The Hydrogen Atom In A Uniform Magnetic Field An Example Of Chaos
Transcript Of The Hydrogen Atom In A Uniform Magnetic Field An Example Of Chaos
THE HYDROGEN ATOM IN A UNIFORM MAGNETIC FIELD  AN EXAMPLE OF CHAOS
Harald FRIEDRICH Technische Universitàt München, PhysikDepartment, 8046 Garching, West Germany
and Dieter WINTGEN MaxPlanckInstitut für Kernphysik, Postfach 103980, 6900 Heidelberg, West Germany and Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
It NORTHHOLLAND  AMSTERDAM
PHYSICS REPORTS (Review Section of Physics Letters) 183, No. 2 (1989) 37—79. NorthHolland, Amsterdam
THE HYDROGEN ATOM IN A UNIFORM MAGNETIC FIELD  AN EXAMPLE OF CHAOS
Harald FRIEDRICH
Technische Universität München, PhysikDepartment, 8046 Garching, West Germany
and
Dieter WINTGEN
MaxPlanckInstitut für Kernphysik, Postfach 103980, 6900 Heidelberg, West Germany and Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
Received May 1989
Contents:
1. Introduction
39
2. The quantum mechanical hydrogen atom in a uniform
magnetic field
40
2.1. Hamiltonian
40
2.2. High field and low field regime
42
2.3. Regime of approximate separability
46
2.4. Comparison of calculated and observed spectra
48
3. Classical dynamics
51
3.1. Scaling
51
3.2. Regularization
52
3.3. Poincaré surfaces of section
53
3.4. Liapunov exponents, periodic orbits and bifurcations 56
4. Quantum mechanical observables and chaos
61
4.1. General remarks on quantum chaos
61
4.2. Energy level statistics
61
4.3. History of the quasiLandau phenomenon
67
4.4. Gutzwiller’s trace formula
68
5. Summary
75
References
76
Abstract: The hydrogen atom in a uniform magnetic field is discussed as a real and physical example of a simple nonintegrable system. The quantum
mechanical spectrum shows a region of approximate separability which breaks down as we approach the classical escape threshold. Classical dynamics depends only on the scaled energy given as the true energy divided by the third root of the square of the field strength. The classical transition from regular motion below the escape threshold to chaos near the escape threshold is accompanied by a corresponding transition in statistical properties of the short ranged quantum spectral fluctuations. Spectral properties involving correlations on a longer range depend
sensitively on systemspecific nonuniversal properties such as the occurrence of prominent periodic classical orbits. Knowledge of the classical periodic orbits leads to a quantitative understanding of the low frequency properties of the quantum spectra as summarized in Gutzwiller’s trace formula. These developments have led to a deeper understanding of the long known “quasiLandau resonances” and other modulations in photoabsorption spectra.
Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 183, No. 2 (1989) 37—79. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 33.00, postage included.
0 3701573/89/$15.05 © Elsevier Science Publishers B.V. (NorthHolland Physics Publishing Division)
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
39
1. Introduction
Although the equations of motion of classical mechanics are strictly deterministic, the actual path along which a complex classical system evolves may depend extremely sensitively on the initial conditions so that the evolution of the system becomes de facto unpredictable. Such a system is said to behave irregularly or chaotically. The fact that classical motion can be irregular has been known at least since the beginning of this century. However in recent years it has become increasingly clear that even seemingly simple systems with few degrees of freedom generally show chaotic behaviour, and advances in computer technology have made it possible to study irregular motion in small systems in considerable detail. This has made chaos one of the fastest growing fields in physics.
Bound classical motion in one spatial degree of freedom is always periodic if it is governed by a time independent Hamiltonian, i.e. if energy is conserved, and hence onedimensional conservative systems cannot be chaotic. The same is true for Ndimensional conservative systems, N> 1, if they are integrable, i.e. if their Hamiltonian can be written in terms of N conserved actions. The simplest conservative systems capable of exhibiting chaos are systems in N = 2 spatial dimensions with no other integral of motion besides the energy. Examples are various types of single particle billards [1—3],the Hénon—Heiles potential [4], various types of anharmonically coupled harmonic or anharmonic oscillators [5, 6J, or the problem of a hydrogen atom in a uniform magnetic field (see fig. 1).
In contrast to the other simple systems mentioned above, the hydrogen atom in a uniform magnetic field is not an abstract model system but a real physical system that can be and has been studied in the laboratory [7, 8]. When we study the regular or chaotic nature of the classical dynamics or look for manifestations of classical chaos in quantum spectra we are, in this example, doing real physics and not only mathematical physics. Our objects of study are sometimes classical trajectories or quantum spectra generated by computer codes, frequently however they are real spectra observed in experiments. In some cases, e.g. for the oscillations in photoabsorption spectra which have long been known under the name of quasiLandau resonances, a deeper appreciation of the classical dynamics, and in particular of the important role of isolated unstable periodic orbits embedded in the chaotic part of the phase space, has led to a deeper understanding of the structure of complex spectra. In particular, we now understand why the experimentally observed quasiLandau peaks are related to closed classical orbits in a way resembling a Bohr—Sommerfeld quantization condition, even in the classically chaotic region, where the orbits are unstable and the observed peaks do not correspond to individually resolved quantum states.
