrogen atmosphere of a neutron starwith a strong magnetic field was elaborated [10, 11]. Thedatabase for astrophysical calculations was created usingthis model in Refs. [12, 13] (see also the review [14]).Quantum-mechanical calculations of the characteris-tics of the one-electron ion (e.g., He + ) that moves in astrong magnetic field were performed in Refs. [15–17],based on the formalism suggested by Bezchastnov [18].The basic difference from the case of a neutral atomis that the ion motion is restricted by the field in thetransverse plane, therefore it is quantized [2, 5]. Thederivation of the practical formulas for such calculationswas described rather sketchy in the above-cited papers.The primary goal of the present text is to expose a de-tailed derivation of such formulas. Our second aim is tosupplement the consideration of the bound-bound radia-tive transitions of a moving one-electron ion, which werestudied in [16, 17], by the bound-free transitions. Third,we derive asymptotic analytic expressions for the bind-ing energies and radiative transition rates in the limitof ultra-strong magnetic field. For the last purpose, weextend the method previously developed by Hasegawa &Howard [19] for a hydrogen atom with infinitely heavynucleus to the present case of a moving one-electron ion.In Sect. II we present the general formalism for treatingone and two charged particles in a uniform magnetic fieldand introduce basic notations. In Sect. III we presenta detailed derivation of the convenient basis of orbitals[18] for treating the problem of two charged particles ina strong magnetic field (as a by-product, we notice somecorrections to Ref. [18] near the end of Sect. III D). Sec-tion IV is devoted to the solution of the Schr¨odinger equa-tion and calculation of the main properties of the two-particle system using the basis constructed in Sect. III. InSect. V we present general formulas for treating interac-tion of such system of particles with radiation, based onthe solution described in Sect. IV. In Sect. VI, followingand extending the method of Hasegawa & Howard [19]and using the theory described in the preceding sections,we derive analytic approximations for wave functions,binding energies, the transverse geometric size, overlapintegrals and oscillator strengths between different quan-tum states, and photoionization cross sections of a one-electron ion. Section VII presents the conclusions. InAppendices we derive some useful supplementary rela-tions and prove some statements from the main text. II. GENERALITIESA. Charged particle in uniform magnetic field
The quantum-mechanical problem of motion of acharged particle in a uniform and constant magnetic fieldwas first solved by Rabi [20] and Landau [21]. In thissubsection we expose this solution for completeness andintroduce some basic notations.Let us consider a motion of a free particle with a pos-itive or negative charge Ze , where e is the elementarycharge, in a uniform magnetic field B . The Hamiltonianequals the kinetic energy operator H (1) = m ˙ r H (1) ⊥ + p z m , H (1) ⊥ = π ⊥ m , (1)where m is the mass, π = m ˙ r = p − Zec A ( r ) (2)is the kinetic momentum, A ( r ) is the vector potentialof the field, and p = − i ~ ∇ is the canonical momentumoperator conjugate to r . In Eq. (1) and hereafter, “ ⊥ ”denotes the “transverse” part, related to the motion inthe xy plane, while B is along the z -axis.A classical particle moves along a spiral around thenormal to the xy plane at the guiding center r c . In quan-tum mechanics, r c is an operator, related to the pseudo-momentum operator k : k = m ˙ r + Zec B × r = p − Zec A ( r ) + Zec B × r , (3) r c = cZeB k × B . (4)The pseudomomentum is a constant of motion in a homo-geneous magnetic field (unlike the canonical momentumand the kinetic momentum, which are not conserved).Coordinate operators of r c commute with H (1) ⊥ , but donot commute with each other: [ x c , y c ] = − i ~ c/ ( ZeB ).Another constant of motion, which generalizes the par-allel component of the orbital momentum l , is l z = ˆ z · (cid:18) r × k + π (cid:19) = c ZeB ( k − π ) , (5)where ˆ z = B /B is the unit vector in the magnetic fielddirection.In general, H (1) ⊥ should be supplemented by ( − B · ˆ µ ),where ˆ µ = g mag ( e/ mc ) ˆ S is the intrinsic magnetic mo-ment of the particle, ˆ S is the spin operator, and g mag is the spin g -factor ( g mag = − . . ~ k z of the lon-gitudinal momentum p z in H (1) are described by the wavefunctions ψ ( r ) = C norm e i k z z Φ ( Z ) ( r ⊥ ) , (6)where C norm is a normalization constant and r ⊥ =( x, y ) = ( r ⊥ cos ϕ, r ⊥ sin ϕ ). The form of the functionΦ ( Z ) ( r ⊥ ) depends on a choice of the gauge for A ( r ). Letus consider the cylindrical gauge A = 12 B × r . (7)Then ∇ · A = 0, A = B r ⊥ , and A · p = p · A = ( B × r ) · p . Therefore,( p · A + A · p ) ψ = − i ~ [ ∇ · ( A ψ ) + A · ∇ ψ ]= − i ~ ( ψ ∇ · A + 2 A · ∇ ψ )= 2 A · p ψ, (8)so that π = (cid:18) p − Zec A (cid:19) = p − Zec (cid:0) p · A + A · p (cid:1) + (cid:18) Zec (cid:19) A = p − Zec Bl z + (cid:18) Ze c (cid:19) B r ⊥ , (9)where l z is defined by Eq. (5). In the cylindrical gauge( k + π ) / p , therefore l z takes the same form as with-out the field.For H (1) ⊥ we have the following eigenvalue problem: (cid:18) p ⊥ m − σω c l z + mω r ⊥ (cid:19) Φ = E ⊥ Φ , (10)where ω c = | Z | eB/mc is the cyclotron frequency and σ ≡ sign Z . Since l z is an integral of motion, a solutionto Eq. (10) can be found separately for each its eigenvalue ~ λ with integer λ , such that it can be written in the polarcoordinate system ( r ⊥ , ϕ ), in the ( x, y ) plane asΦ ( Z ) ( r ⊥ , ϕ ) = f ( r ⊥ ) e i λϕ √ π . (11)Substitution of function (11) into Eq. (10) gives − ~ m r ⊥ dd r ⊥ (cid:18) r ⊥ d f d r ⊥ (cid:19) + (cid:18) ~ λ mr ⊥ + mω r ⊥ (cid:19) f = (cid:18) E ⊥ + σ ~ ω c λ (cid:19) f. (12)Let us introduce notations ρ = r ⊥ a ,Z , κ = E ⊥ ~ ω c + σ λ , (13)where a m ,Z = r ~ mω c = a m p | Z | , (14) and a m = p ~ c/eB is the magnetic length . Then Eq. (12)becomes d f d ρ + 1 ρ d f d ρ + κ ρ − (cid:18) λ ρ + 14 (cid:19) f = 0 . (15)At ρ →
0, it turns into the Cauchy-Euler equationd f d ρ + 1 ρ d f d ρ − λ ρ f = 0 , (16)which has the explicit solution f = A ρ λ/ + A ρ − λ/ .Normalizability of ψ ( r ⊥ , ϕ ) requires that we select a solu-tion with non-negative power, f ∼ ρ | λ | / ∼ r | λ |⊥ at ρ → f = ρ | λ | / e − ρ/ y ( ρ ) in Eq. (15) gives theconfluent hypergeometric equation ρ d y d ρ + ( | λ | + 1 − ρ ) d y d ρ − | λ | + 12 + κ = 0 . (17)Its solution is the Kummer function [22], which providesa finite f at ρ → ∞ only under the condition that κ − | λ | + 12 = n r , (18)where n r is non-negative integer, called radial quantumnumber . Recalling Eq. (13), we obtain E ⊥ = ~ ω c (cid:18) n r + 12 + | λ | − σλ (cid:19) (19)and y ( ρ ) = CL | λ | n r ( ρ ) , (20)where L | λ | n r ( ρ ) is a generalized Laguerre polynomial [22]and C is a normalization constant. Therefore, the nor-malized solutions to Eq. (10) can be written asΦ ( Z ) n,s ( r ⊥ ) = exp(i λϕ ) √ π a m ,Z I n + s,n r ⊥ a ,Z ! , (21)and E ( Z ) ⊥ ,n = ~ ω c (cid:18) n + 12 (cid:19) , (22)where s = σλ, n = n r + | λ | − s I n ′ n ( ρ ) ( n ′ = n + s ) are the normalizedLaguerre functions [23], which are proportional toe − ρ/ ρ ( n ′ − n ) / L n ′ − nn ( ρ ) and are normalized so that Z ∞ I n,s ( ρ ) d ρ = 1 . (24)Explicitly, I n ′ ,n ( ρ ) = e − ρ/ ρ ( n ′ − n ) / n X k =0 ( − k √ n ! n ′ ! ρ k k ! ( n − k )! ( n ′ − n + k )! , if n ′ ≥ n ; (25a) I n ′ ,n ( ρ ) = ( − n ′ − n I n,n ′ ( ρ ) , if n ′ < n . (25b)The number n in Eq. (23) enumerates the Landau energylevels (e.g., Ref. [24]). Its definition implies that, for agiven Landau level, the quantum number s is boundedfrom below: s = − n, − n + 1 , − n = 2 , . . . . (26)The functions Φ ( − n,s ( r ⊥ ), which describe electron mo-tion perpendicular to magnetic field, are often called Lan-dau functions . They satisfy the condition of orthogonal-ity Z R d r ⊥ Φ ( Z ) ∗ n s ( r ⊥ )Φ ( Z ) n s ( r ⊥ ) = δ n n δ s s (27)and completeness X n,s Φ ( Z ) ∗ ns ( r ⊥ , )Φ ( Z ) ns ( r ⊥ , ) = δ ( r ⊥ , − r ⊥ , ) , (28)therefore they form a complete basis in a Hilbert space.By construction, Φ ( Z ) ns ( r ⊥ ) is an eigenfunction of theorbital momentum projection operator l z with eigenvalue λ = σs and an eigenfunction of the squared transversekinetic momentum operator π ⊥ = 2 mH (1) ⊥ with eigen-value m ~ ω c (2 n + 1). Therefore, according to Eq. (5), itis also an eigenfunction of the squared transverse pseu-domomentum operator k with eigenvalue m ~ ω c (2˜ n + 1),where ˜ n = n + σλ = n + s . We can identify the eigenstatesusing the pair of numbers ( n, ˜ n ) instead of ( n, s ) and ac-cordingly to define the Landau functions with modifiedsubscripts (e.g., [17, 18]), F ( Z ) n, ˜ n ( r ⊥ ) = Φ ( Z ) n, ˜ n − n ( r ⊥ ) = e i σ (˜ n − n ) ϕ √ π a m ,Z I ˜ n,n r ⊥ a ,Z ! (29)( σ ≡ sign Z , a m ,Z ≡ a m / p | Z | ). From Eq. (25b) we seethat F ( Z ) n, ˜ n ( r ⊥ ) = ( − ˜ n − n F ( − Z )˜ n,n ( r ⊥ ) = F ( − Z )˜ n,n ( − r ⊥ ) . (30)Let us define cyclic components of any vector a as a ± = a x ± i a y √ , a = a z . (31)The transverse cyclic components of the kinetic mo-mentum operator and of the pseudomomentum opera-tor transform one Landau state | n, ˜ n i ⊥ , characterized by F ( Z ) n ˜ n ( r ⊥ ), into another Landau state: π α | n, ˜ n i ⊥ = − i σα ~ a m p n + 1 / − σα/ | n − σα, ˜ n i , (32) k α | n, ˜ n i ⊥ = i σα ~ a m p ˜ n + 1 / σα/ | n, ˜ n + σα i , (33)where α = ± σ = sign Z . B. Two charged particles in uniform magnetic field
Motion of two particles with masses m − and m + andcharges − e and Ze ( e > Z >
1) in a homogeneousmagnetic field B = (0 , , B ) is governed by the Hamilto-nian H = π − m − + π m + + V C = p − ,z m − + p ,z m + + H ⊥ + V C , (34)where H ⊥ = π − , ⊥ m − + π , ⊥ m + (35)determines the motion of the two non-interacting parti-cles according to Eq. (1), π − = p − + e c B × r − , π + = p + − Ze c B × r + (36)are the kinetic momenta, and V C = − Ze | r − − r + | (37)is the Coulomb potential.Longitudinal motion of the system as a whole can befactorized out by introducing the z -coordinate of center-of-mass Z = ( m + z + + m − z − ) /M , where M = m + + m − ,and the relative coordinate z = z − − z + . The change ofvariables ( z − , z + ) → ( z c . m . , z ) implies ∂∂z ∓ = ∂z c . m . ∂z ∓ ∂∂z c . m . + ∂z∂z ∓ ∂∂z = m ∓ M ∂∂z c . m . ± ∂∂z ; ∂ ∂ z ∓ = (cid:16) m ∓ M (cid:17) ∂ ∂z . m . ± m ∓ M ∂ ∂z c . m . ∂z + ∂ ∂z . Substituting it into the original Hamiltonian (34), we seethat this change of coordinates concerns only the firstand second terms, − ~ m − ∂ ∂z − − ~ m + ∂ ∂z = − ~ M ∂ ∂z . m . − ~ m ∗ ∂ ∂z = P z M + p z m ∗ , (38)where p z = − i ~ ∂/∂z and P z = − i ~ ∂/∂z c . m . are the rela-tive and total longitudinal momenta, respectively, and m ∗ = m + m − /M (39)is the reduced mass. Therefore we can fix the eigenvalueof the total longitudinal momentum P z and write thetotal wave function and energy asΨ P z ( r + , ⊥ , r − , ⊥ , z, z c . m . ) = e i P z z c . m . / ~ ψ ( r + , ⊥ , r − , ⊥ , z ) ,E tot = E + ~ P z M . (40)To find ψ ( r + , ⊥ , r − , ⊥ , z ), it is sufficient to consider thesystem that does not move along B . Thus we will assume P z = 0 hereafter.The wave function ψ and energy E satisfies theSchr¨odinger equation with the Hamiltonian H = p z m ∗ + H ⊥ + V C . (41)As follows from Sect. II A, the Landau functionsΦ ( − n − s − ( r − , ⊥ ) = ( − s − Φ (1) n − + s − , − s − ( r − , ⊥ ) andΦ ( Z ) n + s + ( r + , ⊥ ) are eigenfunctions of the operators π − , ⊥ and π , ⊥ , so that π − , ⊥ Φ ( − n − s − ( r − , ⊥ ) = ~ a (2 n − + 1) Φ ( − n − s − ( r − , ⊥ ) , (42) π , ⊥ Φ ( Z ) n + s + ( r + , ⊥ ) = ~ a ,Z (2 n + + 1) Φ ( Z ) n + s + ( r + , ⊥ ) , (43)where n ± ≥ s ± ≥ − n ± are integer quantumnumbers. Therefore, the transverse energy of two non-interacting particles, corresponding to the Hamiltonian H ⊥ , Eq. (35), is E ⊥ n − ,n + = ~ ω c+ (cid:18) n + + 12 (cid:19) + ~ ω c − (cid:18) n − + 12 (cid:19) , (44)where