The semiclassical resonance spectrum of Hydrogen in a constant magnetic field

38 Pages·0.45 MB·English

by  Gregor Tanner

The semiclassical resonance spectrum of
Hydrogen in a constant magnetic (cid:12)eld
(cid:3)
Gregor Tanner
Niels Bohr Institute, Blegdamsvej 17,
DK-2100 Copenhagen (cid:31), Denmark
yz
6 Kai T. Hansen
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Fakult(cid:127)at fu(cid:127)r Physik, Universita(cid:127)t Freiburg
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Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany

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Jo(cid:127)rg Main
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1 Institut fu(cid:127)r Theoretische Physik I, Ruhr-Universita(cid:127)t Bochum,

D-44780 Bochum, Germany
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0 December 12, 1995
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Abstract
y
d
Wepresentthe(cid:12)rstpurelysemiclassicalcalculationof theresonancespec-
-
o trum in the Diamagnetic Kepler problem (DKP), a hydrogen atom in a con-
a
h stant magnetic (cid:12)eld with Lz =0. The classical system is unbound and com-
(cid:0)2=3
c pletely chaotic for a scaled energy (cid:15) (cid:24) EB larger than a critical value
(cid:15)c >0. The quantum mechanical resonances can in semiclassical approxima-
tion be expressed as the zeros of the semiclassical zeta function, a product
over all the periodic orbits of the underlying classical dynamics. Intermit-
tency originating from the asymptotically separable limit of the potential
at large electron{nucleus distance causes divergences in the periodic orbit
formula. Using a regularisation technique introduced in [G. Tanner and D.
Wintgen, Phys. Rev. Lett. 75, 2928 (1995)] together with a modi(cid:12)ed cycle
expansion, we calculate semiclassical resonances, both position and width,
which are in good agreement with quantum mechanical results obtained by
the method of complex rotation. The method also provides good estimates
(cid:3)
email: [email protected]
y
email: [email protected]
z
Present address: NORDITA, Blegdamsvej 17, DK-2100 Copenhagen (cid:31), Denmark. Also at:
Physics Department, University of Oslo, Box 1048, Blindern, N-0316 Oslo, Norway.
x
email: [email protected]
1
for the bound state spectrum obtained here from the classical dynamics of a
scattering system. A quasi Einstein{Brillouin{Keller (QEBK) quantisation
is derived that allows for a description of the spectrum in terms of approxi-
mate quantum numbers and yields the correct asymptotic behaviour of the
Rydberg{like series converging towards the di(cid:11)erent Landau thresholds.
PACS numbers: 0545, 0365.Sq, 31.15.Gy, 32.30
The hydrogen atom in a uniform magnetic (cid:12)eld has become one of the most im-
portant examples for studying the correspondence between quantum mechanics and
classical chaos. The Hamiltonianis known to a high accuracy and furthermore, the
system is experimentally accessible in the laboratory. The angular momentum in
the direction of the magnetic (cid:12)eld is conserved, which reduces the classical system
to a problem with two degrees of freedom. The classical (cid:13)ow in phase space covers
a wide range of Hamiltonian dynamics reaching from bound, nearly integrable be-
haviour to completely chaotic and unbound motion by varying one parameter, the
scaled energy (cid:15).
In the 1980's, the system served as a catalyst for quantum chaos. Modulations
in the absorption spectra of highly excited hydrogen atoms in a magnetic (cid:12)eld, the
so{called quasi Landau levels [1, 2], could be assigned to classical trajectories [3, 4]
and could be understood in the framework of the semiclassicalperiodic orbit theory
[5, 6, 7]. This strikingly simple answer to a long outstanding question led to a
completely new viewpoint in analysing quantum spectra in general. We refer the
reader to the review articles and article collections [8, 9, 10, 11, 12].
So far, the main interest in the literature laid on understanding quantum man-
ifestations of the system such as the quantum spectra or the wave functions, in
terms of the underlying classical mechanics. E(cid:11)ective numerical techniques to
solve Schro(cid:127)dinger's equation directly (neglecting relativistic and (cid:12)nite mass ef-
fects) have been developed in the recent years, both for the bound state spectra
[13, 14, 15, 16, 17] and resonances in the continuum [18, 19]. Periodic or closed
orbits appear in the Fourier transformed energy { or photo ionisation { spectra ob-
tained from experiments or from full quantum calculations. Classical orbits can in
some cases be identi(cid:12)ed as \scars" inquantum eigenfunctions [20] and as prominent
peaks in the corresponding Husimi distributions [21].
Less e(cid:11)ort has been undertaken in the opposite direction, i.e. in quantising hy-
drogen in a magnetic (cid:12)eld directly in terms of classical entities only. The density of
quantum states [22], photo absorption spectra [23, 24] or the evolution of Rydberg
wave packets [25] can formally be written as sum over periodic or closed orbits.
A brute force application of the orbit sums using the shortest classical orbits only
yields the coarse grained structure of the spectrum [26] or the time evolution of
Rydberg wave packets on short time scales [25], respectively. To resolve the (cid:12)ne
structure of the spectrum and individual highly excited quantum states, the infor-
mation of the classical dynamics on long time scales is needed. The classical orbit