The aim of this article is to give a review of recent work on the hydrogen atom in a uniform magnetic field, paying special attention to the occurrence of chaos in the classical dynamics and its manifestation
L//J ~
C 67
Fig. 1. (a) Examples of single particle billards showing chaotic classical dynamics; a point particle moves freely in the enclosed area and is reflected
by the shaded boundaries. (b) Equipotential lines of the Hénon—Heiles potential f41.
40
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field— an example of chaos
in observed or observable quantum spectra. Chapter 2 contains a description of the system and a general discussion of the properties of its quantum mechanical spectrum, including a comparison between calculated and observed spectra. Chapter 3 contains a detailed discussion of the classical dynamics while chapter 4 discusses how the nature of the classical dynamics, in particular the occurrence of regular and chaotic motion and the existence of periodic orbits, manifests itself in the quantum spectra.
2. The quantum mechanical hydrogen atom in a uniform magnetic field
2.1. Hamiltonian
The hydrogen atom in a uniform magnetic field is accurately described over a wide range of field strengths B by the simple nonrelativistic singleparticle Hamiltonian
H p2I2me — e2Ir + wl~+ ~mew2(x2+ y2).
(1)
The direction of the field is taken as the zdirection and me is the reduced mass of electron and nucleus. The frequency w in (1) is half the cyclotron frequency
w = = eB/2mec.
(2)
At a field strength of
B= B
3cIh3 2.35 X i09 G = 2.35 X i05 T,
0 = m~e
the oscillator energy 11w equals the Rydberg energy ~2= mee4/(2112)
(3) 13.6 eV. In terms of the
dimensionless field strength parameter
y=BIB
0l1wIPJ~,
(4)
relativistic corrections [91to the simple model defined by (1) are negligible for fields w3ith> yi0<”1,0w’~h.Oerne
tnheis otthheerphrianncdip,atlhequeaffnetcutms ofnuspminb—ero. rbEitffceocutsplrienlgat[e1d0]tcoanthbee tnweoglbeocdteyd (fnourcflieeuldssawndithelyenctron) center of
mass motion in the presence of an external magnetic field have been investigated by several authors [10—14]I.t is possible to separate a generalized field strength dependent momentum, which replaces the
center of mass momentum of the fieldfree twobody system. For fields with y > 100 the internal dynamics is considerably influenced by the center of mass motion, but for a vanishing transversal component of the conserved generalized momentum the effect can be accounted for accurately by a constant energy shift which depends only on the magnetic field strength and the azimuthal quantum number m [13].
The azimuthal quantum number m is a good quantum number as is parity, which is frequently expressed in terms of the zparity ir defined with respect to reflection at the xyplane which is perpendicular to the direction of the magnetic field. In each m~subspace of Hilbert space the Schrödinger equation defined by the Hamiltonian (1) remains nonseparable in the two coordinates z,
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
41
parallel, and p = + y2, perpendicular, to the field. Leaving out the contribution of the normal Zeeman term wl~because it is constant, the stationary Schrödinger equation in a given m~subspace is (in atomic units, which we will use throughout unless stated otherwise):
p
—
+
+
— (p2 + z2)hI2]~(p, z) = E~(p,z),
(5)
2 pop Op Oz p
where ~!‘p(, z) is the cylindrical radial part of the full singleelectron wave function ~P,
tII(p, z, 4) (1 I/~)W(p, z) eim4 .
(6)
Thus the physical problem is that of a particle moving in an effective twodimensional potential
V(p, z) =
+ m212p2 — ii~p2+ z2.
(7)
The potential (7) is illustrated in fig. 2 for m = 0. Attempts to solve the Schrodinger equation (5) have a long history. Early accurate numerical
calculations of the energies of lowlying states are due to Praddaude [15], Smith et al. [16] and Simola and Virtamo [17]. First accurate numerical calculations going beyond the lowest three or four states in each m’~subspace were performed by Clark and Taylor [18] who calculated energies, wave functions and oscillator strengths up to and beyond the onset of the nmixing regime (see section 2.2) at a field strength of y = 2 x i0~ (corresponding to B = 4.7 T).
Review articles dealing wholly or in part with the problem of a hydrogen atom in a uniform magnetic field have been written by Garstang [10], Bayfield [19], Kleppner et al. [20], Gay [21], Clark et al. [22], Delande et a!. [23] and Clark [24]. Highly accurate values for the ground state energy have been given by Le Guillou and ZinnJustin [25], and Rösner et al. [26] have given a comprehensive list of the energies of the lowest four or five states in various m~subspaces at arbitrary field strengths (see also Rech et al. [27], Cho et a!. [28], and Liu and Starace [29]). Some recent approaches to solve the problem for Rydberg states in moderate fields involve higher order perturbation theory [30—32], diagonalization in symmetry adapted basis sets [30, 33, 34], adiabatic semiclassical methods [35, 36], and an adiabatic quantum approach [140]. At very high fields, y > 1, the diamagnetic term proportional to y2 dominates the entire spectrum and convergent expansions in the Landau basis are practicable (see section 2.2). In this region complete calculations of bound states [37] and extensive studies of continuum states have been undertaken [31, 38—41].
p
~
Fig. 2. Equipotential lines of the potential (7) for azimuthal quantum number m 0.
42
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
2.2. High field and low field regime
At very high fields the Schrödinger equation (5) is best solved [15—173,7, 38] by expanding the wave function ~I’(p,z) in Landau states PNm(P), which are the normalized radial parts of the twodimensional harmonic oscillator wave functions corresponding to the oscillator energy 11w = y
~P(p,z) = ~, cPNm(p)~PN(z).