ω c − = eBm − c , ω c+ = ZeBm + c . (45)Since the eigenenergy (44) is degenerate with respect toquantum numbers s + and s − , there is a manifold of rep-resentations of eigenfunctions, one of the simplest being ψ n + ,s + ,n − ,s − ( r + , ⊥ , r − , ⊥ ) = Φ ( Z ) n + s + ( r + , ⊥ )Φ ( − n − s − ( r − , ⊥ ) . (46)In addition, the wave function (46) is an eigenfunction ofoperators k − , ⊥ and k , ⊥ , where k − = p − − e c B × r − , k + = p + + Ze c B × r + (47)are pseudomomentum operators, and an eigenfunctionof the operators of the z -projections of the angular mo-menta l − ,z and l + ,z . Therefore, any set of commut-ing operators ( π − , ⊥ , π , ⊥ , k − , k ), ( π − , ⊥ , π , ⊥ , k − , l + ,z ),( π − , ⊥ , π , ⊥ , l − ,z , k ), or ( π − , ⊥ , π , ⊥ , l − ,z , l + ,z ) can beused for determination of the quantum numbers n + , s + , n − , s − .However, the basis (46) is not optimal, because theabove sets of operators do not commute with our Hamil-tonian (41), which means that the four quantum num-bers n + , s + , n − , s − are not “good” in the presence of theCoulomb potential V C .On the other hand, there are operators, which com-mute with each other and with the Hamiltonian H : thetotal pseudomomentum k tot = k − + k + , (48) which is conserved due to translational invariance of H accompanied by the translational gauge transformationsof A [24], and the longitudinal component of the totalangular momentum L z = l − ,z + l + ,z (which is conservedbecause the potential V C is rotationally invariant). For acharged system, the transverse Cartesian components of k tot ( k tot ,x and k tot ,y ) do not commute with each otherand cannot be fixed simultaneously. Instead of selectingone of them, we may consider the square of pseudomo-mentum k . Thus we look for common eigenfunctions ofcommuting operators π − , ⊥ , π ⊥ , k , ⊥ , and L z , whichwill serve as a basis, corresponding to two integrals ofmotion of our system, because two operators of this setcommute with the Hamiltonian.Unlike the system of particles without external fields,the center-of-mass coordinates of the system of chargedparticles in a magnetic field cannot be completely elim-inated from the Hamiltonian. When an external mag-netic field is present, the collective behaviors of a neutralsystem of charged particles, such as an atom, and of acharged system, such as an atomic ion, are very different.In the former case, the collective motion is free whereas inthe latter case, a cyclotron motion arises. This differenceappears clearly in the detailed mathematical study ofAvron et al. [25]. For a neutral system, one can performso called pseudoseparation of the collective motion, afterwhich the resulting Hamiltonian for the internal degreesof freedom depends on the eigenvalues of the collectivepseudomomentum. For a charged system, the number ofintegrals of motion is less than the number of degrees offreedom, therefore the collective and individual coordi-nates and momenta cannot be separated. Thus the setof operators ( K , L z ) is not exclusively associated withthe collective motion but involves both the collective andinternal degrees of freedom.One can, however, perform an approximate separa-tion in the form H = H + H + H , where Hamil-tonians H and H describe the motion of quasipar-ticles corresponding to the collective and internal de-grees of freedom, respectively, and operator H couplesthe internal and collective phase coordinates. An ex-ample of such approximate separation was presented bySchmelcher & Cederbaum [26, 27], who applied a canon-ical transformation of variables to the Hamiltonian writ-ten in terms of the center-of-mass and relative coordi-nates of the particles. Baye and Vincke [5] developed ageneral framework for approximate separations. They in-troduced a parametrized approximate collective integralof motion, which in the two-particle case has the form C ( α ) = k tot , ⊥ − ( Z − e/c ) B × ( α r + , ⊥ + α r − , ⊥ ) , where α = 1 − α and α is a free parameter.In the following we will assume that m + ≫ m − . Bayeand Vincke [5] have shown that in this case any choiceof α will provide the approximate separation as longas α = O ( m − /m + ). The simplest choice α = 0 hasbeen introduced by Baye [28] and successfully used byBaye and Vincke [29] in the calculations of the collectivemotion corrections for atomic ions.However, since the separation of the collective motionis only approximate, there is no particular advantage inselecting the center of mass for describing the coordinateof the quasiparticle corresponding to the collective mo-tion. An equally reasonable choice can be just the coordi-nate of the heavy particle (the nucleus). Using the latterchoice, Bezchastnov [18] derived a basis of eigenstates ofthe transverse Hamiltonian H ⊥ , whose elements are alsoeigenstates of squared total pseudomomentum K , total z -projection of the angular momentum L z , and squaredkinetic momenta of each particle, π , ⊥ and π − , ⊥ . InSect. III we present a more detailed and physically trans-parent derivation of the same basis with some corrections. III. TRANSVERSE BASISA. Canonical transformations
Let us pass from variables ( r + , ⊥ , r − , ⊥ ) to variables( R ⊥ , r ⊥ ), where R ⊥ = r + , ⊥ , r ⊥ = r − , ⊥ − r + , ⊥ . (49)Hereafter we will also use the three-dimensional vectors R and r , assuming the z -components R z = 0 and r z = z ,so that r = r − − r + . The canonical momenta transformas p + = P − p , p − = p . (50)In the new variables, the transverse Hamiltonian (35)becomes H ⊥ = 12 m + (cid:18) P ⊥ − p ⊥ − Ze c B × R (cid:19) + 12 m − (cid:16) p ⊥ + e c B × r + e c B × R (cid:17) . (51)The total pseudomomentum [Eq. (48)] becomes k tot = P + ( Z − e c B × R − e c B × r . (52)We look for a unitary transformation, which will allowus to separate motion of quasiparticles with the totalcharge of the system, ( Z − e , and with the electroncharge, − e . For this aim, we should transform the firstbracket so as to add ( e/ c ) B × R in it, and the sec-ond bracket so as to subtract ( e/ c ) B × R . The uni-tary transformation operator can be written in the form U = exp( − i φ ), where φ is a Hermitian operator to bedetermined. The wave function is transformed as ψ = U ψ ′ , ψ ′ = U † ψ , and the Hamiltonian as H = U H ′ U † , H ′ = U † HU (symbol U † denotes the Hermitian adjointto U ). It is easy to check that for our purpose we can take U = exp (cid:18) − i e ~ c ( B × r ) · R (cid:19) = exp (cid:18) − i e ~ c ( B × r − ) · r + (cid:19) = exp (cid:18) i e ~ c ( B × r + ) · r − (cid:19) . (53)This operator performs a gauge transformation and shiftsthe momenta: U † P U = P + e c B × r , U † p U = p − e c B × R . (54)This shift cancels the third term in Eq. (52), and thetotal pseudomomentum becomes K = U † k tot U = P + ( Z − e c B × R . (55)The total angular momentum L = l − + l + = r + × p + + r − × p − = R × P + r × p (56)retains its form after the transformation: U † L U = L . The operators of kinetic momenta of the nucleus andthe electron become respectively U † π + U = Π − k , U † π − U = π . (57)where Π = P − ( Z − e c B × R , π = p + e c B × r , k = p − e c B × r . (58)Therefore the transformed transverse Hamiltonian equals H ′⊥ = U † H ⊥ U = 12 m + ( Π ⊥ − k ⊥ ) + π ⊥ m − . (59)We will consider Π and π as kinetic momenta of quasi-particles with charges ( Z − e and ( − e ), and k as apseudomomentum of the latter quasiparticle. By anal-ogy with Eq. (5), the z -projection of the total angularmomentum [Eq. (56)] equals L z = c Z − eB ( K ⊥ − Π ⊥ ) + c eB ( π ⊥ − k ⊥ ) . (60)Expanding the brackets in Eq. (59) and substituting k ⊥ = π ⊥ − (2 eB/c ) l z , we obtain H ′⊥ = H ′ + H ′ + H ′ , (61)where H ′ = Π ⊥ m + , H ′ = π ⊥ m ∗ , (62) H ′ = − m + Π ⊥ · k ⊥ − eBm + c l z . (63)Here, we have introduced the z -projection l z of the an-gular momentum of the second (negative) quasiparticleby analogy with Eq. (5). The term m − Π ⊥ · k ⊥ in H ′ couples together the motion of the two quasi-particles.Equivalently, k and π might be considered as the ki-netic momentum and pseudomomentum of a quasiparti-cle with charge + e . Accordingly, the transverse Hamil-tonian can be rewritten as H ′⊥ = H ′ + H ′′ + H ′′ , (64)where H ′ is the same as in Eq. (62) and H ′′ = k ⊥ m ∗ , H ′′ = − m + Π ⊥ · k ⊥ + eBm − c l z . (65)Comparing the last formula with Eq. (63), we see that H ′′ = H ′ + ( eB/m ∗ c ) l z . The latter decomposition (64)was used in Ref. [18]. We prefer to use the former de-composition, Eq. (61), because H ′ → m + → ∞ ,ensuring asymptotic decoupling for massive ions.Operators H ′ and H ′ ( H ′′ ) have the form of the Hamil-tonian of free quasiparticles with charges ( Z − e and e ( − e ) and masses m + and m ∗ , respectively. and H ′ ( H ′′ )couples them together. According to Sect. II A, the eigen-functions of H ′ are Φ ( Z − N,S ( R ⊥ ), N ≥ S ≥ − N , andthe eigenenergies are independent of S , E ,N = ~ ω c1 (cid:18) N + 12 (cid:19) , (66)where ω c1 = ( Z − eBm + c = Z − Z ω c+ . (67)Analogously, the eigenfunctions of H ′ are Φ (1) n,s ( r ⊥ ), n ≥ s ≥ − n , and the eigenenergies are independent of s , E ,n = ~ ω c2 (cid:18) n + 12 (cid:19) , (68)where ω c2 = eBm ∗ c = (cid:18) m − m + (cid:19) ω c − . (69) B. Creation and annihilation operators
Instead of S and s , it is sometimes convenient to usequantum numbers ˜ N = N + S and ˜ n = n + s . As fol-lows from Eq. (21), an interchange of n with ˜ n or N with˜ N does not affect the modulus of a single-particle eigen-function. We will use these quantum numbers to specifyquantum states | N, ˜ N i and | n, ˜ n i of the two quasiparti-cles, described by Hamiltonians H ′ and H ′ , respectively,in the ( xy )-plane.Let us consider the cyclic components (31) of the op-erators of the kinetic momenta and pseudomomenta of the quasiparticles. According to Eqs. (32) and (33), theoperators ˆ a = i a m ~ π − , ˆ˜ a = − i a m ~ k +1 , (70a)ˆ b = i a m ,Z − ~ Π +1 , ˆ˜ b = − i a m ,Z − ~ K − (70b)lower the quantum numbers n, ˜ n, N, ˜ N by one, as follows:ˆ a | n, ˜ n i = √ n | n − , ˜ n i , (71a)ˆ˜ a | n, ˜ n i = √ ˜ n | n, ˜ n − i , (71b)ˆ b | N, ˜ N i = √ N | N − , ˜ N i , (71c)ˆ˜ b | N, ˜ N i = p ˜ N | N, ˜ N − i . (71d)Their Hermitian adjoint operatorsˆ a † = − i a m ~ π +1 , ˆ˜ a † = i a m ~ k − , (72a)ˆ b † = − i a m ,Z − ~ Π − , ˆ˜ b † = i a m ,Z − ~ K +1 (72b)raise the quantum numbers n, ˜ n, N, ˜ N by one:ˆ a † | n, ˜ n i = √ n + 1 | n + 1 , ˜ n i , (73a)ˆ˜ a † | n, ˜ n i = √ ˜ n + 1 | n, ˜ n + 1 i , (73b)ˆ b † | N, ˜ N i = √ N + 1 | N + 1 , ˜ N i , (73c)ˆ˜ b † | N, ˜ N i = p ˜ N + 1 | N, ˜ N + 1 i . (73d)As far as ˆ a, ˆ˜ a, ˆ b, ˆ˜ b can be considered as annihilation oper-ators for excitations in n, ˜ n, N, ˜ N , their Hermitian adjointoperators can be considered as the creation operators.It is noteworthy that π ± = p ± ± i eB c r ± , (74) k ± = p ± ∓ i eB c r ± , (75)Π ± = P ± ∓ i( Z − eB c R ± , (76) K ± = P ± ± i( Z − eB c R ± . (77)Therefore, r ± = ± i ceB ( k ± − π ± ) , (78) R ± = ∓ i c ( Z − eB ( K ± − Π ± ) , (79)so that r +1 = a m (ˆ a † − ˆ˜ a ) , r − = a m (ˆ a − ˆ˜ a † ) , (80a) R +1 = a m ,Z − (ˆ b − ˆ˜ b † ) , R − = a m ,Z − (ˆ b † − ˆ˜ b ) . (80b)It is also useful to consider the circular componentsof the kinetic momentum of the nucleus π + . Accordingto Eq. (32), these operators change the nucleus Landaunumber n + by ± π + , +1 | n + , ˜ n + i = − i ~ a m ,Z √ n + | n + − , ˜ n + i ⊥ , (81a) π + , − | n + , ˜ n + i = i ~ a m ,Z p n + + 1 | n + + 1 , ˜ n + i ⊥ . (81b)The canonical transformation of these operators withaccount of Eq. (57) and Eqs. (70), (72) gives U † π + , +1 U = Π +1 − k +1 = − i ~ a m (cid:16) √ Z − b + ˆ˜ a (cid:17) , (82a) U † π + , − U = Π − − k − = i ~ a m (cid:16) √ Z − b † + ˆ˜ a † (cid:17) . (82b) C. Good quantum numbers
The effective quasiparticle Hamiltonians can be writ-ten in terms of the creation and annihilation operators(Sect. III B) as H ′ = ~ ω c1 (cid:18) ˆ b † ˆ b + 12 (cid:19) , H ′ = ~ ω c2 (cid:18) ˆ a † ˆ a + 12 (cid:19) . (83)Using the expressions l z = ~ (ˆ˜ a † ˆ˜ a − ˆ˜ a † ˆ˜ a ) and Π ⊥ · k ⊥ = − ~ a √ Z − a † ˆ b + ˆ˜ a ˆ b † ) , (84)we can write the coupling operator H ′ in Eq. (63) as H ′ = ~ ω c1 √ Z − a † ˆ b + ˆ b † ˆ˜ a ) + ~ ω c+ (ˆ˜ a † ˆ˜ a − ˆ a † ˆ a ) . (85)Equations (83) and (85) do not contain operators b ′ and b ′† , which determine the square of total transversepseudomomentum K ⊥ = 2 ~ a ( Z − (cid:18) ˆ˜ b † ˆ˜ b + 12 (cid:19) , (86)which confirms that K ⊥ is an integral of motion([ H ′⊥ , K ⊥ ] = 0) and the related quantum number ˜ N = h ˆ˜ b † ˆ˜ b i is conserved. In other words, ˜ N is a good quantumnumber.For the z -projection of the total angular momentum,Eq. (60) gives L z = ~ (ˆ˜ b † ˆ˜ b − ˆ b † ˆ b + ˆ a † ˆ a − ˆ˜ a † ˆ˜ a ) . (87)The eigenvalues of L z equal ~ L = ~ ( S − s ). It is easy tocheck that [ H ′⊥ , L ′ z ] = 0. Therefore, L is a good quantumnumber. From Eq. (87) we see that L = ˜ N − N − ˜ n + n. (88)Finally, using Eqs. (83), (85), and the expression π ⊥ = ~ a (2ˆ a † ˆ a + 1) , (89)we can check that [ π ⊥ , H ′⊥ ] = 0. It means that n = h a † a i is a good quantum number (as long as we disregard V C ). D. Transverse basis states
The results of Sect. III C allow us to consider eigen-states of the transverse Hamiltonian H ′⊥ with fixed num-bers ˜ N , n, L . On the other hand, the quantum numbers S and s = S − L , or equivalently N = ˜ N − S and ˜ n = n + s are not well defined, because l z does not commute with H ′ . Since the eigenfunctions F ( Z − N, ˜ N ( R ⊥ ) F ( − n, ˜ n ( r ⊥ ) ofthe states | N, ˜ N , n, ˜ n i ⊥ = | N, ˜ N i ⊗ | n, ˜ n i with fixed˜ N and n (recall that | N, ˜ N i and | n, ˜ n i are the eigen-states of H ′ and H ′ , respectively) form a complete basisin the Hilbert space of functions of ( R ⊥ , r ⊥ ), the eigen-states of H ′⊥ can be looked as superpositions of states | N, ˜ N , n, ˜ n i ⊥ with different N and ˜ n . Taking into ac-count the constraint N + ˜ n = N , where N ≡ ˜ N − L + n ,we can write the eigenfunction Ψ ′ = U Ψ of H ′⊥ asΨ ′ ˜ N,n,L ( R ⊥ , r ⊥ ) = N X ˜ n =0 C ˜ n F ( Z − N − ˜ n, ˜ N ( R ⊥ ) F ( − n, ˜ n ( r ⊥ ) , (90)where C ˜ n are some constants, which may depend on˜ N , n, L . The eigenfunctions of the initial Hamiltonian H ⊥ areΨ ˜ N,n,L ( r + , ⊥ , r − , ⊥ ) = U N X ˜ n =0 C ˜ n F ( Z − N − ˜ n, ˜ N ( r + , ⊥ ) F ( − n, ˜ n ( r − , ⊥ − r + , ⊥ ) , (91)where U is given by Eq. (53).By construction, Ψ ˜ N,n,L ( r + , ⊥ , r − , ⊥ ) is an eigenfunc- tion of L z and K for any coefficients C ˜ n . Let us considerits transformation under the action of operators π ± , ⊥ [Eq. (36)]. Equation (89) gives π ⊥ | N, ˜ N , n, ˜ n i = ~ a (2 n + 1) | N, ˜ N , n, ˜ n i . (92)On the other hand, according to Eq. (57), π ⊥ = U † π − U .Therefore, Ψ ˜ N,n,L ( r + , ⊥ , r − , ⊥ ) is an eigenfunction of π − , ⊥ with the appropriate eigenvalue ( ~ /a m ) (2 n + 1) for any set of C ˜ n , which means that n − = n . Therefore,we can write N = ˜ N − L + n − (93)The operator π , ⊥ , being transformed according toEq. (57), can be expressed using Eq. (84) as( Π ⊥ − k ⊥ ) = ~ a h Z − (cid:16) ˆ b † ˆ b + 1 (cid:17) + 2ˆ˜ a † ˆ˜ a + 1 + 2 √ Z − a † ˆ b + ˆ b † ˆ˜ a ) i . (94)From this equation, taking into account that N = N − ˜ n , we obtain( Π ⊥ − k ⊥ ) | N, ˜ N, n, ˜ n i ⊥ = 2 ~ a (cid:20) (cid:18) ( Z − N − ( Z − n + Z (cid:19) |N − ˜ n, ˜ N , n, ˜ n i ⊥ + √ Z − p ( N − ˜ n )(˜ n + 1) |N − ˜ n − , ˜ N, n, ˜ n + 1 i ⊥ + √ Z − p ( N − ˜ n + 1) ˜ n |N − n + 1 , ˜ N, n, ˜ n − i ⊥ (cid:21) . (95)Using this relation with Eq. (90) and changing the summation index ˜ n so as to collect together the homogeneousterms with F ( Z − N − ˜ n, ˜ N ( R ⊥ ) F ( − n, ˜ n ( r ⊥ ), we obtain( Π ⊥ − k ⊥ ) Ψ ′ ˜ N,n,L ( R ⊥ , r ⊥ ) = 2 ~ a N X ˜ n =0 (cid:20) (cid:18) ( Z − N − ( Z − n + Z (cid:19) C ˜ n + √ Z − p ( N − ˜ n + 1) ˜ n C ˜ n − + √ Z − p ( N − ˜ n )(˜ n + 1) C ˜ n +1 (cid:21) F ( Z − N − ˜ n, ˜ N ( R ⊥ ) F ( − n, ˜ n ( r ⊥ ) . (96)Comparing Eq. (96) with Eq. (90), we see that Ψ ′ ˜ N,n,L ( R ⊥ , r ⊥ ) will be an eigenfunction of ( Π ⊥ − k ⊥ ) = U † π , ⊥ U under the condition that the coefficients C ˜ n satisfy the relation [41] √ Z − hp ( N − ˜ n + 1)˜ n C ˜ n − + p ( N − ˜ n )(˜ n + 1) C ˜ n +1 i = (cid:2) ( Z −
2) ˜ n − ( Z − N + Zn + (cid:3) C ˜ n , (97)where n + = 0 , , , . . . , N . Imposing this relation, we see from Eq. (96) that π , ⊥ Ψ ˜ N,n,L ( r + , ⊥ , r − , ⊥ ) = 2 ~ a ,Z (cid:18) n + + 12 (cid:19) Ψ ˜ N,n,L ( r + , ⊥ , r − , ⊥ ) , (98)Each value of n + corresponds to an eigenvector { C ˜ n } (˜ n = 0 , , . . . , N ). We see that the numbers ˜ N , n = n − ,and L affect the eigenvalue problem only in combination N = ˜ N + n − − L . Therefore each eigenvector may bemarked by only two numbers N and n + . These eigen-vectors are orthonormal, N X k =0 C ( N ,n ′ + ) k C ( N ,n + ) k = δ n ′ + ,n + , (99) and satisfy the completeness condition, N X n + =0 C ( N ,n + ) k ′ C ( N ,n + ) k = δ k ′ k , (100)therefore they form an orthonormal basis in a ( N + 1)-dimensional vector space.Thus we have built a basis of eigenstates of four oper-0ators K ⊥ , L z , π , and π − , such that | ˜ N , L, n − , n + i ⊥ , = N X k =0 C ( N ,n + ) k |N − k, ˜ N , n − , k i ⊥ , (101) described by wave functionsΨ ˜ N,L,n − ,n + ( r + , ⊥ , r − , ⊥ ) = U † N X k =0 C ( N ,n + ) k F ( Z − N − k, ˜ N ( r + , ⊥ ) F ( − n − ,k ( r − , ⊥ − r + , ⊥ ) . (102)They are characterized by 4 discrete quantum numbers, related to the four degrees of freedom for motion of the 2charged particles perpendicular to the magnetic field: K ⊥ | ˜ N , L, n − , n + i ⊥ , = ~ a ,Z − (2 ˜ N + 1) | ˜ N, L, n − , n + i ⊥ , , (103) L z | ˜ N , L, n − , n + i ⊥ , = ~ L | ˜ N, L, n − , n + i ⊥ , , (104) π − | ˜ N, L, n − , n + i ⊥ , = ~ a (2 n − + 1) | ˜ N, L, n − , n + i ⊥ , , (105) π | ˜ N, L, n − , n + i ⊥ , = ~ a ,Z (2 n + + 1) | ˜ N, L, n − , n + i ⊥ , , (106)˜ N ≥ , n − ≥ , L ≤ ˜ N + n − , ≤ n + ≤ N = ˜ N + n − − L. (107)According to Eqs. (35), (105), and (106), H ⊥ | ˜ N , L, n − , n + i ⊥ , = E ⊥ n − ,n + | ˜ N , L, n − , n + i ⊥ , , (108)where E ⊥ n − ,n + is defined by Eq. (44). This basis is orthonormal, h ˜ N ′ , L ′ , n ′− , n ′ + | ˜ N , L, n − , n + i ⊥ , = δ ˜ N ′ ˜ N δ L ′ L δ n ′− n − δ n ′ + n + (109)and complete, X ˜ N,L,n − ,n + Ψ ∗ ˜ N,L,n − ,n + ( r ′ + , ⊥ , r ′− , ⊥ )Ψ ˜ N,L,n − ,n + ( r + , ⊥ , r − , ⊥ ) = δ ( r ′ + , ⊥ − r + , ⊥ ) δ ( r ′− , ⊥ − r − , ⊥ ) . (110)Using Eq. (30), one can rewrite Eq. (102) in the form [42]Ψ ˜ N,L,n − ,n + ( r + , ⊥ , r − , ⊥ ) = U † N X ˜ n =0 ( − n − − ˜ n C ( N ,n + )˜ n F ( Z − N − ˜ n, ˜ N ( r + , ⊥ ) F (1)˜ n,n − ( r ⊥ ) , (111)which is equivalent to the representation used in Ref. [18]. E. Recurrence relations for the basis coefficients
Equation (97) can be rewritten in the form p ( N − ˜ n + 1) ˜ n C ( N ,n + )˜ n − + p ( N − ˜ n )(˜ n + 1) C ( N ,n + )˜ n +1 = Z (˜ n + n + − N ) − n + N√ Z − C ( N ,n + )˜ n (112)and used as a recurrence relation to calculate the coefficients C ( N ,n + )˜ n for given values of N and n + . One can startthe recurrent procedure from an arbitrary value of the initial coefficient to (for instance C ( N ,n + )0 = 1 for the upward1recurrence of C ( N ,n + ) N = 1 for the downward recurrence), and then scale them by a single number factor to satisfy thenormalization relation (99), N X ˜ n =0 h C ( N ,n + )˜ n i = 1 , (113)The upward recurrence (that is, starting from ˜ n = 0 to higher ˜ n is stable under the condition that it is performed for n + ≥ N − Z , whereas the downward recurrence (starting from ˜ n = N to lower ˜ n ) is stable for n + < N − Z [43].Supplementary recurrence relations for coefficients C ( N ,n + )˜ n are derived in Appendix A. IV. SOLUTION OF THE SCHR ¨ODINGER EQUATIONA. Expansion on the transverse basis
We are looking for eigenfunctions of Hamiltonian H in Eq. (41). The potential V C commutes with the transformedtotal pseudomomentum K [Eq. (55)] and the total angular momentum L [Eq. (56)]. Therefore, we may considerstates with definite K and L z , that is to fix ˜ N and L . However, V C does not commute with squared kinetic momenta π ± . Let us expand the states with fixed ˜ N and L over the complete basis of states constructed in Sect. III D: ψ κ ( r + , ⊥ , r − , ⊥ , z ) = X n − ,n + Ψ ˜ N,L,n − ,n + ( r + , ⊥ , r − , ⊥ ) g n − n + ; κ ( z ) . (114)Here κ is a compound quantum number, which is assigned to the considered quantum state and includes ˜ N and L .Let us substitute Eq. (114) into the Schr¨odinger equation Hψ κ = E κ ψ κ , (115)multiply both sides by Ψ ∗ ˜ N,L,n − ,n + ( r + , ⊥ , r − , ⊥ ), and integrate over r + , ⊥ , r − , ⊥ . Using Eqs. (34), (108), and (109), weobtain (cid:18) p z m ∗ − E k κ (cid:19) g n − n + ; κ ( z ) = − ∞ X n ′− =0 N X n ′ + =0 V ( ˜ N − L ) n ′− ,n ′ + ; n − ,n + ( z ) g n ′− n ′ + ; κ ( z ) , (116)where E k κ = E κ − E ⊥ n − ,n + (117)is the energy corresponding to the relative motions along z and V ( ˜ N − L ) n ′− ,n ′ + ; n − ,n + ( z ) = N ′ X k ′ =0 N X k =0 C ( N ′ ,n ′ + ) k ′ C ( N ,n + ) k Z R F ( Z − ∗N ′ − k ′ , ˜ N ( R ⊥ ) F ( Z − N − k, ˜ N ( R ⊥ ) d R ⊥ × Z R F ( − ∗ n ′− ,k ′ ( r ⊥ ) − Ze √ r + z F ( − n − ,k ( r ⊥ ) d r ⊥ (118)is an effective one-dimensional potential, N ′ = ˜ N − L + n ′− , N = ˜ N − L + n − . Since the Coulomb potential does notcontain R ⊥ , the first integral in Eq. (118) equals δ N ′ − k ′ , N − k = δ k ′ ,n ′− − n − + k . Thus, using Eq. (29). we obtain V ( ˜ N − L ) n ′− ,n ′ + ; n − ,n + ( z ) = N X k = k min C ( N ′ ,n ′ + ) n ′− − n − + k C ( N ,n + ) k Z R Φ ( − ∗ n ′− ,s ( r ⊥ ) − Ze p r ⊥ + z Φ ( − n − ,s ( r ⊥ ) d r ⊥ , (119) k min = max(0 , n − − n ′− ).Since the transverse basis is complete, the infinite system (116) is equivalent to the Schr¨odinger equation (115).Truncating the sum (114), one obtains a finite system, which solves Eq. (115) approximately. The same finite systemof equations can be obtained from the variational principle on the truncated basis.2 B. Calculation of effective potentials
It is convenient to define reduced (dimensionless) potentials through the relations V ( ˜ N − L ) n ′− ,n ′ + ; n − ,n + ( z ) = − Ze a m √ v ( ˜ N − L ) n ′− ,n ′ + ; n − ,n + ( ζ ) , ζ ≡ za m √ , (120) v ( ˜ N − L ) n ′− ,n ′ + ; n − ,n + ( ζ ) = ˜ N − L X s = s min C ( N ′ ,n ′ + ) n ′− + s C ( N ,n + ) n − + s v n ′− ,n − ; s ( ζ ) , (121)where s min = − min( n − , n ′− ), and v n ′ ,n ; s ( ζ ) = Z ∞ I n + s,n ( ρ ) I n ′ + s,n ′ ( ρ ) p ρ + ζ d ρ. (122)The functions v n ′ ,n ; s ( ζ ) belong to the class of effective potentials studied in Ref. [7]. Thanks to the relation v n ′ ,n ; s ( ζ ) = v n ′ r ,n r ; | s | ( ζ ), where n ′ r = n + ( s − | s | ) / n r = n + ( s − | s | ) / | n ′ , s i and | n, s i , it is sufficient to consider only non-negative subscripts of thesefunctions. Using Eq. (25), we obtain v n,k ; s ( ζ ) = p n !( n + s )! k !( k + s )! n X l =0 l !( n − l )!( l + s )! k X m =0 ( − m + l ( s + l + m )! m !( k + m )!( s + m )! v s + l + m ( ζ ) , (123)and v s ( ζ ) = 1 s ! Z ∞ x s e − x d x p ζ + x . (124)A change of variable brings the latter integral to v s ( ζ ) = 1 s ! Z ∞ ζ ( x − ζ ) s e ζ − x d x √ x = s X l =0 ( − l s ! l !( s − l )! ζ l e ζ Γ( s − l + 1 / , ζ ) , (125)where Γ( a, x ) is the incomplete gamma function. Atsmall or moderate ζ , v s ( ζ ) can be calculated using therecurrence relation v s +1 ( ζ ) = (2 s + 1) v s ( ζ )+ ζ s + 1 (cid:2) v s − ( ζ ) − v s ( ζ ) (cid:3) , (126) v ( ζ ) = √ π e ζ erfc( | ζ | ) , (127) v ( ζ ) = 1 − ζ v ( ζ ) + | ζ | . (128)Here, erfc( ζ ) is the complementary error function, whichcan be calculated using, e.g., an expansion in power seriesat small | ζ | and continued fractions at | ζ | & | ζ | or s , however, the recurrencerelation (128) fails because of round-off errors in positiveand negative terms, which nearly annihilate. In this case,one can use the asymptotic formula v s ( ζ ) ∼ p ζ + 1 + s (cid:18) s ( ζ + 1 + s ) (cid:19) . (129) C. Adiabatic approximation