(8)
This leads to a set of coupled channel equations for the wave functions 111(z) in the various Landau channels:
(~~  Ny + (E  Em))~(Z)  V~N.(z)~(z)= 0.
(9)
In (9) Em = (~m+~l)y12 is the zeropoint energy of the lowest Landau state and defines the real ionization threshold in the corresponding m~subspace — as opposed to the zerofield threshold at E = 0 in eq. (5). The potentials in (9) are defined by
J V~N(z)= dp P~m(P) —1 2 ~Nm(P)
(10)
0
and do not depend on the sign of the azimuthal quantum number m. Analytic expressions for V~N(z)
have been given by Friedrich and Chu [38]. Asymptotically (large zi) the diagonal potentials are given
by
V~N(z)=— 1 (i 2N+H + 1 ~ +O(z4)),
(11)
where b = ~/~7i3s the oscillator width of the Landau states. An efficient numerical procedure for
calculating the potentials and solving the coupled equations (9) for various magnetic field strengths can
be obtained by exploiting the fact that the potentials only depend on the field strength via a universal
scaling factor 1/b [31, 42].
The formulation (9) of the Schrödinger equation (5) shows that we have, in each m~subspace, a
system of coupled Coulombic channels which are labelled by the Landau quantum number N =
0, 1, 2
The channel thresholds
E~~E~+Ny
(12)
lie Ny above the real ionization threshold Em in the respective subspace. The coupling potentials V~N(z) fall off relatively slowly and are asymptotically proportional to z 2INN I—i
For extremely large field strengths the energy y needed to excite a Landau oscillation perpendicular to the magnetic field becomes very large and the problem approaches that of a onedimensional hydrogen atom parallel to the field [43]. The bound states are then dominated by the contribution from the N = 0 channel, in which the motion of the electron perpendicular to the field is given by the lowest
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
43
Landau state ~O,m’ and they form a nondegenerate Rydberg series converging to the respective
ionization threshold Em [37]
Ep=E~~1I2(V~/hp)2,
(13)
where t’ runs from zero to infinity for positive zparity states and from one to infinity for negative zparity states. The quantum defect parameters ~ in (13) are all negative and converge to zero in the limit of infinite field strengths; note that the lowest state in each m~subspace of positive zparity corresponds to ii =0 and becomes infinitely bound in this limit.
At sufficiently high field strengths, y ~ 1, Landau excited states corresponding to N> 0 all lie above the ionization threshold and form Rydberg series of autoionizing resonances which converge to the respective Landau channel thresholds (12). These resonances can autoionize by deexcitation of Landau oscillations perpendicular to the field [38]. The spectrum of bound and autoionizing states in the
= 0~subspace is illustrated in fig. 3 for y = 2, y = 1, and ‘y = 0.5. As the field strength is reduced from values around y = 1, the Rydberg series associated with the different Landau channels begin to overlap and interfere. Near y = 0.3 the lowest autoionizing resonances, which are characterized by a jump through ~r of the asymptotic phase shift ~ of the open channel (N = 0) wave function, cross the ionization threshold and become perturbations of the Rydberg
series of bound states, which are characterized by a jump through unity of the corresponding quantum defects p..~,.This is illustrated in fig. 4 for the m~= 0 + subspace. As the field strength is reduced further, more and more Rydberg series overlap and the spectrum becomes increasingly complicated. As long as not too many Rydberg series overlap, the problem may be treated with the techniques of multichannel
8 E~Em (Ry) 
6
— —
Nr2


1=2.0
4

—~—
N=2
——~~—— —
N=1 
2  
N=2
—~—
Nr1
0  —_
N=t N=0
 

N=0
~

—
N~0
2 
—
Fig. 3. Spectrum of bound and autoionizing states in the m” = 0~subspace for field strength parameters y = 2.0, 1.0, 0.5 (from ref. [42]).
44
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
I
~ou~d~:s~aIs
I
m’~= 0~
I•
31
‘in
I
I
I
I
~0.2 X~0.3I
I
1~I~
Y~0.5
I
I’
1.2
1.0
•I
I
I
I
I
0.2
0
0.2
0.4
0.6 0.8
~.0
E/R
Fig. 4. Quantum defects ~s of bound states and asymptotic phase shifts of the open channel (N = 0) wave function for various values of the field strength parameter y (from Friedrich and Chu [381).(E is the energy relative to threshold in Rydbergs.)
quantum defect theory [44, 45]. Figure 5 illustrates the multichannel structure of the spectrum in the m~= 1 + subspace at y = 0.04. The top half of the figure shows the quantum defects of the bound states (left half) and the asymptotic phase shifts of the open channel (N = 0) wave functions in the region between the ionization threshold Em and the inelastic threshold EmN=l = Em + y (right half). The perturber of the bound states and the resonances above threshold form a Rydberg series whose energies
4 V~f) 5 6 810
__________________________
6 ~~ ~‘~‘
I 006
0,04
I II
0
I
I
I
002
0
0.02
E~Em
o
I~
I
II I
0.04 0.06
I
~=io ~
2
,=~
*
12~2
I—
——1.0
v
3
12)
Veff
4
5 6 810
Fig. 5. Spectrum in the m” = 1 * subspace at y = 0.04. The top half shows quantum defects of the perturbed Rydberg series of bound states (E < E,,,) and the asymptotic phase shifts of the open channel (N = 0) wave function for energies between the ionization threshold and the inelastic threshold (first Landau threshold). The bottom half of the figure shows the quantum defects of the (perturbed) Rydberg series which consists of the boundstate perturber in the left hand part of the top half and the resonances in the right hand part of the top half. The horizontal bars show the absolute widths of the perturber and the resonances, while the vertical bars show the same widths multiplied by the third power of the effective quantum number v~with respect to the inelastic threshold. These renormalized widths would be roughly constant in an unperturbed Rydberg
series of autoionizing resonances. (From ref. [421(.E)nergies are in Rydbergs.)
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
45
E~converge to the inelastic, the N = 1, threshold and can be characterized by a series of second order
quantum
defects
2): ~j~
E~= EmN=l — 1/2(~~— 2~I~(2))2.
(14)
This series of perturbers and resonances is in turn perturbed by a resonance associated with the Rydberg series converging to the N = 2 Landau channel threshold, and this affects the energies, as shown by the jump through unity of the second order quantum defects 142) in the lower part of fig. 5, as
well as the widths of the perturbers or resonances. The essential features of the spectrum at y = 0.04 can be accurately described in the framework of threechannel quantum defect theory, the important channels being the N=0, N= 1, and N=2 Landau channels [31, 42, 46, 47].
At still lower field strengths the number of interfering Landau channels becomes so large that a
description in terms of the Landau basis (8) becomes impracticable. For lowlying bound states at low fields the spectrum is close to that of the fieldfree hydrogen atom and the effect of the diamagnetic interaction can be treated perturbatively [48—52].
At vanishing field strength the spectrum in each m” subspace is degenerate due to the 0(4) symmetry of the pure Coulomb problem. This can be expressed in the conservation of the Runge—Lenz vector
A = (—2h2m~E)”2[(pX L) — (mee2/r)r].
(15)
For small but finite values of the field strength the quantity
.Z=4A2—5A~
(16)
is an invariant up to first order in ~,2 In the perturbative regime it is appropriate to label the eigenstates of 1~by an index k starting at k = 0 for the maximum eigenvalue within a given nmanifold of states
degenerate in the zerofield limit. In a given rn’T subspace k runs from 0 (for ~ = +1) or 1 (for IT = —1) to its maximum value n — rn — 1 within a given nmanifold. For positive eigenvalues s of (16) the eigenstates have approximately °A(3) symmetry and are almost eigenstates of the angular momentumtype operator A = (Ar, A~,l~)with eigenvalues A(A + 1) of A2. These states are called rotator states and the eigenvalue A is related to the label k by A = n — 1 — k. The energy shifts of the rotator states are given approximately in the perturbative regime by [48, 51]
= ~y2n2(~A(A + 1) + ~n2 — 3m2 + ~).
(17)
The eigenfunctions of 1~with negative eigenvalues s have approximately 0(2) ® 0(2) symmetry and can be described by a twodimensional harmonic oscillator with anharmonic corrections. The approximate energy shifts of these vibrational states are, again to first order in y2:
= ~gy2n2[(2o+ Im~+ 1)2V’~n— 3(2o + m~+ 1)2 — m2 + 1],
(18)
where o = 0, 1,2,... is given by 2o = n — rn~— 1 — k or 2o = n — m~—2— k, depending on whether n — mi — k is odd or even [48, 51]. The accuracy of the formulae (17), (18) has been tested by Wintgen and by Wunner [31, 52].
46
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
2.3. Regime of approximate separability
The classification of states by the principal quantum number n and the “intrashell label” k is definitely meaningful in the 1mixing regime where the diamagnetic interaction is strong enough to break rotational symmetry but still so weak that nmanifolds of states originating from different principal quantum numbers n are separated in energy. It is not a priori clear whether such a classification remains meaningful in the nmixing regime where neighbouring nmanifolds overlap. The onset of the nmixing regime occurs, when the lowest (k = n — mi — 1) vibrator state of the (n + 1)manifold meets the highest (k = 0 or k = 1) rotator state of the nmanifold. From the leading terms in (17) and (18) the onset of the nmixing regime is given by
y2n716/5.
(19)
It was observed in numerical diagonalizations of the Hamiltonian (1) that near degeneracies occurred at the onset of nmixing, the actual magnitudes of the avoided crossings between the lowest state of the n + 1 manifold and the highest state of the nmanifold decreasing exponentially with n [53, 54]. This was interpreted as evidence for an additional hidden symmetry related to a further constant of motion and even as evidence for the existence of an approximately separable representation of the Hamiltonian. An additional (approximate) constant of motion valid for low fields was recognized by Solov’ev [48]and Herrick [51]and is given by the combination (16) of components of the Runge—Lenz vector. Solov’ev explained the small anticrossings of states as being a consequence of the fact that the states involved in the crossing are not only approximately eigenstates of the adiabatic invariant (16) with very different eigenvalues, but also correspond to different classes (rotator and vibrator) of eigenstates.
The above results stimulated an intensive search to find an approximately separable representation of the Hamiltonian. Such a representation was explicitly constructed by Wintgen and Friedrich [33]via a
I
E(crrc’)
I
m~
112 113 
n=31 K=28
n = 30 K= 10
nK53301
I
I
—

45
46
4.7
4.8
B IT)
Fig. 6. Energy eigenvalues obtained by sequential diagonalization. The dashed lines show the results of diagonalizing the Hamiltonian within subspaces of states characterized by a given value of the intrashell label k. The solid lines show the exact results obtained by allowing kmixing.