In strong magnetic fields, such that ~ ω c − ≫ Z Ha(where Ha= m − e / ~ is the atomic unit of energy),the system of equations (116) approximately splits intoseparate subsystems with fixed n − . Indeed, since themagnetic length a m is small at large B , the denomina-tor in Eq. (119) remains nearly constant over the rangewhere the Landau functions Φ in the integral are notsmall. Since the Landau functions are orthogonal, theintegral is small for n ′− = n − . Therefore the potentials V ( ˜ N − L ) n ′− ,n ′ + ; n − ,n + ( z ) with n ′− = n − only weakly couple thesubsystems with different fixed n − numbers.A solution with fixed n − may be called adiabatic ap-proximation with respect to the electron motion, or “e-adiabatic approximation” for short. By analogy with thewell known adiabatic approximation for electron motionin a strong magnetic field with a stationary Coulombpotential, it assumes that the Coulomb potential affectsonly the motion along the magnetic field, whereas therelatively fast motion in the transverse plane is governed3 FIG. 1: Absolute values of the reduced effective potentials v ( ˜ N − L ) n ′− ,n ′ + ; n − ,n + ( ζ ) for He + ( Z = 2), n ′− = n − = 0 and fourlowest values of N = ˜ N − L (different line styles, explainedby the legend in panel (a)), n + (different colors, as marked inpanel (a)), and n ′ + = 0 , , , by the magnetic field alone. In each subsystem, the ef-fective potential is given by Eq. (121) with n ′− = n − .Examples of these effective potentials for n − = 0 areshown in Fig. 1. We see that the effective potentials v ( N )0 ,n ′ + ; 0 ,n + ( ζ ) with n ′ + = n + are relatively small. Neglect-ing these small potentials results in the full adiabatic ap-proximation, where the system of equations (116) is splitinto separate equations with fixed n − and n + : (cid:18) − ~ m ∗ d d z − E k κ (cid:19) g κ ( z ) = Ze a m √ v ( ˜ N − L ) n − ,n + ( ζ ) g κ ( z ) , (130)where ζ ≡ z/a m √ v ( ˜ N − L ) n − ,n + ( ζ ) ≡ v ( ˜ N − L ) n − ,n − ; n + ,n + ( ζ ) (131)and E k κ is given by by Eq. (117). Due to the symmetryof the effective potential, v ( ζ ) = v ( − ζ ), it is sufficientto solve Eq. (130) for z > E k κ <
0, form the discretespectrum. We number these solutions by ν = 0 , , , . . . ,with even and odd ν corresponding to the symmetric andantisymmetric longitudinal wave functions, respectively: g κ ( z ) = ( − ν g κ ( z ). In the continuum, with E k κ > g κ ( z ) at anyenergy. For example, we can use the real solutions, whosebehaviors at | z | → ∞ is analogous to the usual Coulomb functions (e.g., [30]) g real κ ( z ) ∼ sin[ φ n − ,n + ,E ( z )] + R E, ± cos[ φ n − ,n + ,E ( z )] , (132)where φ n − ,n + ,E ( z ) = k κ z + m ∗ e ~ k κ ln( k κ z ) (133)is the z -dependent part of the phase of the wave functionat z → + ∞ , k κ = k n − ,n + ,E = (2 m ∗ E k κ ) / / ~ (134)is the wavenumber, and E k κ = E − E ⊥ n − ,n + [Eq. (117)]. D. Coupled channel formalism
In a strong magnetic field, the adiabatic approxima-tion is a convenient starting point for solving the fullsystem (116) by iterations, so that the leading longi-tudinal wave function in Eq. (116) remains the onecorresponding to the starting adiabatic solution (cf.Ref. [7]). Then the numbering of the quantum statescan be the same as in the adiabatic approximation, | κ i = | ˜ N , L, n − , n , ν i for the discrete spectrum and | κ i = | ˜ N , L, n − , n , E, ±i for the continuum. Here, n − and n are the values of n − and n + for the leadingterm in expansion (114).
1. Bound-state wave functions
For the bound states, the energies E κ are determinedas the eigenenergies of the system of equations (116), and ν = 0 , , , . . . corresponds to the longitudinal degree offreedom and controls the parity of the wave function.For the continuum, the energy E and the parity ( ± ) arefixed arbitrarily. Since ˜ N and L enter Eq. (116) only incombination ( ˜ N − L ), the “longitudinal wave functions” g n − n + ; κ ( z ) can be numbered as g n − n + ; N ,ν ( z ) for the dis-crete spectrum and g n − n + ; N ,E, ± ( z ) for the continuum.The degeneracy in ˜ N , at a fixed N , reflects the transla-tional invariance. Indeed, different ˜ N correspond to dif-ferent mean values of the squared total pseudomomentumprojection on the ( xy )-plane, h k , ⊥ i , which according toEqs. (48) and (4) is proportional to the squared sum ofthe guiding centers, h ( r c , − + r c , + ) i , measured from thechosen gauge axis, which can be freely changed by thegauge transformation A ( r ) → (1 / B × ( r − r A ) witharbitrary r A (cf. Ref. [7]).Accordingly, for the physical problems that do not in-volve explicit positions of the guiding centers in space,including the present case of a single ion in a uniformfield, one can identify the ion states by four quantumnumbers instead of five: | κ i = |N , n − , n , ν i .4
2. Continuum wave functions
For the continuum, we construct the basis by analogywith the R -matrix formalism [30]. Let I o be the total number of open channels at given E , i.e., number of suchpairs n − , n + that E ⊥ n − ,n + < E . In this case, numbers n ± mark a selected open channel, defined for E > E ⊥ n − ,n by asymptotic conditions at z → + ∞ g real n o − ,n o+ ; n − ,n , N ,E, ± ( z ) ∼ C norm n δ n o − n − δ n o+ n sin[ φ n o − ,n o+ ; κ ( z )] + R n o − ,n o+ ; n − ,n ; N ,E, ± cos[ φ n o − ,n o+ ; κ ( z )] o , (135)where the pairs ( n o − , n o+ ) relate to different open chan-nels ( E > E ⊥ n o − ,n o+ ), φ n o − ,n o+ ; E ( z ) is the z -dependent partof the phase of the wave function at z → + ∞ , de-fined in Eq. (133), and C norm is a normalization con-stant. The quantities R n o − ,n o+ ; n − ,n + ; N ,E, ± with differ-ent pairs of ( n o − , n o+ ) and ( n − , n + ), corresponding to theopen channels, constitute the reactance matrix R N ,E, ± ,which has dimensions I o × I o . For the closed channels ,defined by the inequality E < E ⊥ n c − ,n c+ , one should imposethe boundary conditions g n c − ,n c+ ; n − ,n , N ,E, ± ( z ) → z → ∞ .The set of solutions, defined by Eqs. (116) and (135),constitute a complete set of I o independent real basisfunctions. If the wave functions are normalized by thecondition Z R d r + , ⊥ Z R d r − , ⊥ Z z max − z max d z | ψ κ ( r + , ⊥ , r − , ⊥ , z ) | = 1 , (136) where z max is half the normalization length, then theorthonormality of the transverse basis [Eq. (109)] leadsto the condition X n − ,n + Z z max − z max | g n − n + ; κ ( z ) | d z = 1 . (137)In the adiabatic approximation, a single term of the sumis retained in Eq. (137) for each open channel.One can compose the basis of outgoing waves, appro-priate to photoionization, by analogy with the case of theH atom in Refs. [31, 32]. The basis of outgoing waves withdefinite z -parity is defined by the asymptotic conditionsat z → ∞ g out n o − ,n o+ ; n − ,n , N ,E, ± ( z ) ∼ C norm n δ n o − n − δ n o+ n e i φ n − ,n ,E ( z ) − S ∗ n o − n o+ ; n − ,n ; N ,E, ± e − i φ n o − n o+ ( z ) o , (138)where S n o − n o+ ; n − n + ; N ,E, ± are the elements of the unitary scattering matrix S N ,E, ± = (1 + i R N ,E, ± )(1 − i R N ,E, ± ) − .The basis of outgoing waves is obtained from the real basis by transformation g out n ′′− n ′′ + ; n − n + ; N ,E, ± ( z ) = 2i X n ′− n ′ + (cid:2) (1 + i R N ,E, ± ) − (cid:3) n − n + ; n ′− n ′ + g real n ′′− n ′′ + ; n ′− n ′ + ; N ,E, ± ( z ) . (139)Here, pairs ( n − , n + ) and ( n ′− , n ′ + ) run over open channels, but ( n ′′− , n ′′ + ) run over all (open and closed) channels. Asfollows from the orthonormality of the transverse basis, the normalization integral on the left-hand side of Eq. (136)equals 2 z max | C norm | h S N ,E, ± S †N ,E, ± i = 4 z max | C norm | , (140)where we have used the unitarity of the S-matrix. Then according to Eq. (136) | C norm | = (4 z max ) − / . (141)After the orthonormal basis of outgoing waves have been obtained at given N and E for each z -parity, with g out n ′− n ′ + ; n − n + ; N ,E, ± ( z ) = ± g out n ′− n ′ + ; n − n + ; N ,E, ± ( − z ), we can easily construct solutions for electron waves propagating at z → ±∞ in a definite open channel ( n − , n ) for an arbitrary L ≤ N − n − . These solutions are given by Eq. (114)with coefficients g n ′− n ′ + ; κ ( z ) = 1 √ (cid:16) g out n ′− n ′ + ; n − ,n ; N ,E, + ( z ) ± g out n ′− n ′ + ; n − ,n ; N ,E, − ( z ) (cid:17) , (142)5where the sign + or − represents electron escape in the positive or negative z direction, respectively. Waves incomingfrom z → ±∞ are given by the complex conjugate of Eq. (142). E. Geometric sizes of the ions
In order to evaluate populations of different bound levels and the collision frequency ν coll in the plasma environment,it is useful to calculate the geometric sizes of the ions. For this aim we use the root-mean-square longitudinal andtransverse sizes, |h κ | z | κ i| / and |h κ | r ⊥ | κ i| / , by analogy with Ref. [33]. Let us use the basis expansion (114) forcalculation of these matrix elements. For the longitudinal mean squared size, using the orthonormality condition(109), we obtain h κ | z | κ i = X n − ,n + Z z max − z max g ∗ n − n + ; κ ( z ) g n − n + ; κ ( z ) z d z. (143)For the mean squared transverse size, using Eq. (80), we obtain h κ | r ⊥ | κ i = 2 h κ | r +1 r − | κ i = 2 a h κ | (ˆ a † − ˆ˜ a ) (ˆ a − ˆ˜ a † ) | κ i . (144)Then, using Eqs. (71a), (71b), (73a), and (73b), we obtain h κ | r ⊥ | κ i = 2 a ∞ X n − =0 N X n ′ + =0 N X n + =0 " N X k =0 ( n − + k + 1) C ( N ,n + ) k C ( N ,n ′ + ) k Z z max − z max g ∗ n − n ′ + ; κ ( z ) g n − n + ; κ ( z ) d z − N X k =0 p ( n − + 1)( k + 1) C ( N ,n + ) k C ( N +1 ,n ′ + ) k Z z max − z max g ∗ n − +1 ,n ′ + ; κ ( z ) g n − n + ; κ ( z ) d z − N X k =0 p n − k C ( N ,n + ) k C ( N − ,n ′ + ) k − Z z max − z max g ∗ n − − ,n ′ + ; κ ( z ) g n − n + ; κ ( z ) d z . (145) V. INTERACTION WITH RADIATIONA. Radiative transitions
In presence of an electromagnetic wave, its vector po-tential A rad should be added to the vector potential ofthe constant magnetic field A in Eqs. (2) and (3). Thenthe total Hamiltonian of the system of an electron, a nu-cleus, a constant magnetic field, and radiation can bewritten as H tot = 12 m + (cid:18) π + − Zec A rad , ⊥ ( r + ) (cid:19) + 12 m − (cid:16) π − + ec A rad , ⊥ ( r − ) (cid:17) − Ze √ r + z + X q γ ~ ω q ˆ c † q γ ˆ c q γ , (146)where q , α , and ω q are the photon wavevector, polariza-tion index, and frequency, respectively, c † q γ and ˆ c q γ are the photon creation and annihilation operators, A rad ( r ) = X q γ (cid:18) π ~ c ω q V (cid:19) / (cid:0) e q γ ˆ c q γ e i q · r + e ∗ q γ ˆ c † q γ e − i q · r (cid:1) (147)is the electromagnetic field operator (in the Schr¨odingerrepresentation), the subscripts ⊥ and z mark the trans-verse and longitudinal vector components with respect tothe magnetic field direction, and V is the normalizationvolume (see, e.g., Refs. [23, 38]).The operator of interaction with radiation is obtainedby expanding the brackets in Eq. (146). We will usethe Coulomb gauge, ∇ · A rad = 0, and the transverseapproximation assuming q · e q γ = 0. Then operator A rad commutes with π ± , so that A rad · π ± + π ± · A rad =2 A rad · π ± .Neglecting nonlinear effects (i.e., terms proportionalto A ), we obtain the following operator that couplesinternal degrees of freedom to radiation: V int = em − c A rad , ⊥ ( r − ) · π − − Zem + c A rad , ⊥ ( r + ) · π + + em − c A rad , z ( r − ) p − ,z − Zem + c A rad , z ( r + ) p + ,z . (148)6Using Eq. (147), we can rewrite Eq. (148) as V int = s π ~ ω q V X q γ (cid:16) ˆ c q γ e q γ · j eff + e ∗ q γ · j † eff ˆ c † q γ (cid:17) , (149)where j eff = e i q · r ( e ˙ r − − Ze ˙ r + ) (150)is the effective current operator ( ˙ r ± = π ± /m ± ).Let us consider an absorption of a photon with a givenwavevector q and polarization α . The initial state is | i i = f q γ (1) | ψ i i and the final state is | f i = f q γ (0) | ψ f i ,where | ψ i,f i denotes the state of the system of chargedparticles and f q γ ( N ) is the function of photon number[23]. According to the Fermi’s golden rule, probabilityof transition, per unit time, from initial quantum state f q γ (1) | i i to final state f q γ (0) | f i with absorption of onephoton is given byd w i → f = 2 π ~ (cid:12)(cid:12) h ψ f | f † q γ (0) V int f q γ (1) | ψ i i (cid:12)(cid:12) × δ ( E f − E i − ~ ω q ) d ν f , (151)where E i , E f , and ~ ω q are the energies of the initialstate, final state, and absorbed photon, respectively,and d ν f is the density of final states. Taking into ac-count the properties of the photon creation and anni-hilation operators [23], ˆ c q γ f q γ ( N ) = √ N f q γ ( N − c † q γ f q γ ( N ) = √ N + 1 f q γ ( N + 1), where f q γ ( N ) is thefunction of photon number, and performing normaliza-tion of transition rate (151) by the photon flux c/ V , wearrive at the differential cross sectiond σ i → f, q γ = 4 π ω q c (cid:12)(cid:12) h ψ f | e q γ · j eff | ψ i i (cid:12)(cid:12) δ ( E f − E i − ~ ω q ) d ν f . (152)If the final state belongs to the discrete spectrum, thefinal energies E f are distributed within a narrow bandaround the value E f = E i + ~ ωf i, where ωf i is a centralvalue of the transition frequency. Then integration ofEq. (152) over the final states gives σ i → f, q γ = 4 π ~ ω q c (cid:12)(cid:12) h ψ f | e q γ · j eff | ψ i i (cid:12)(cid:12) ∆ fi ( ω q − ω fi ) , (153)where ∆ fi ( ω q − ω fi ) describes the profile of the spectralline, which is normalized so that R ∆ fi ( ω ) d ω = 1 . If the final state belongs to the continuum and wavefunctions are normalized according to Eq. (136), thend ν f = z max π m ∗ ~ k f d E f , (154) k f = ~ − q m ∗ E k f , E k f ≡ E f − E ⊥ n − ,f n ,f , (155)with E f − E ⊥ n − ,f n ,f >