(From ref. [331.)
Harald FRIEDRICH Technische Universitàt München, PhysikDepartment, 8046 Garching, West Germany
and Dieter WINTGEN MaxPlanckInstitut für Kernphysik, Postfach 103980, 6900 Heidelberg, West Germany and Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
It NORTHHOLLAND  AMSTERDAM
PHYSICS REPORTS (Review Section of Physics Letters) 183, No. 2 (1989) 37—79. NorthHolland, Amsterdam
THE HYDROGEN ATOM IN A UNIFORM MAGNETIC FIELD  AN EXAMPLE OF CHAOS
Harald FRIEDRICH
Technische Universität München, PhysikDepartment, 8046 Garching, West Germany
and
Dieter WINTGEN
MaxPlanckInstitut für Kernphysik, Postfach 103980, 6900 Heidelberg, West Germany and Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
Received May 1989
Contents:
1. Introduction
39
2. The quantum mechanical hydrogen atom in a uniform
magnetic field
40
2.1. Hamiltonian
40
2.2. High field and low field regime
42
2.3. Regime of approximate separability
46
2.4. Comparison of calculated and observed spectra
48
3. Classical dynamics
51
3.1. Scaling
51
3.2. Regularization
52
3.3. Poincaré surfaces of section
53
3.4. Liapunov exponents, periodic orbits and bifurcations 56
4. Quantum mechanical observables and chaos
61
4.1. General remarks on quantum chaos
61
4.2. Energy level statistics
61
4.3. History of the quasiLandau phenomenon
67
4.4. Gutzwiller’s trace formula
68
5. Summary
75
References
76
Abstract: The hydrogen atom in a uniform magnetic field is discussed as a real and physical example of a simple nonintegrable system. The quantum
mechanical spectrum shows a region of approximate separability which breaks down as we approach the classical escape threshold. Classical dynamics depends only on the scaled energy given as the true energy divided by the third root of the square of the field strength. The classical transition from regular motion below the escape threshold to chaos near the escape threshold is accompanied by a corresponding transition in statistical properties of the short ranged quantum spectral fluctuations. Spectral properties involving correlations on a longer range depend
sensitively on systemspecific nonuniversal properties such as the occurrence of prominent periodic classical orbits. Knowledge of the classical periodic orbits leads to a quantitative understanding of the low frequency properties of the quantum spectra as summarized in Gutzwiller’s trace formula. These developments have led to a deeper understanding of the long known “quasiLandau resonances” and other modulations in photoabsorption spectra.
Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 183, No. 2 (1989) 37—79. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 33.00, postage included.
0 3701573/89/$15.05 © Elsevier Science Publishers B.V. (NorthHolland Physics Publishing Division)
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
39
1. Introduction
Although the equations of motion of classical mechanics are strictly deterministic, the actual path along which a complex classical system evolves may depend extremely sensitively on the initial conditions so that the evolution of the system becomes de facto unpredictable. Such a system is said to behave irregularly or chaotically. The fact that classical motion can be irregular has been known at least since the beginning of this century. However in recent years it has become increasingly clear that even seemingly simple systems with few degrees of freedom generally show chaotic behaviour, and advances in computer technology have made it possible to study irregular motion in small systems in considerable detail. This has made chaos one of the fastest growing fields in physics.
Bound classical motion in one spatial degree of freedom is always periodic if it is governed by a time independent Hamiltonian, i.e. if energy is conserved, and hence onedimensional conservative systems cannot be chaotic. The same is true for Ndimensional conservative systems, N> 1, if they are integrable, i.e. if their Hamiltonian can be written in terms of N conserved actions. The simplest conservative systems capable of exhibiting chaos are systems in N = 2 spatial dimensions with no other integral of motion besides the energy. Examples are various types of single particle billards [1—3],the Hénon—Heiles potential [4], various types of anharmonically coupled harmonic or anharmonic oscillators [5, 6J, or the problem of a hydrogen atom in a uniform magnetic field (see fig. 1).
In contrast to the other simple systems mentioned above, the hydrogen atom in a uniform magnetic field is not an abstract model system but a real physical system that can be and has been studied in the laboratory [7, 8]. When we study the regular or chaotic nature of the classical dynamics or look for manifestations of classical chaos in quantum spectra we are, in this example, doing real physics and not only mathematical physics. Our objects of study are sometimes classical trajectories or quantum spectra generated by computer codes, frequently however they are real spectra observed in experiments. In some cases, e.g. for the oscillations in photoabsorption spectra which have long been known under the name of quasiLandau resonances, a deeper appreciation of the classical dynamics, and in particular of the important role of isolated unstable periodic orbits embedded in the chaotic part of the phase space, has led to a deeper understanding of the structure of complex spectra. In particular, we now understand why the experimentally observed quasiLandau peaks are related to closed classical orbits in a way resembling a Bohr—Sommerfeld quantization condition, even in the classically chaotic region, where the orbits are unstable and the observed peaks do not correspond to individually resolved quantum states.
The aim of this article is to give a review of recent work on the hydrogen atom in a uniform magnetic field, paying special attention to the occurrence of chaos in the classical dynamics and its manifestation
L//J ~
C 67
Fig. 1. (a) Examples of single particle billards showing chaotic classical dynamics; a point particle moves freely in the enclosed area and is reflected
by the shaded boundaries. (b) Equipotential lines of the Hénon—Heiles potential f41.
40
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field— an example of chaos
in observed or observable quantum spectra. Chapter 2 contains a description of the system and a general discussion of the properties of its quantum mechanical spectrum, including a comparison between calculated and observed spectra. Chapter 3 contains a detailed discussion of the classical dynamics while chapter 4 discusses how the nature of the classical dynamics, in particular the occurrence of regular and chaotic motion and the existence of periodic orbits, manifests itself in the quantum spectra.
2. The quantum mechanical hydrogen atom in a uniform magnetic field
2.1. Hamiltonian
The hydrogen atom in a uniform magnetic field is accurately described over a wide range of field strengths B by the simple nonrelativistic singleparticle Hamiltonian
H p2I2me — e2Ir + wl~+ ~mew2(x2+ y2).
(1)
The direction of the field is taken as the zdirection and me is the reduced mass of electron and nucleus. The frequency w in (1) is half the cyclotron frequency
w = = eB/2mec.
(2)
At a field strength of
B= B
3cIh3 2.35 X i09 G = 2.35 X i05 T,
0 = m~e
the oscillator energy 11w equals the Rydberg energy ~2= mee4/(2112)
(3) 13.6 eV. In terms of the
dimensionless field strength parameter
y=BIB
0l1wIPJ~,
(4)
relativistic corrections [91to the simple model defined by (1) are negligible for fields w3ith> yi0<”1,0w’~h.Oerne
tnheis otthheerphrianncdip,atlhequeaffnetcutms ofnuspminb—ero. rbEitffceocutsplrienlgat[e1d0]tcoanthbee tnweoglbeocdteyd (fnourcflieeuldssawndithelyenctron) center of
mass motion in the presence of an external magnetic field have been investigated by several authors [10—14]I.t is possible to separate a generalized field strength dependent momentum, which replaces the
center of mass momentum of the fieldfree twobody system. For fields with y > 100 the internal dynamics is considerably influenced by the center of mass motion, but for a vanishing transversal component of the conserved generalized momentum the effect can be accounted for accurately by a constant energy shift which depends only on the magnetic field strength and the azimuthal quantum number m [13].
The azimuthal quantum number m is a good quantum number as is parity, which is frequently expressed in terms of the zparity ir defined with respect to reflection at the xyplane which is perpendicular to the direction of the magnetic field. In each m~subspace of Hilbert space the Schrödinger equation defined by the Hamiltonian (1) remains nonseparable in the two coordinates z,
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
41
parallel, and p = + y2, perpendicular, to the field. Leaving out the contribution of the normal Zeeman term wl~because it is constant, the stationary Schrödinger equation in a given m~subspace is (in atomic units, which we will use throughout unless stated otherwise):
p
—
+
+
— (p2 + z2)hI2]~(p, z) = E~(p,z),
(5)
2 pop Op Oz p
where ~!‘p(, z) is the cylindrical radial part of the full singleelectron wave function ~P,
tII(p, z, 4) (1 I/~)W(p, z) eim4 .
(6)
Thus the physical problem is that of a particle moving in an effective twodimensional potential
V(p, z) =
+ m212p2 — ii~p2+ z2.
(7)
The potential (7) is illustrated in fig. 2 for m = 0. Attempts to solve the Schrodinger equation (5) have a long history. Early accurate numerical
calculations of the energies of lowlying states are due to Praddaude [15], Smith et al. [16] and Simola and Virtamo [17]. First accurate numerical calculations going beyond the lowest three or four states in each m’~subspace were performed by Clark and Taylor [18] who calculated energies, wave functions and oscillator strengths up to and beyond the onset of the nmixing regime (see section 2.2) at a field strength of y = 2 x i0~ (corresponding to B = 4.7 T).
Review articles dealing wholly or in part with the problem of a hydrogen atom in a uniform magnetic field have been written by Garstang [10], Bayfield [19], Kleppner et al. [20], Gay [21], Clark et al. [22], Delande et a!. [23] and Clark [24]. Highly accurate values for the ground state energy have been given by Le Guillou and ZinnJustin [25], and Rösner et al. [26] have given a comprehensive list of the energies of the lowest four or five states in various m~subspaces at arbitrary field strengths (see also Rech et al. [27], Cho et a!. [28], and Liu and Starace [29]). Some recent approaches to solve the problem for Rydberg states in moderate fields involve higher order perturbation theory [30—32], diagonalization in symmetry adapted basis sets [30, 33, 34], adiabatic semiclassical methods [35, 36], and an adiabatic quantum approach [140]. At very high fields, y > 1, the diamagnetic term proportional to y2 dominates the entire spectrum and convergent expansions in the Landau basis are practicable (see section 2.2). In this region complete calculations of bound states [37] and extensive studies of continuum states have been undertaken [31, 38—41].
p
~
Fig. 2. Equipotential lines of the potential (7) for azimuthal quantum number m 0.
42
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
2.2. High field and low field regime
At very high fields the Schrödinger equation (5) is best solved [15—173,7, 38] by expanding the wave function ~I’(p,z) in Landau states PNm(P), which are the normalized radial parts of the twodimensional harmonic oscillator wave functions corresponding to the oscillator energy 11w = y
~P(p,z) = ~, cPNm(p)~PN(z).
(8)
This leads to a set of coupled channel equations for the wave functions 111(z) in the various Landau channels:
(~~  Ny + (E  Em))~(Z)  V~N.(z)~(z)= 0.