0. In this case the cross sectionof photoabsorption takes the form σ i → f, q γ = 4 πz max m ∗ ~ k f ω q c (cid:12)(cid:12) h ψ f | e q γ · j eff | ψ i i (cid:12)(cid:12) . (156) The above equations do not include the photoninteraction with magnetic moments ˆ µ ± of the particles.For transitions without spin-flip, the latter interactioncan be taken into account by supplementing the op-erator e q γ · j eff by the term − i( q × e ) · ( ˆ µ − + ˆ µ + )(cf. [35]), whereas operators ( q × e ) × ˆ µ ± are responsiblefor spin-flip transitions (cf. [36]). The correspondingcontributions to the transition matrix elements areproportional to q and prove to be of the same order ofmagnitude as the first-order corrections ( ∝ q ) to thedipole approximation that we use below. Therefore,these terms will be neglected in the dipole approximation. B. Dipole approximation
In the dipole approximation for the matrix elements ofradiative transitions, e i q · r is replaced by 1. For bound-bound and bound-free transitions this is justified, pro-vided that the mean bound-state size l is much smallerthan q − = ω q /c . Since the binding energy E b is bythe order of magnitude ∼ Ze /l , this requirement trans-lates into ~ ω q ≪ α − E b /Z , where α f = e / ~ c is the finestructure constant. In this approximation, the effectivecurrent (150) takes the form j eff = em − π − − Zem + π + . (157)Using the commutation relation π ± m ± = i ~ [ H , r ± ] , (158)where H is the field-free Hamiltonian given by Eq. (34),we can transform “velocity form” of the matrix elementin Eq. (152) to the “length form” (cf., e.g., Ref. [37]) h ψ f | j eff | ψ i i = − i ~ ( E f − E i ) D fi = i ω q D fi . (159)Here, D fi = h ψ f | D | ψ i i is the matrix element of the elec-tric dipole moment, D = Ze r + − e r − = ( Z − e R − e r . (160)For the bound-bound transitions, substitution ofEq. (159) into Eq. (153) gives σ i → f, q γ = 4 π ω q ~ c | e q γ · D fi | ∆ fi ( ω q − ω fi ) . (161)For the bound-free transitions, substitution of Eq. (159)into Eq. (156) gives σ i → f, q γ = 4 z max πm ∗ ω q ~ k f c | e q γ · D fi | . (162)In the cyclic coordinates (31), we have e q γ · D fi = X α = − e q γ, − α D fi,α . (163)7Using Eq. (80), we can write the transverse (right and left) cyclic components of the dipole operator D as D +1 = ( Z − e a m ,Z − (ˆ b − ˆ˜ b † ) − ea m (ˆ a † − ˆ˜ a ) = e a m (cid:2) √ Z − b − ˆ˜ b † ) − ˆ a † + ˆ˜ a (cid:3) , (164) D − = ( Z − ea m ,Z − (ˆ b † − ˆ˜ b ) − ea m (ˆ a − ˆ˜ a † ) = e a m (cid:2) √ Z − b † − ˆ˜ b ) − ˆ a + ˆ˜ a † (cid:3) , (165)whereas the longitudinal component D = − ez does not affect the transverse states of motion. From these equationsand Eqs. (71), (73), and (82) we see that D α transforms each pure transverse state | N, ˜ N , n, ˜ n i ⊥ into the superpositionof such states, D +1 | N, ˜ N , n, ˜ n i ⊥ = ea m √ Z − (cid:2) √ N | N − , ˜ N i − p ˜ N + 1 | N, ˜ N + 1 i (cid:3) ⊗ | n, ˜ n i − ea m | N, ˜ N i ⊗ (cid:2) √ n + 1 | n + 1 , ˜ n i − √ ˜ n | n, ˜ n − i (cid:3) , (166) D − | N, ˜ N , n, ˜ n i ⊥ = ea m √ Z − (cid:2) √ N + 1 | N + 1 , ˜ N i − p ˜ N | N, ˜ N − i (cid:3) ⊗ | n, ˜ n i − ea m | N, ˜ N i ⊗ (cid:2) √ n | n − , ˜ n i − √ ˜ n + 1 | n, ˜ n + 1 i (cid:3) . (167)It follows that D α transforms a state with a definite L =˜ N − N + n − ˜ n into a state with L ′ = L + α . This entailsthe selection rule L f = L i + α. (168)It reflects conservation of the total angular momentumof the entire system, comprising an electron, a nucleus,and a photon. As a consequence, in the right-hand sideof Eq. (162) we can use the expansion | e q γ · D fi | = X α = − | e q γ, − α | | D fi,α | . (169)Terms e − α e ∗− α ′ D fi,α D ∗ fi,α ′ with α = α ′ are absent, be-cause D α and D α ′ transform a pure quantum state | ψ i i into states | ψ f i and | ψ f ′ i with different z -projections ofthe angular momentum, so that D fi,α and D fi,α ′ cannotbe non-zero simultaneously. Thus Eqs. (161) and (162)can be written as σ i → f, q γ = X α = − | e q γ,α | σ (bb , bf) i → f,α ( ω q ) (170)where σ bb i → f,α ( ω ) = 4 π ω ~ c | D fi, − α | ∆ fi ( ω − ω fi ) , (171) σ bf i → f,α ( ω ) = 4 z max πm ∗ ω ~ c k f | D fi, − α | (172)for the bound-bound and bound-free transitions, respec-tively.Neglecting the Doppler broadening, we can model∆ fi ( ω ) in Eq. (171) by the Lorentz-Cauchy profile,∆ fi ( ω ) = 1 π ν eff ( ω − ω fi ) + ν , (173) where ν eff is an effective damping frequency. In the sim-plest approximation, ν eff = ν coll + ν rad , where ν coll is aneffective frequency of collisions of a given ion with plasmaparticles, and ν rad = 4 ω fi ~ c X α = − | D fi,α | (174)is the natural radiative width (cf., e.g., [38]).For practical computations, it is convenient to considerthe continuum wave functions normalized so that the am-plitude of the outgoing wave in a selected open channelequals 1 at infinity. Then the factor 4 z max in Eq. (172)drops out from the numerical code, being canceled by thesquared normalization constant (141).Kopidakis et al. [35] defined a dimensionless interac-tion operator, which can be written asˆ M = 2 ~ e e q γ · j eff . (175)The correspondence between the “velocity form” and“length form” of this operator has been discussed inRef. [31] regarding the problem of a hydrogen atom ina strong magnetic field. It was also employed in Ref. [32]for treatment of the bound-free transitions of a hydrogenatom moving in the magnetic field. Using the dipole ap-proximation, we can rewrite Eq. (172) in the same formas Eq. (7) of [32]: σ bf i → f,α ( ω ) = πα f z max k f Ha ~ ω m ∗ m e | M fi, − α | = 2 πα f Ry ~ ω s Ry E k f m ∗ m e z max a B | M fi, − α | , (176)where M fi,α = i( ~ ω/ Ry) D fi,α /ea B is the respectivecyclic component of the dimensionless matrix element8 h ψ f | e q γ · ˆ M | ψ i i , a B = ~ /m e e is the Bohr radius,Ha = 2Ry = e /a B is the Hartree energy unit, Ry beingthe Rydberg energy and m e the electron mass.For the bound-bound transitions, it is customary todefine dimensionless oscillator strengths (e.g., [19]) f fi,α = ~ ω Ry (cid:12)(cid:12)(cid:12)(cid:12) D fi, − α ea B (cid:12)(cid:12)(cid:12)(cid:12) . (177)In these notations, Eq. (171) can be written as σ bb i → f,α ( ω ) = 2 π e m e c f fi,α ∆ fi ( ω ) . (178) C. Expansion on the transverse basis
Let us use the basis expansion (114) for calculation ofthe matrix elements D fi,α = h ψ f | D α | ψ f i . For the longi-tudinal polarization ( α = 0), using the orthonormaliza-tion condition (109), we obtain D fi, = − eδ ˜ N f ˜ N i δ L f L i X n − ,n + Z z max − z max g ∗ n − n + ; κ f ( z ) g n − n + ; κ i ( z ) z d z. (179)In the adiabatic approximation, we are left with the onlyterm with n − = n − ,i = n − ,f and n + = n + ,i = n + ,f , while transitions with n − ,i = n − ,f or n + ,i = n + ,f areforbidden. Beyond the adiabatic approximation, the lat-ter transitions are allowed, but | D fi, | is small comparedto the case without changing n − and n + . For the ini-tial and final states with definite z -symmetry, the corre-sponding selection rule follows: D fi, is non-zero only fortransitions between the state of opposite z -symmetry. Inparticular, bound-bound transitions between states with ν i and ν f of the same parity (both even or both odd) aredipole forbidden.For the circular polarizations α = ±
1, using Eq. (82),we can transform Eqs. (164) and (165) to D +1 ea m = i a m ~ U † π + , +1 U − ˆ a † − √ Z − b † , (180) D − ea m = − i a m ~ U † π + , − U − ˆ a − √ Z − b. (181)The first term in each of these equations shifts the quan-tum number n + by ± n − according to Eqs. (71a), (71b),(73a) and (73b), while the last term shifts the number ˜ N according to Eqs. (71c), (71d), (73d), and (73d). In allthe cases the selection rule (168) holds. Thus the trans-verse basis states (101) are transformed as D +1 | ˜ N , L, n − , n + i ⊥ , /e a m = √ Z √ n + | ˜ N, L + 1 , n − , n + − i ⊥ , − p n − + 1 | ˜ N , L + 1 , n − + 1 , n + i ⊥ , −√ Z − p ˜ N + 1 | ˜ N + 1 , L + 1 , n − , n + i ⊥ , , (182) D − | ˜ N , L, n − , n + i ⊥ , /e a m = √ Z p n + + 1 | ˜ N , L − , n − , n + + 1 i ⊥ , − √ n − | ˜ N , L − , n − − , n + i ⊥ , −√ Z − p ˜ N | ˜ N − , L − , n − , n + i ⊥ , . (183)Let us substitute the basis expansion (102) into D fi,α = h ψ f | D α | ψ i i with α = ± L f = L i ± N f = ˜ N i , taking into account the orthogonality relation (109), we obtain D fi, +1 ea m = √ Z N X n + =0 √ n + ∞ X n − =0 L n − ,n + − n − ,n + ( κ f | κ i ) − ∞ X n − =0 p n − + 1 N X n + =0 L n − +1 ,n + ; n − ,n + ( κ f | κ i ) , (184) D fi, − ea m = √ Z N X n + =0 p n + + 1 ∞ X n − =0 L n − ,n + +1; n − ,n + ( κ f | κ i ) − ∞ X n − =0 √ n − N X n + =0 L n − − ,n + ; n − ,n + ( κ f | κ i ) , (185)where N ≡ ˜ N i − L i + n − and L denotes the longitudinaloverlap integral, L n ′− ,n ′ + ; n − ,n + ( κ ′ | κ ) = Z z max − z max g ∗ n ′− ,n ′ + ; κ ′ g n − ,n + ; κ d z. (186)The last term in each of equations (182) and(183) corresponds to transitions with L f = L i + α and ˜ N f = ˜ N i + α , which leave ( ˜ N − L ) un- changed. The corresponding dipole matrix elementsequal ea m p ( Z − N i , N f ) h ψ f | ψ i i = 0, because ψ i and ψ f are orthogonal. The absence of such transitionsagrees with the degeneracy of the problem in ˜ N , dis-cussed above. Thus it is sufficient to study only thetransitions between states with ˜ N = 0; the results fornon-zero ˜ N are then obtained by adding ˜ N to both L i and L f .9 VI. APPROXIMATE SOLUTIONS FORULTRA-STRONG FIELDS
In this section we consider an approximate treatmentof the hydrogenlike ion in the full adiabatic approxima-tion (Sect. IV C), using the method previously developedby Hasegawa & Howard [19] for a strongly magnetized Hatom.
A. Wave functions and eigenenergies
Let us find an approximate solution to the Schr¨odingerequation in the adiabatic approximation, Eq. (130), forthe bound states. Following Hasegawa & Howard [19],we find asymptotic solutions at small and large z andmatch them at an intermediate point z = Ca λ m , < λ < / , (187)where C and λ are constants, independent of z and a m .The matching is provided by equating the logarithmicderivatives η ( z ) = g ′ κ ( z ) g κ ( z ) (188)of the interior and exterior solutions at z = z . Thebounds on λ in Eq. (187) ensure that z/a m → ∞ and z | ln a m | → a m → z > z , which isneeded for validation of the approximate exterior solu-tion (Sect. VI A 1), while z /a → z < z , asrequired for the validity of the approximate exterior so-lution (Sect. VI A 2).