(9)
In (9) Em = (~m+~l)y12 is the zeropoint energy of the lowest Landau state and defines the real ionization threshold in the corresponding m~subspace — as opposed to the zerofield threshold at E = 0 in eq. (5). The potentials in (9) are defined by
J V~N(z)= dp P~m(P) —1 2 ~Nm(P)
(10)
0
and do not depend on the sign of the azimuthal quantum number m. Analytic expressions for V~N(z)
have been given by Friedrich and Chu [38]. Asymptotically (large zi) the diagonal potentials are given
by
V~N(z)=— 1 (i 2N+H + 1 ~ +O(z4)),
(11)
where b = ~/~7i3s the oscillator width of the Landau states. An efficient numerical procedure for
calculating the potentials and solving the coupled equations (9) for various magnetic field strengths can
be obtained by exploiting the fact that the potentials only depend on the field strength via a universal
scaling factor 1/b [31, 42].
The formulation (9) of the Schrödinger equation (5) shows that we have, in each m~subspace, a
system of coupled Coulombic channels which are labelled by the Landau quantum number N =
0, 1, 2
The channel thresholds
E~~E~+Ny
(12)
lie Ny above the real ionization threshold Em in the respective subspace. The coupling potentials V~N(z) fall off relatively slowly and are asymptotically proportional to z 2INN I—i
For extremely large field strengths the energy y needed to excite a Landau oscillation perpendicular to the magnetic field becomes very large and the problem approaches that of a onedimensional hydrogen atom parallel to the field [43]. The bound states are then dominated by the contribution from the N = 0 channel, in which the motion of the electron perpendicular to the field is given by the lowest
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
43
Landau state ~O,m’ and they form a nondegenerate Rydberg series converging to the respective
ionization threshold Em [37]
Ep=E~~1I2(V~/hp)2,
(13)
where t’ runs from zero to infinity for positive zparity states and from one to infinity for negative zparity states. The quantum defect parameters ~ in (13) are all negative and converge to zero in the limit of infinite field strengths; note that the lowest state in each m~subspace of positive zparity corresponds to ii =0 and becomes infinitely bound in this limit.
At sufficiently high field strengths, y ~ 1, Landau excited states corresponding to N> 0 all lie above the ionization threshold and form Rydberg series of autoionizing resonances which converge to the respective Landau channel thresholds (12). These resonances can autoionize by deexcitation of Landau oscillations perpendicular to the field [38]. The spectrum of bound and autoionizing states in the
= 0~subspace is illustrated in fig. 3 for y = 2, y = 1, and ‘y = 0.5. As the field strength is reduced from values around y = 1, the Rydberg series associated with the different Landau channels begin to overlap and interfere. Near y = 0.3 the lowest autoionizing resonances, which are characterized by a jump through ~r of the asymptotic phase shift ~ of the open channel (N = 0) wave function, cross the ionization threshold and become perturbations of the Rydberg
series of bound states, which are characterized by a jump through unity of the corresponding quantum defects p..~,.This is illustrated in fig. 4 for the m~= 0 + subspace. As the field strength is reduced further, more and more Rydberg series overlap and the spectrum becomes increasingly complicated. As long as not too many Rydberg series overlap, the problem may be treated with the techniques of multichannel
8 E~Em (Ry) 
6
— —
Nr2


1=2.0
4

—~—
N=2
——~~—— —
N=1 
2  
N=2
—~—
Nr1
0  —_
N=t N=0
 

N=0
~

—
N~0
2 
—
Fig. 3. Spectrum of bound and autoionizing states in the m” = 0~subspace for field strength parameters y = 2.0, 1.0, 0.5 (from ref. [42]).
44
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
I
~ou~d~:s~aIs
I
m’~= 0~
I•
31
‘in
I
I
I
I
~0.2 X~0.3I
I
1~I~
Y~0.5
I
I’
1.2
1.0
•I
I
I
I
I
0.2
0
0.2
0.4
0.6 0.8
~.0
E/R
Fig. 4. Quantum defects ~s of bound states and asymptotic phase shifts of the open channel (N = 0) wave function for various values of the field strength parameter y (from Friedrich and Chu [381).(E is the energy relative to threshold in Rydbergs.)
quantum defect theory [44, 45]. Figure 5 illustrates the multichannel structure of the spectrum in the m~= 1 + subspace at y = 0.04. The top half of the figure shows the quantum defects of the bound states (left half) and the asymptotic phase shifts of the open channel (N = 0) wave functions in the region between the ionization threshold Em and the inelastic threshold EmN=l = Em + y (right half). The perturber of the bound states and the resonances above threshold form a Rydberg series whose energies
4 V~f) 5 6 810
__________________________
6 ~~ ~‘~‘
I 006
0,04
I II
0
I
I
I
002
0
0.02
E~Em
o
I~
I
II I
0.04 0.06
I
~=io ~
2
,=~
*
12~2
I—
——1.0
v
3
12)
Veff
4
5 6 810
Fig. 5. Spectrum in the m” = 1 * subspace at y = 0.04. The top half shows quantum defects of the perturbed Rydberg series of bound states (E < E,,,) and the asymptotic phase shifts of the open channel (N = 0) wave function for energies between the ionization threshold and the inelastic threshold (first Landau threshold). The bottom half of the figure shows the quantum defects of the (perturbed) Rydberg series which consists of the boundstate perturber in the left hand part of the top half and the resonances in the right hand part of the top half. The horizontal bars show the absolute widths of the perturber and the resonances, while the vertical bars show the same widths multiplied by the third power of the effective quantum number v~with respect to the inelastic threshold. These renormalized widths would be roughly constant in an unperturbed Rydberg
series of autoionizing resonances. (From ref. [421(.E)nergies are in Rydbergs.)