1. Exterior solution for bound states At ζ → ∞ all effective potentials V ( N )0 ,n + ( z ) converge tothe 1D Coulomb potential, so that Eq. (130) is replacedby − ~ m ∗ d g κ d z − Ze | z | g κ = E k κ g κ . (189)The well known solution to this equation for a boundstate (i.e., for E k κ <
0, so that lim z →∞ g κ ( z ) = 0) is g ext ( z ) = C W W ˜ ν, (2 z/ ˜ νa ∗ ) , (190)where W ˜ ν, ( x ) is the Whittaker function [22], C W is anormalization constant, a ∗ = ~ m ∗ Ze , (191)is the “effective Bohr radius”, ˜ ν is the “effective principalquantum number” defined through the relation ǫ κ ≡ | E k κ | Ry ∗ ≡ ν , (192) where Ry ∗ ≡ Z e m ∗ ~ = Z m ∗ m e Ry (193)is the “effective Rydberg energy”.At small (but non-zero) argument x = 2 z/ ˜ νa ∗ andnon-integer ˜ ν , the Whittaker function can be expandedas [19] W ˜ ν, ( x ) = − Γ(˜ ν ) (cid:26) (cid:2) ˜ νx − (˜ νx ) (cid:3) cos ˜ νπ + (cid:20) − νx (cid:18) ln x ˜ ν + 12˜ ν + ψ (˜ ν ) − ψ (1) − ψ (2) (cid:19)(cid:21) sin ˜ νππ + O (˜ ν x ln x ) (cid:27) , (194)where ψ ( x ) ≡ d ln Γ( x )d x (195)is the digamma function [22]; ψ (1) = ψ (2) − − γ E ,where γ E = 0 . . . . is the Euler-Mascheroni constant.If z were zero, ˜ ν would be integer. At non-zero z ,with a m →
0, ˜ ν tends to integer values in such a waythat sin ˜ νπ tends to zero logarithmically (as will be seenfrom the solutions below), that is slower than z → z cot ˜ νπ →
0, and from Eq. (194) we obtain W ˜ ν, (cid:18) z ˜ νa ∗ (cid:19) = Γ(˜ ν ) sin ˜ νππ (cid:2) O (cid:0) a − λ m cot ˜ νπ (cid:1)(cid:3) . (196)Taking the derivative in Eq. (194) and dividing byEq. (196), we obtain the logarithmic derivative in theform η ext = dd z ln W (cid:18) z ˜ νa ∗ (cid:19) ≈ − a ∗ (cid:18) ln 2 za ∗ + π cot ˜ νπ + 2 γ E − Θ(˜ ν ) (cid:19) , (197)where z ∼ z ≪ a ∗ andΘ(˜ ν ) = ln ˜ ν − ν − ψ (˜ ν ) = ln ˜ ν + 12˜ ν − ψ (1 + ˜ ν ) . (198)
2. Interior solution for even states
Now let us consider | z | ≪ z . In terms of the dimen-sionless coordinate ζ (120), the Schr¨odinger equation inthe adiabatic approximation [Eq. (130)] becomesd g κ d ζ + 2 / a m a ∗ v ( N )0 ,n + ( ζ ) g κ − (cid:18) a m a ∗ (cid:19) ǫ κ g κ = 0 , (199)where ǫ κ is the dimensionless eigenvalue defined inEq. (192). We solve this equation approximately, us-ing the perturbation theory with respect to the small0 FIG. 2: Dependence of the quantities J N n + [Eq. (203)] on quantum number N (left panel, for n + = 0, 1, 2, 3, and 4) and onquantum number n + (right panel, for N from 0 to 20) for Z = 2 (dots connected by solid lines) and Z = 6 (triangles connectedby dashed lines). parameter a m /a ∗ separately for the even and odd states.Since in this section we are interested in the bound states, E k κ <
0, we can safely set n − = 0. Then N = ˜ N − L and v ( N )0 ,n + = N X k =0 (cid:16) C ( N ,n + ) k (cid:17) Z ∞ I k, ( ρ ) p ρ + ζ d ρ, (200)where, according to Eq. (25), I k, ( ρ ) = ρ k/ e − ρ/ /k !, andcoefficients C ( N ,n + ) k are given by the relation (112) andnormalization (113).Let us consider even states. In the first approximation(linear in a m /a ∗ ), the last term in Eq. (199) drops out,and the solution is proportional to1 − / a m a ∗ Z ζ d ζ ′ Z ζ ′ d ζ ′′ v ( N )0 ,n + ( ζ ′′ )The logarithmic derivative for the interior solution be-comes η int ( z ) ≈ − a ∗ Z z/a m √ v ( N )0 ,n + ( ζ ) d ζ. (201)As shown in Appendix B, this expression leads to η int = − a ∗ (cid:20) z a m + ln 2 + γ E − J N n + + O (cid:18) a z (cid:19)(cid:21) , (202) where H k = P kn =1 n − is the k th harmonic number, J = 0, and for N ≥ J N n + = N X k =1 H k (cid:16) C ( N ,n + ) k (cid:17) . (203)In the particular case N = 0, which corresponds tothe non-moving ion, Eq. (202) reduces to the result ofHasegawa & Howard [19]. For N = 1, we obtain J = Z − Z , J = 1 Z . (204)The dependence of the quantities J N n + on the quantumnumbers N and n + is shown in Fig. 2.
3. Binding energies of even states
By equating η int (202) to η ext (197) at z = z we obtainthe following equation for the effective quantum number˜ ν : π cot ˜ νπ + Θ(˜ ν ) = ln a ∗ a m + ln 2 − γ E − J N n + | sin ˜ νπ | ≪ ν ) is defined by Eq. (198). So-lution of Eq. (205) gives the longitudinal energies of theeven states through Eq. (192).The lowest-energy state for each N and n + (so called tightly bound state ) corresponds to ˜ ν ≪
1. In this case,1
FIG. 3: Energy levels of the helium ion in strong magnetic fields.
Left panel : Dependence of the energies on magnetic fieldstrength. The numbers at the curves are N and n + , which mark the discrete states | κ i = |N , n − , n + , ν i with n − = 0 and ν = 0(i.e., only tightly-bound states are considered). Right panel : Dependence of the energies on N for the three smallest values of n + at B = 2 . × G. the series expansion ([22], 6.3.14, 23.2), ψ (1 + ˜ ν ) = − γ E + ( π /
6) ˜ ν − ( π /
90) ˜ ν + . . . leads to an approxi-mate relation Θ(˜ ν ) ≈ ln ˜ ν + 1 / (2˜ ν ) + γ E − ( π /
6) ˜ ν. Tothe same accuracy up to O (˜ ν ), π cot π ˜ ν ≈ ˜ ν − − π ˜ ν/ ν = ln γ ∗ ν − γ E − J N n + + π ν, (206)where we have introduced the magnetic-field parameter γ ∗ = a ∗ a = ~ BZ m ∗ e c = BB ∗ , (207)which should be large in the considered approximation.Here, B ∗ ≡ Z m ∗ e c/ ~ ≈ . × ( m ∗ /m e ) Z G.Equation (206) can be easily solved numerically. For ex-ample, as long as ln( γ ∗ / > γ E + J N n + , one can useiterations of the form˜ ν i +1 = w (cid:18) ln γ ∗ ν i − γ E − J N n + + π ν i (cid:19) − +(1 − w ) ˜ ν i (208)with an appropriate weight w , starting from1˜ ν = ln γ ∗ − γ E − J N n + . (209) Having tried different values of w at different γ ∗ we foundthat a good choice of w , which provides a quick conver-gence of the iterations, can be roughly approximated as w = (1 + 4˜ ν / ) / (1 + 15˜ ν / ).In the ground state, N = n + = 0, and Eq. (206) with-out the last (linear in ˜ ν ) term reproduces the result ofHasegawa and Howard [19]. However, by means of a com-parison with the numerical results from Ref. [16] we foundthat even at the superstrong field strengths B ∼ G,encountered in magnetars, a solution of such a truncatedequation produces an unacceptable error ∼
200 eV inthe ground-state energy. The linear term in Eq. (206)considerably improves the accuracy.In Fig. 3 we show the dependence of energy E κ = E k κ + ~ ω c+ n + on the magnetic field strength B and on thequantum number N (the left and right panels, respec-tively). The “electron cyclotron” quantum number n − must equal zero in the displayed energy range. The zero-point energy E ⊥ , is dismissed, because it does not affectthe binding, being the same for the bound and unboundstates. It is of interest to compare the B -dependences ofour analytic estimates of energy shown in the left panelwith Fig. 2 in Ref. [16], which shows the analogous de-pendences computed numerically. We can notice a qual-itative difference at B < G, where different energybranches overlap in our figure. This difference is caused2by the fact that our analytic estimates have a good accu-racy only at superstrong fields. At such high fields, how-ever, the branches corresponding to non-zero n + mergeinto continuum, because E k κ increases slower (logarithmi-cally) than the energy of transverse excitations n + ~ ω c+ .Ultimately, at B & × G only the branch with n + = 0 survives, while the states with n + > ν is not small corresponds to the“loosely-bound” or “hydrogenlike” states. As long as thelogarithm on the right-hand side of Eq. (205) is large,this equation can be satisfied only if cot ˜ νπ is also large(tends to infinity at a m → ν = ν k δ ν k , ν k = 2 , , , . . . , (210)where the even numbers ν k enumerate the even statesand the quantum defect δ ν k is given by1 δ ν k ≈ π cot πδ ν k ≈ (cid:16) ln γ ∗ − γ E − J N n + (cid:17) + Θ (cid:16) ν k (cid:17) , (211)where function Θ is defined by Eq. (198).The “tightly-bound” solution described above[Eq. (206)] corresponds formally to ν k = 0. One shouldnote that the approximations (206) and (211) are basedon the condition that the right-hand side in Eq. (205) islarge, implying that J N n + ≪ ln γ ∗ . One can show (seeAppendix C) that J N ∼ ln N at N ≫
1. Therefore,the necessary condition of the applicability of the ap-proximations (206) and (211) for the states with n + = 0is ln( γ ∗ / N ) ≫
1. If this is not the case, one has to useEq. (205) instead of the approximations (206) or (211).
4. Odd bound states
For the odd-parity states, the longitudinal wave func-tion g κ ( z ) tends to zero at z →
0. Therefore it is smallin the region | z | . a m where the effective potentials V ( N )0 ,n + ( z ) substantially differ from the 1D Coulomb poten-tial. In the first approximation with respect to the smallparameter a m /a ∗ = γ − / ∗ , one can use the 1D Coulombpotential instead of the true effective potential. Thus theproblem is reduced to finding odd-parity solutions for the1D H atom. Formally it corresponds to setting z = 0 and˜ ν to integers in Sect. VI A 1. The review of the theory ofthe 1D H atom is given in Ref. [39]. The odd-parity solu-tions are well-behaved, because the singularity of the po-tential term in the Schr¨odinger equation (189) is finite forsuch wave functions. The Whittaker functions (190) withinteger ˜ ν are expressed through the generalized Laguerrepolynomials, W ˜ ν, ( x ) = ( − ˜ ν x e − x/ L ν ( x ) / ˜ ν , and thenormalized wave functions become g ν k ( z ) ≈ g (0) ν k ( z ) = r a ∗ ˜ ν ˜ ν ! z exp (cid:18) − | z | ˜ νa ∗ (cid:19) L ν (cid:18) | z | ˜ νa ∗ (cid:19) , (212) where ˜ ν = ( ν k + 1) / , , , . . . , and ν k is an odd longi-tudinal quantum number. This solution does not dependon the state of motion of the ion. Such dependence can berevealed by the perturbation theory. To the first order, E k κ = − ~ m ∗ a ∗ ˜ ν + Z ∞−∞ δV ( N ) n + ( z ) h g (0) ν k ( z ) i d z, (213)where δV ( N ) n + ( z ) = V ( N )0 ,n + ( z ) + Ze | z | . (214)is the perturbation potential. Since the integrand is non-negative, the energy levels are shifted upwards.
5. Continuum states
For positive longitudinal energies, the exterior solutionis obtained by analogy with Sect. VI A 1 with a replace-ment ˜ ν → i˜ ν , where ˜ ν is defined by Eq. (192) with posi-tive E k κ . According to this definition,˜ ν = ( a ∗ k κ ) − , (215)where k κ is the absolute value of the outgoing wavenum-ber given by Eq. (134). Instead of Eq. (190) we have twolinearly independent solutions with asymptotes W ± i˜ ν (cid:18) ∓ z ˜ νa ∗ (cid:19) ∼ e ˜ νπ/ exp (cid:20) ± i (cid:18) k κ z + ln | k κ z | ka ∗ (cid:19)(cid:21) , (216)at z → ∞ . Then the longitudinal part g κ ( z ) of a realwave function of definite z -parity, ψ κ ( r + , ⊥ , r − , ⊥ , z ) = ± ψ κ ( r + , ⊥ , r − , ⊥ , − z ), normalized according to Eq. (137),is approximated at z > z by g real κ ( z ) = e − ˜ νπ/ √ z max (cid:20) W i˜ ν (cid:18) − z ˜ νa ∗ (cid:19) e i θ + W − i˜ ν (cid:18) z ˜ νa ∗ (cid:19) e − i θ (cid:21) , (217)where the phase factor e ± i θ has to be determined from thematching conditions at z = z . In Eq. (217), the prefac-tor is obtained assuming that the normalization integralover the interval [ − z , z ] is negligibly small comparedwith the total normalization integral over [ − z max , z max ]. B. Overlap integrals between even states
In the exterior region z > z , assuming that at leastone of the longitudinal wave functions g κ ( z ) or g κ ′ ( z )belongs to discrete spectrum (tends to zero at z → ∞ ),the use of the Schr¨odinger equation and integration byparts leads to the identity Z ∞ z g κ ( z ) g κ ′ ( z ) d z = ~ m ∗ g κ ( z ) g κ ′ ( z ) η − η ′ E k κ − E k κ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = z , (218)3where η and η ′ are the corresponding logarithmic deriva-tives (188). In the limit E k κ → E k κ ′ we have Z ∞ z g κ ( z ) = ~ m ∗ g κ ( z ) d z ∂η∂E k κ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = z . (219)The latter equation is used for normalization of the over-lap integral (218). At a m →
0, we have z →
0, hence theintegral over the interior region | z | < z can be neglected.Then we have the normalized overlap integral L ( κ ′ | κ ) ≈ R ∞ z g κ ( z ) g κ ′ ( z ) d z hR ∞ z g κ ( z ) d z i / hR ∞ z g κ ′ ( z ) d z i / . (220)Using Eq. (218) for the numerator and Eq. (219) for thedenominator on the right-hand side, we obtain |L ( κ ′ | κ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η − η ′ E k κ − E k κ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂η∂E k κ ∂η ′ ∂E k κ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = z . (221)Here, L is defined by Eq. (186), where the subscripts aresuppressed because only the terms with n ± = n ± and n ′± = n ′ ± survive in the adiabatic approximation.Equations (192) and (197) give ∂η∂E κ = 2 m ∗ a ∗ ˜ ν ~ ∂η∂ ˜ ν = 2 m ∗ a ∗ ˜ ν ~ (cid:18) π sin ˜ νπ + dΘ(˜ ν )d˜ ν (cid:19) , (222)where function Θ is defined by Eq. (198). From Eq. (222),using Eqs. (202), (205) and the identity 1 / sin x = 1 +cot x , we obtain ∂η∂E κ = 2 m ∗ a ∗ ˜ ν ~ G N ,n + , ˜ ν , (223)where G N ,n + , ˜ ν ≡ "(cid:18)
12 ln γ ∗ − γ E − J N n + + Θ(˜ ν ) (cid:19) + π + dΘ(˜ ν )d˜ ν (cid:21) . (224)Although η and η ′ depend on z , their difference doesnot. From Eq. (202) we obtain the approximation η − η ′ = J N ′ n ′ + − J N n + a ∗ . (225)In summary, |L ( κ ′ | κ ) | is given by Eqs. (221), (225), (223),and (192) as function of ˜ ν . In its turn, ˜ ν is determinedby Eq. (205).
1. The case of tightly bound states
For the tightly bound states ( ν k = ν ′k = 0), we have0 < ˜ ν ≪ < ˜ ν ′ ≪
1. In this particular case Eqs. (197) and (198) yield ∂η∂E k κ = m ∗ a ∗ ~ (cid:2) ˜ ν + 2˜ ν + O (˜ ν ) (cid:3) . (226)Substitution of Eqs. (225) and (226) into Eq. (221) withthe use of Eq. (192) gives L ( κ ′ | κ ) ≈ J N ′ n ′ + − J N n + ) (cid:18) ν ′ ) − ν (cid:19) (cid:0) ˜ ν + 2˜ ν (cid:1) (cid:2) (˜ ν ′ ) + 2(˜ ν ′ ) (cid:3) . (227)Using Eq. (209) for ˜ ν and ˜ ν ′ , we substitute1˜ ν − ν ′ ) ≈ (cid:16) J N ′ n ′ + − J N n + (cid:17) ln γ ∗ , (228)˜ ν + 2˜ ν ≈ (cid:16) ln γ ∗ (cid:17) − + 2 (cid:16) ln γ ∗ (cid:17) − ≈ (˜ ν ′ ) + 2(˜ ν ′ ) (229)and thus obtain L ( κ ′ | κ ) ≈ (cid:18) γ ∗ / (cid:19) − ∼ − γ ∗ . (230)We see that the overlap integral does not depend onquantum numbers in this lowest-order approximation,valid at γ ∗ → ∞ . The asymptotic fractional accuracyof this approximation can be estimated from comparisonof Eq. (206) with Eq. (209) as ∼ [ln(ln γ ∗ )] / ln γ ∗ .
2. Overlaps of tightly bound states with other even states
For the overlap integral between a tightly bound state( ν k = 0, 0 < ˜ ν ≪
1) and an even loosely-bound state( ν ′k = 2, 4, . . . , ˜ ν ′ & | κ i = |N , , n − , n + , i and substituteEq. (223) for the second state | κ ′ i = | n ′ + , N ′ , ν ′k i . Recall-ing Eq. (192), we finally obtain L ( κ ′ | κ ) = 2(˜ ν ′ ) ( J N ′ n ′ + − J N n + ) √ ǫ κ G N ′ ,n ′ + , ˜ ν ′ √ ǫ κ − ǫ κ ′ , (231)where G N ′ ,n ′ + , ˜ ν ′ and J N n + are defined in Eqs. (203) and(224), respectively.
3. The bound-free case
Equations (218) and (219) give the relation (cid:16)R ∞ z g κ ( z ) g κ ′ ( z ) d z (cid:17) R ∞ z g κ ( z ) d z = ~ m ∗ g κ ′ ( z ) ∂η/∂E κ η ′ − ηE k κ ′ − E k κ ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = z . (232)In the case where the initial state κ belongs to the dis-crete spectrum but the final state κ ′ belongs to the con-tinuum, taking into account that z → γ ∗ → ∞ , we4can use the leading term of Eq. (217) at small z . It isgiven by Eq. (196) with ˜ ν replaced by i˜ ν ′ . Using the ex-pression for the gamma function of imaginary argument(cf. [22], 6.1.29)Γ(i˜ ν ′ ) = r π ˜ ν ′ sinh ˜ ν ′ π e − i φ ˜ ν ′ , (233)where φ ˜ ν ′ = π ν ′ + 12i [ln Γ(1 − i˜ ν ′ ) − ln Γ(1 + i˜ ν ′ )] , (234)we obtain g κ ′ ( z ) ≈ g κ ′ (0) = e − ˜ ν ′ π z max sinh ˜ ν ′ π ˜ ν ′ π S κ ′ = 1 − e − π ˜ ν ′ π ˜ ν ′ z max , (235)where S κ ′ ≡ sin ( φ ˜ ν ′ − θ ) (236)and θ is the undetermined phase factor in Eq. (217).To find the factor S κ ′ , we use the matching condition η ′ ext = η ′ int . The derivative of the wave function (217 canbe written asd g κ ′ d z = e − π ˜ ν ′ / √ z max Re (cid:20) e i θ dd z W i˜ ν ′ (cid:18) − z ˜ ν ′ a ∗ (cid:19)(cid:21) . (237)At z → z W i˜ ν ′ (cid:18) − z ˜ ν ′ a ∗ (cid:19) = e − i φ ˜ ν ′ r π ˜ ν ′ sinh π ˜ ν ′ (cid:20) cosh ˜ ν ′ π + i π sinh ˜ ν ′ π (cid:18) ln 2 za ∗ − Re Θ(i˜ ν ′ ) + 2 γ E (cid:19) (cid:21) , (238)where function Θ is given by Eq. (198). Substitution ofthe last expression into Eq. (237) givesd g κ ′ d z = − a ∗ e − ˜ ν ′ π/ √ z max (cid:20)(cid:18) cosh ˜ ν ′ π Im Θ(i˜ ν ′ ) (cid:19) cos( π ˜ ν ′ − θ )+ sinh ˜ ν ′ ππ (cid:18) ln 2 z a ∗ − Re Θ(i˜ ν ′ ) + 2 γ E (cid:19) sin( π ˜ ν ′ − θ ) (cid:21) . (239)Here, Im Θ(i˜ ν ′ ) = Im (cid:18) ln i˜ ν ′ − ν ′ − ψ (i˜ ν ′ ) (cid:19) = π − coth π ˜ ν ′ ) , (240)where in the last equality we have used the relation ψ (i˜ ν ′ ) = 1 / (2˜ ν ′ ) + ( π/
2) coth π ˜ ν ′ ) and have chosen thevalue of Im ln i˜ ν ′ ) = π/
2. Dividing Eq. (239) by Eq. (235)and using Eq. (240), we thus obtain the logarithmicderivative η ′ ext at small z = z : η ′ ext = − a ∗ (cid:20) ln 2 z a ∗ + π sign ˜ ν ′ − e − π ˜ ν ′ cot( φ ˜ ν ′ − θ )+2 γ E − Re Θ(i˜ ν ′ )] , (241) From Eq. (198) we obtainRe Θ(i˜ ν ′ ) = ln ˜ ν ′ − Re ψ (1 + i˜ ν ′ ) . (242)An efficient way of calculating Re Θ(i˜ ν ′ ) is presented inAppendix D.By matching of η ′ ext , which is given by Eq. (241), to η ′ int , which is given by Eq. (202), we obtainsign ˜ ν ′ cot( φ ˜ ν ′ − θ ) = 1 − e − π ˜ ν ′ π h ln γ ∗ − γ E − J N ′ n ′ + + 2 Re Θ(i˜ ν ′ ) i . (243)Now the factor S κ ′ (236) is provided in the explicit formby the identity 1 / sin ( φ ˜ ν ′ − θ ) = 1 + cot ( φ ˜ ν ′ − θ ): S κ ′ = − e − π ˜ ν ′ π ! h ln γ ∗ − γ E + J N ′ n ′ + + 2 Re Θ(i˜ ν ′ ) i (cid:27) − . (244)If the initial state is tightly bound ( ν k = 0), then wecan use Eq. (226) for the factor ( ∂η/∂E κ ) in Eq. (232).In the lowest approximation with respect to ˜ ν ≪ ν ) term in this equa-tion: ∂η/∂E k κ ≈ m ∗ a ∗ ˜ ν/ ~ . For the functions g κ and g ′ κ of equal parity, the denominator on the left-handside of Eq. (232) is approximately R z max z g κ ( z ) d z ≈ R ∞−∞ g κ ( z ) d z = because of the normalization con-dition (137), while the numerator is (cid:18)Z ∞ z g κ ( z ) g κ ′ ( z ) d z (cid:19) ≈ (cid:20) L ( κ ′ | κ ) (cid:21) = 14 L ( κ ′ | κ ) , (245)for the same normalization. Thus Eq. (232) can berewritten as L ( κ ′ | κ ) = ~ g κ ′ ( z ) m ∗ a ∗ ˜ ν ( η ′ − η ) | z = z E k κ ′ − E k κ ! (246)Using also Eq. (235) for g κ ( z ) and Eq. (225) for ( η ′ − η )in Eq. (232), we obtain L ( κ ′ | κ ) = ~ m ∗ a ∗ − e − π ˜ ν ′ πz max ˜ ν ′ ˜ ν J N ′ n ′ + − J N n + E k κ ′ − E k κ ! S κ ′ . (247)We recall that ˜ ν and ˜ ν ′ are defined by Eq. (192) andEq. (215) respectively, the numbers J N n + and J N ′ n ′ + aregiven by Eq. (203), a ∗ is defined by Eq. (191), and S κ ′ isprovided by Eqs. (244) and (242). C. Transverse geometric size
In the adiabatic approximation, Eq. (145) reduces to h κ | r ⊥ | κ i = 2 a N X k =0 ( k + 1 + n − ) (cid:16) C ( N ,n + ) k (cid:17) . (248)5This equation shows that the transverse size of the ionincreases with increasing N . In classical physics, thisincrease corresponds to the action of the electric field,induced in the reference frame comoving with the ion.The forces on the nucleus and the electron, caused bythis field, have opposite directions and therefore tend tostretch the ion along the radius. Since on the average N is proportional to the square of transverse momen-tum of the transverse motion of the ion as a whole, thisstretching tends to enhance with an increase of N .One can show (see Appendix E) that N X k =0 k (cid:16) C ( N ,n + ) k (cid:17) = Z − Z ( N − n + ) + n + Z . (249)In the most important case of the helium ion, the right-hand side of Eq. (249) reduces to N / n − = 0, forthese states we obtain h κ | r ⊥ | κ i = 2 a (cid:18) Z − Z ( N − n + ) + n + Z (cid:19) (250)In particular, for the helium ion ( Z = 2) h κ | r ⊥ | κ i = (2 + N ) a is independent of n + . D. Radiative transitions for circular polarization
In the adiabatic approximation, expressions (184) and(185) for the circular components of the dipole matrixelement for the radiative transitions from state | i i = | κ i to state | f i = | κ ′ i with n − = n ′− = 0 reduce to D fi, − α ea m = q Z n max+ L ( κ ′ | κ ) δ n ′ + ,n + + α δ N ′ , N + α , (251)where α = ± n max+ = max( n + , n ′ + ), and the overlapintegral L ( κ ′ | κ ) is given by Eq. (221).For the transitions between tightly-bound states ( ν ′k = ν k = 0), Eqs. (177), (251), and (230) give the followingapproximate expression for the oscillator strength to theleading order in 1 / ln γ ∗ : f fi,α = ~ ω Ry a a Z n max+ (cid:18) − γ ∗ (cid:19) δ n ′ + ,n + + α δ N ′ , N + α , (252)and Eq. (178) yields σ bb i → f,α ( ω ) = 4 π α f a Zn max+ (cid:18) − γ ∗ (cid:19) ω ∆ fi ( ω − ω fi ) δ n ′ + ,n + + α δ N ′ , N + α , (253)where α f is the fine structure constant. Figure 4 presents some examples of the transition energies and oscilla-tor strengths for radiative transitions between different tightly-bound states in the dipole adiabatic approximation,according to Eq. (252).For the bound-free transitions, substitution of Eq. (251) into Eq. (172) gives σ bf i → f,α ( ω ) = 4 πz max m ∗ ωZe a n max+ ~ ck f |L ( κ ′ | κ ) | δ n ′ + ,n + + α δ N ′ , N + α , (254)where k f ≡ k κ ′ is the longitudinal wavenumber defined by Eq. (155). Using Eq. (247), we obtain σ bf i → f,α ( ω ) = 2 Zn max+ ~ ω a e m ∗ c k f a ∗ ˜ ν − e − π ˜ ν ′ ˜ ν ′ S κ ′ J N ′ n ′ + − J N n + E k f − E k i ! δ n ′ + ,n + + α δ N ′ , N + α . (255)Taking into account definitions of γ ∗ (207), ˜ ν ′ (215), and α f = e / ~ c , we can write Eq. (255) in the form σ bf i → f,α ( ω ) = 2 α f Zn max+ ~ ωm ∗ γ ∗ ˜ ν (cid:20) − exp (cid:18) − πa ∗ k f (cid:19)(cid:21) S κ ′ J N ′ n ′ + − J N n + E k f − E k i ! δ n ′ + ,n + + α δ N ′ , N + α . (256)Using notations (192) and (193), we can also rewrite it as σ bf i → f,α ( ω ) = 4 α f n max+ a ~ ω Ry ∗ (cid:20) − exp (cid:18) − π √ ǫ f (cid:19)(cid:21) S f √ ǫ i (cid:18) J N ′ n ′ + − J N n + ǫ f + ǫ i (cid:19) δ n ′ + ,n + + α δ N ′ , N + α . (257)Here we have used the relation k f a ∗ = √ ǫ f , which follows from Eqs. (155), (191), (192), and (193). Note that both6 FIG. 4: Resonance transition energies E f − E i (upper panels)and oscillator strengths f fi,α (lower panels) for transitionsbetween tightly-bound states | i i = |N , n − , n + , ν i ( n − = 0, ν = 0) and | f i = |N ′ , n ′− , n ′ + , ν ′ i ( n ′− = 0, ν ′ = 0) with N ′ = N + 1 and n ′ + = n + + 1 in the adiabatic approximationaccording to Eq. (252), as functions of magnetic field strength B , for N = 0 (solid lines), 1 (long-dash-dot lines). 2 (short-dash-dot lines), 3 (long dashes), and 4 (short dashes). Leftpanels: n + = 0; right panels: n + = N . The lines are termi-nated at the points where the final state crosses the continuumand becomes autoionizing ( E f = 0). ǫ i and ǫ f are positive by definition. The energy conserva-tion law requires that E κ ′ = E κ + ~ ω . Therefore, accord-ing to Eqs. (117) and (44), for the allowed dipole transi-tions ( n ′ + = n + + α ) we have ~ ω/ Ry ∗ = ǫ f + ǫ i + 2 αγ ∗ .Examples of the photoionization cross sections, givenby Eq. (257) for the circular polarizations α = ±
1, arepresented in Figs. 5 – 8. Figures 5, 6, and 7 correspondto the field strengths B = 10 , 10 , and 10 G, respec-tively. For α = +1, cross sections for the four smallestvalues of n + and the four smallest possible values of N at each n + are shown. For α = −
1, there are no lineswith n + = 0, because absorption of photons with thispolarization by such states is forbidden in the adiabaticdipole approximation. In Fig. 8 for B = 5 × G, only n + = 0 and n + = 1 are considered, because the stateswith n + > E i at such strongfield. Although they can be treated as bound states inthe adiabatic approximation, they actually belong to thecontinuum and can autoionize due to admixtures of theLandau orbitals with smaller n + .Each cross section in Figs. 5 – 8 decreases with increas-ing photon energy ~ ω above the photoionization thresh-old ~ ω thr = − E i = | E k i | + α ~ ω c+ , where the longitudinal FIG. 5: Photoionization cross sections σ bf i → f,α ( ω ) for differentinitial tightly-bound states | i i = |N , n − , n + , ν i ( n − = 0, ν =0) at B = 10 G in the adiabatic approximation accordingto Eq. (257) for the right ( α = +1, the left panel) and left( α = −
1, right panel) circular polarizations as functions ofthe photon energy ~ ω in units of Thomson cross section σ T =(8 π/ e /m e c ) . The results are displayed for initial stateswith quantum numbers n + = 0 (solid lines), 1 (dot-dashedlines), 2 (dashed lines), and 3 (dotted lines) and N = n + , n + +1 , n + + 2 , n + + 3 (lines of the same type from top to bottomfor each n + ).FIG. 6: The same as in Fig. 5, but for B = 10 G. FIG. 7: The same as in Fig. 5, but for B = 10 G.FIG. 8: The same as in Fig. 5, but for B = 5 × G andonly for n + = 0 and 1. energies E k i are calculated according to the approxima-tion (206). The cross sections for the circular polarizationbecome smaller with increasing magnetic field strength B , in agreement with the decrease of the geometric trans-verse cross section of the ion, which is proportional to a ∝ B − (cf. Sect. VI C). The photoionization crosssections also become smaller with increasing N at fixed n + and ω . VII. CONCLUSIONS
We performed a systematic derivation of practicalequations for computing the basic characteristics of aone-electron ion in different quantum states in a strongmagnetic field: its binding energies, geometric sizes, os-cillator strengths of bound-bound transitions, and pho-toionization cross sections. These quantities are nec-essary ingredients for construction of models of atmo-spheres of neutron stars with strong magnetic fields un-der the conditions where one-electron ions can contributesubstantially into the atmospheric opacities. We didnot assume that the atomic nucleus is infinitely mas-sive or fixed in space, but considered the full quantum-mechanical two-body problem. This is especially im-portant in sufficiently warm atmospheres with suffi-ciently strong magnetic fields, where the thermal mo-tion of the ions cannot be decoupled from their inter-nal quantum-mechanical structure and the Rabi-Landauquantization of both the electron and the nucleus mustbe taken into account. The obtained results generallyconfirm, somewhat correct and extend the previouslypublished quantum-mechanical studies of an one-electronion, which moves in a quantizing magnetic field.In addition, we performed an approximate analytictreatment of the problem in the adiabatic approximationand derived explicit asymptotic expressions for the bind-ing energies, transverse geometric sizes, and cross sec-tions of absorption of radiation, polarized transversely tothe magnetic field. We expect that these analytic expres-sions can be useful in the case of superstrong magneticfields, typical for magnetars.