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
45
E~converge to the inelastic, the N = 1, threshold and can be characterized by a series of second order
quantum
defects
2): ~j~
E~= EmN=l — 1/2(~~— 2~I~(2))2.
(14)
This series of perturbers and resonances is in turn perturbed by a resonance associated with the Rydberg series converging to the N = 2 Landau channel threshold, and this affects the energies, as shown by the jump through unity of the second order quantum defects 142) in the lower part of fig. 5, as
well as the widths of the perturbers or resonances. The essential features of the spectrum at y = 0.04 can be accurately described in the framework of threechannel quantum defect theory, the important channels being the N=0, N= 1, and N=2 Landau channels [31, 42, 46, 47].
At still lower field strengths the number of interfering Landau channels becomes so large that a
description in terms of the Landau basis (8) becomes impracticable. For lowlying bound states at low fields the spectrum is close to that of the fieldfree hydrogen atom and the effect of the diamagnetic interaction can be treated perturbatively [48—52].
At vanishing field strength the spectrum in each m” subspace is degenerate due to the 0(4) symmetry of the pure Coulomb problem. This can be expressed in the conservation of the Runge—Lenz vector
A = (—2h2m~E)”2[(pX L) — (mee2/r)r].
(15)
For small but finite values of the field strength the quantity
.Z=4A2—5A~
(16)
is an invariant up to first order in ~,2 In the perturbative regime it is appropriate to label the eigenstates of 1~by an index k starting at k = 0 for the maximum eigenvalue within a given nmanifold of states
degenerate in the zerofield limit. In a given rn’T subspace k runs from 0 (for ~ = +1) or 1 (for IT = —1) to its maximum value n — rn — 1 within a given nmanifold. For positive eigenvalues s of (16) the eigenstates have approximately °A(3) symmetry and are almost eigenstates of the angular momentumtype operator A = (Ar, A~,l~)with eigenvalues A(A + 1) of A2. These states are called rotator states and the eigenvalue A is related to the label k by A = n — 1 — k. The energy shifts of the rotator states are given approximately in the perturbative regime by [48, 51]
= ~y2n2(~A(A + 1) + ~n2 — 3m2 + ~).
(17)
The eigenfunctions of 1~with negative eigenvalues s have approximately 0(2) ® 0(2) symmetry and can be described by a twodimensional harmonic oscillator with anharmonic corrections. The approximate energy shifts of these vibrational states are, again to first order in y2:
= ~gy2n2[(2o+ Im~+ 1)2V’~n— 3(2o + m~+ 1)2 — m2 + 1],
(18)
where o = 0, 1,2,... is given by 2o = n — rn~— 1 — k or 2o = n — m~—2— k, depending on whether n — mi — k is odd or even [48, 51]. The accuracy of the formulae (17), (18) has been tested by Wintgen and by Wunner [31, 52].
46
H. Friedrich and D. Wintgen, The hydrogen atom in a uniform magnetic field — an example of chaos
2.3. Regime of approximate separability
The classification of states by the principal quantum number n and the “intrashell label” k is definitely meaningful in the 1mixing regime where the diamagnetic interaction is strong enough to break rotational symmetry but still so weak that nmanifolds of states originating from different principal quantum numbers n are separated in energy. It is not a priori clear whether such a classification remains meaningful in the nmixing regime where neighbouring nmanifolds overlap. The onset of the nmixing regime occurs, when the lowest (k = n — mi — 1) vibrator state of the (n + 1)manifold meets the highest (k = 0 or k = 1) rotator state of the nmanifold. From the leading terms in (17) and (18) the onset of the nmixing regime is given by
y2n716/5.
(19)
It was observed in numerical diagonalizations of the Hamiltonian (1) that near degeneracies occurred at the onset of nmixing, the actual magnitudes of the avoided crossings between the lowest state of the n + 1 manifold and the highest state of the nmanifold decreasing exponentially with n [53, 54]. This was interpreted as evidence for an additional hidden symmetry related to a further constant of motion and even as evidence for the existence of an approximately separable representation of the Hamiltonian. An additional (approximate) constant of motion valid for low fields was recognized by Solov’ev [48]and Herrick [51]and is given by the combination (16) of components of the Runge—Lenz vector. Solov’ev explained the small anticrossings of states as being a consequence of the fact that the states involved in the crossing are not only approximately eigenstates of the adiabatic invariant (16) with very different eigenvalues, but also correspond to different classes (rotator and vibrator) of eigenstates.
The above results stimulated an intensive search to find an approximately separable representation of the Hamiltonian. Such a representation was explicitly constructed by Wintgen and Friedrich [33]via a
I
E(crrc’)
I
m~
112 113 
n=31 K=28
n = 30 K= 10
nK53301
I
I
—

45
46
4.7
4.8
B IT)
Fig. 6. Energy eigenvalues obtained by sequential diagonalization. The dashed lines show the results of diagonalizing the Hamiltonian within subspaces of states characterized by a given value of the intrashell label k. The solid lines show the exact results obtained by allowing kmixing.
(From ref. [331.)