Acknowledgments
The work of A.P. was partially supported bythe Russian Foundation for Basic Research andDeutsche Forschungsgemeinschaft according to the re-search project 19-52-12013.8
AppendicesA. Supplementary relations for C ( N ,n + ) k Let us consider the operator
B ≡ √ Z (cid:16) √ Z − b + ˆ˜ a (cid:17) , (A1)where ˆ b and ˆ˜ a are defined by Eqs. (70) and (72)). It follows from Eqs. (71) and (73) that[ B , B † ] = 1 . (A2)Besides, from Eqs. (70 and (72) we see that B † B = a Z ~ ( Π ⊥ − k ⊥ ) − . (A3)Therefore, the eigenvalues of the operator B † B on the transverse basis states [Sect. III D] equal n + : B † B Ψ ′ ˜ N,n,L = n + Ψ ′ ˜ N,n,L . (A4)Taking the commutation relation (A2) into account, we conclude that B † raises n + by one, while B decreases n + byone (it can be proved explicitly using Eqs. [81]). On the other hand, using the explicit form of the basis wave functions(102) and the definition of B , we obtain B | ˜ N ′ , L, n − , n + i = N X k =0 F ( Z − N − k − , ˜ N ( r + , ⊥ ) F ( − n − ,k ( r − , ⊥ − r + , ⊥ ) r Z − Z √N − kC ( N ,n + ) k + √ k + 1 √ Z C ( N ,n + ) k +1 ! . (A5)Therefore √ n + C ( N − ,n + − k = r Z − Z √N − k C ( N ,n + ) k + r k + 1 Z C ( N ,n + ) k +1 (A6)In the same way, by considering B † | ˜ N ′ , L, n − , n + i , we find that p n + + 1 C ( N +1 ,n + +1) k = r Z − Z √N + 1 − k C ( N ,n + ) k + r kZ C ( N ,n + ) k − . (A7)Furthermore, let us consider operator ˜ B ≡ √ Z (cid:16) ˆ b − √ Z − a (cid:17) . (A8)It is also easy to see that [ ˜ B , ˜ B † ] = 1 and˜ B † ˜ B = ˆ b † ˆ b + ˆ˜ a † ˆ˜ a − a ,Z ~ ( Π ⊥ − k ⊥ ) + 12 . (A9)Therefore, the eigenvalues of the operator B † B on the transverse basis states equal N . Taking into account thecommutation relations, we obtain that ˜ B † and ˜ B are the creation and annihilation operators with respect to thequantum number N + ˜ n − n + = N − n + . By analogy with the case of operator B above, we obtain the followingrecurrent relations: p N − n + C ( N − ,n + ) k = r N − kZ C ( N ,n + ) k − r Z − Z √ k + 1 C ( N ,n + ) k +1 , (A10) p N − n + + 1 C ( N +1 ,n + ) k = r N + 1 − kZ C ( N ,n + ) k − r Z − Z √ k C ( N ,n + ) k − . (A11)9In the particular case n + = 0, relation (A6) gives √N − k √ Z − C ( N , k = −√ k + 1 C ( N , k +1 , which after k iterationsyields C ( N , k = ( − k ( Z − k/ vuut k ! k Y p =1 ( N − k + p ) C ( N , . (A12)Here, C ( N , can be found using Eq. (A11) at k = n + = 0, C ( N +1 , = C ( N , / √ Z , which yields C ( N , = C (0 , /Z N / = 1 /Z N / . The result can be written in the form C ( N , k = ( − k ( Z − k/ Z N / s N ! k ! ( N − k )! ≡ ( − k ( Z − k/ Z N / s(cid:18) N k (cid:19) . (A13)Now, using Eq. (A7), we can find C ( N ,n + ) k with different n + . B. Proof of Equation (202)
The substitution of Eq. (200) into Eq. (201) gives η int ( z ) ≈ − a ∗ Z z/a m √ d ζ N X k =0 k ! (cid:16) C ( N ,n + ) k (cid:17) Z ∞ d ρ ρ k e − ρ p ρ + ζ . (B1)Interchanging the summation and integration orders, we obtain η int ( z ) ≈ − a ∗ N X k =0 k ! (cid:16) C ( N ,n + ) k (cid:17) Z ∞ ρ k e − ρ (cid:20) ln (cid:16) ζ + p ρ + ζ (cid:17) −
12 ln ρ (cid:21) d ρ = − a ∗ N X k =0 (cid:16) C ( N ,n + ) k (cid:17) (cid:20) ln ζ + 1 k ! Z ∞ ρ k e − ρ ln (cid:16) p ρ/ζ (cid:17) d ρ − k ! Z ∞ ρ k e − ρ ln ρ d ρ (cid:21) . (B2)The first integral in the square brackets can be evaluated at ζ ≫ k ! ln 2 + O ( ζ − ). The last integral equals ([34],4.352) Z ∞ ρ k e − ρ ln ρ d ρ = − k ! ψ ( k ) = k ! ( H k − γ E ) , (B3)where ψ ( k ) is the digamma function (195) and H k = P kn =1 n − is the k th harmonic number. Taking into accountthe normalization of C ( N ,n + ) k [Eq. (99)] and setting z = z , we obtain Eq. (202). C. Estimate of J N at large N Let us consider Eq. (203) at n + = 0 and N ≫ J N = 1 Z N N X k =1 H k ( Z − k (cid:18) N k (cid:19) , (C1)where we have used Eq. (A13) for (cid:16) C ( N , k (cid:17) . According to the Stirling’s approximation for factorials, (cid:18) N k (cid:19) ∼ (cid:18) k/ N ) k/ N (1 − k/ N ) − k/ N + o (1) (cid:19) N . (C2)This function is strongly peaked at k ≈ N /
2. Therefore, we can take out H k at k ≈ N / N ≫ J N ∼ H [ N / Z N N X k =1 (cid:18) N k (cid:19) ( Z − k ≈ H [ N / Z N N X k =0 (cid:18) N k (cid:19) ( Z − k = H [ N / (1 + ( Z − N Z N = H [ N / . (C3)0Now from the double inequality [40] 12( k + 1) < H k − ln k − γ E < k (C4)we have H [ N / = ln N − ln 2 + γ E + O (1 / N ) and J N ∼ ln N + O (1) . (C5) D. Calculation of Re
Θ(i˜ ν ) The function Re Θ(i˜ ν ), which is given by Eq. (242), can be represented as (e.g., [22], 6.3.17)Re Θ(i˜ ν ) = ln ˜ ν + γ E − ∞ X k =1 (˜ ν ) k [ k + (˜ ν ) ] = ln ˜ ν + γ E − ∞ X k =1 k (1 + ǫk ) , (D1)where ǫ is the dimensionless longitudinal energy defined by Eq. (192). At large ǫ (small ˜ ν ) the series on the right-hand side converges well, but with decreasing ǫ the convergence becomes progressively slower, which can be easilyunderstood from the fact that the series diverges logarithmically at ǫ →
0. At this limit, one can use the formula(e.g., [22], 6.3.19) Re ψ (1 + i˜ ν ) = ln ˜ ν + ∞ X n =1 ( − n − B n n ˜ ν n , (D2)where B n are the Bernoulli numbers. However, due to the asymptotic nature of the latter formula, its accuracyrapidly worsens with increasing ǫ at any fixed number of terms.In this appendix we propose a method to calculate Re Θ(i˜ ν ) with keeping the number of terms of the sum inEq. (D1) reasonably small at intermediate ǫ . Let us consider the integral Z ∞ K f ǫ ( x ) d x = 12 ln (cid:18) ǫK (cid:19) , where f ǫ ( x ) ≡ x (1 + ǫx ) . (D3)According to the first mean value theorem for integrals, ∃ ξ k : k < ξ k < k + 1 , Z k +1 k f ǫ ( x ) d x = f ǫ ( k + 1) − f ǫ ( k )2 − f ′′ ǫ ( ξ k ) . (D4)Assuming that K ∈ N , we can write Z ∞ K f ǫ ( x ) d x = ∞ X k = K Z k +1 k f ǫ ( x ) d x = f ǫ ( K )2 + ∞ X k = K +1 f ǫ ( k ) − R K , where R K = 112 ∞ X k = K f ′′ ǫ ( ξ k ) . (D5)Using explicit f ǫ ( x ) in Eq. (D3), one can show that2 ξ f ǫ ( ξ ) < f ′′ ǫ ( ξ ) < ξ f ǫ ( ξ ) (D6)for any ξ > ǫ >
0. Therefore,0 < R K < ∞ X k = K ξ k (1 + ǫ ξ k ) < ∞ X k = K k (1 + ǫk ) < ∞ X k = K k (1 + ǫK ) = 12 K (1 + ǫ K ) . (D7)Equations (D3) and (D5) allow us to rewrite the sum on the right-hand side of Eq. (D1) as ∞ X k =1 k (1 + ǫk ) = K − X k =1 k (1 + ǫk ) + 12 K (1 + ǫK ) + 12 ln (cid:18) ǫK (cid:19) + R K . (D8)1Recalling that ˜ ν = 1 / √ ǫ and using Eqs. (D8) and (D7), we transform Eq. (D1) intoRe Θ(i˜ ν ) = γ E − K X k =1 k + ǫk + 12 K (1 + ǫK ) −
12 ln (cid:18) ǫ + 1 K (cid:19) − aK (1 + ǫK ) , (D9)where 0 < a < /
2. This transformation allows us to greatly reduce the number K of terms in the sum, that areneeded to attain a required accuracy. For example, to reproduce four digits of Re Θ(i˜ ν ) = − . ν = 1, wemust retain more than 300 terms in the original formula (D1), while K = 15 suffices in Eq. (D9) with a = 0 and only K = 7 with a = 0 . ν ) using Eq. (D9) with K = 3 + [11˜ ν ] and a = 0 .
25 at ˜ ν < ǫ > / ν ≥ ν ) ≈ − ǫ − ǫ − ǫ − ǫ − ǫ − ǫ − ǫ − ǫ . (D10)This recipe ensures that both absolute and fractional errors of calculated Re Θ(i˜ ν ) are less than 10 − for any ˜ ν . E. Proof of Equation (249)
Let us rewrite equations (A6) and (A10), respectively, as √ k + 1 C ( N ,n + ) k +1 = p Z n + C ( N − ,n + − k − p ( Z − N − k ) C ( N ,n + ) k , (E1)( Z − √ k + 1 C ( N ,n + ) k +1 = p ( Z − N − k ) C ( N ,n + ) k − p Z ( Z − N − n + ) C ( N − ,n + ) k . (E2)Taking the sum of the left and the right parts of these equations, we exclude the term p ( Z − N − k ) C ( N ,n + ) k and,having divided both parts by Z , obtain √ k + 1 C ( N ,n + ) k +1 = r n + Z C ( N − ,n + − k − r Z − Z p N − n + C ( N − ,n + ) k . (E3)Dropping the term with k = 0 (which equals zero) from the sum in the left-hand side of Eq. (249) and shifting thesummation index k to k + 1, we can write N X k =0 k (cid:16) C ( N ,n + ) k (cid:17) = N − X k =0 ( k + 1) (cid:16) C ( N ,n + ) k +1 (cid:17) . (E4)The substitution of Eq. (E3) gives N X k =0 k (cid:16) C ( N ,n + ) k (cid:17) = N − X k =0 r n + Z C ( N − ,n + − k − r Z − Z p N − n + C ( N − ,n + ) k ! = n + Z N − X k =0 (cid:16) C ( N − ,n + − k (cid:17) − Z p ( Z − n + ( N − n + ) N − X k =0 C ( N − ,n + − k C ( N − ,n + ) k + Z − Z ( N − n + ) N − X k =0 (cid:16) C ( N − ,n + ) k (cid:17) . (E5)According to the orthonormality relation (99), the first and third sums on the right-hand side equal one, and thesecond sum equals zero. Thus we are left with Eq. (249).2 [1] Demidov, I. V. Helium Ion in a Superstrong MagneticField , M.Sc. thesis (in Russian), Saint Petersburg Aca-demic University, 2018.[2] Johnson, B. R., Hirschfelder, J. O., & Yang K. H. Inter-action of atoms, molecules and ions with constant electricand magnetic fields,
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