sums, however, diverge due to the exponential growth of the number of orbits in
2
chaotic systems and appropriate resummation techniques have to be developed to
overcome this problem.
Q
Starting point is here the spectral determinant D(E) = n(E(cid:0)En), which can
in semiclassical approximation be written as product over all periodic orbits of the
system, the so{called semiclassical zeta function [27]. An analytic continuation of
the product formula may be given by expanding the product and regrouping terms
with the help of a symbolic description of the (cid:13)ow in form of a cycle expansion
[28, 29]. This method has been shown to work for strictly hyperbolic classical
dynamics, where all periodic orbits are unstable and the Lyapunov exponents of
the orbits, i.e. the logarithm of the largest eigenvalue of the Monodromy matrix
divided by the period, are strictly bound away from zero.
The Diamagnetic Kepler problem (DKP) corresponding to conserved angular
momentum Lz = 0, (with the magnetic (cid:12)eld in z direction), is known to be com-
pletely chaotic in the sense that all periodic orbits are unstable for values of the
scaled energy (cid:15) > (cid:15)c = 0:328782::: [30]. The classical motion is unbound in this
regime and the (cid:13)ow in phase space can be described in form of a complete ternary
symbolic dynamics [31, 30]. The DKP is, however, not hyperbolic even for energies
(cid:15) > (cid:15)c. The potential becomes separable in the limit where the classical electron is
far from the nucleus, causing regular, but unstable, dynamics in this phase space
region. As a consequence, the Lyapunov exponents of periodic orbits extending
far into the regular region of phase space tend to zero. This behaviour, known as
intermittency in the chaos-literature, is typically for the neighbourhood of marginal
stable orbits or stable islands and is thus generic in Hamiltonian systems.
The regular limit of the classical motion causes divergences in the semiclassical
periodic orbit expressions. This kind of divergence is di(cid:11)erent from the well known
convergence problem of semiclassical expressions due to the exponential growth of
the number of periodic orbits. Divergences introduced through intermittency can
be regularised by (cid:12)rst summing over periodic orbit contributions in the regular
classical regime alone. An analytic continuation of each of the various sums can be
given explicitly [32]. This procedure is equivalent to a cycle expansion of the zeta
function in terms of an in(cid:12)nite alphabet [29]. We can (cid:12)nally present the (cid:12)rst purely
semiclassical quantisation of the resonance spectrum of DKP at (cid:12)xed (cid:15) > (cid:15)c. The
regular limit of the dynamics can be identify to cause the Rydberg series structure
at the various Landau thresholds. The high excited Rydberg resonances can be well
described by a modi(cid:12)ed Einstein{Brillouin{Keller quantisation.
The article is written self-consistently in such a way that it contains all the
information needed to understand the classical, the quantum and the semiclassical
aspects of the problem. The classical system together with the symbolic dynamics
is introduced in section 1. Asymptotic expressions for the actions and stability
exponents in the separable limit of the potential are derived here. In section 2,
we explain the method of complex rotation, which is an e(cid:14)cient tool to treat the
full quantum problem in cases of resonances in the continuum. In the last section,
we present our new semiclassical quantisation, and results are compared with full
quantum calculations.
3
1 Classical dynamics
The non relativistic classical Hamiltonianfor the hydrogen atom in a uniform mag-
netic (cid:12)eld with (cid:12)eld strength B along the z{axis is given as
2 2
p e 1
2 2 2
H = (cid:0) + me! (x +y )+!Lz: (1)
2me r 2
The z { component of the angular momentum Lz is conserved and we will restrict
ourself to the problem Lz = 0 in the following. We work in the in(cid:12)nite nucleus
mass approximation and me denotes the mass of the electron. The frequency ! =
eB=2mec is half the cyclotron frequency. The Hamiltonian (1) in atomic units and
for Lz = 0 has the form
2 2
pz p(cid:26) 1 1 2 2
H = E = + (cid:0) p + (cid:13) (cid:26) (2)
2 2
2 2 (cid:26) +z 8
2 3
with the magnetic (cid:12)eld strength (cid:13) = B=B0 written in units of B0 = mee c=h(cid:22) =
5
2:35(cid:1)10 T. The radial distance from the z{axis is given by the coordinate (cid:26). The
system is bounded for E < 0, while for E > 0 almost all trajectories escape to
z = (cid:6)1 with nonzero kinetic energy.
Introducing the scaling transformation
(cid:0)2=3 1=3
r = (cid:13) ~r; p = (cid:13) p~; (3)
yields the new Hamiltonian
2 2
H~ = (cid:15) = (cid:13)(cid:0)2=3H = p~(cid:26) + p~z (cid:0) p 1 + 1(cid:26)~2; (4)
2 2
2 2 (cid:26)~ +z~ 8
which is independent of (cid:13). The classical dynamics is now controlled by one param-
(cid:0)2=3
eter only, the scaled energy (cid:15) = E(cid:13) . The classical action along a trajectory
scales with the magnetic (cid:12)eld like
Z
S = pdq = (cid:13)(cid:0)1=3S~ (5)
with (cid:15) (cid:12)xed.
The Hamiltonian (4) is singular at ~r = 0 yielding singular equations of motions
at this point. For solving the equations of motionnumerically, it is more convenient
to switch to semi-parabolic coordinates and momenta [8]
2 2
(cid:23) = r~(cid:0)z~; (cid:22) = r~+z~; p(cid:23) = d(cid:23)=d(cid:28); p(cid:22) = d(cid:22)=d(cid:28); (6)
with a new rescaled time (cid:28)
2 2 (cid:0)1
d(cid:28) = dt=2r~= ((cid:23) +(cid:22) ) dt: (7)
The Hamiltonian (4) in semi-parabolic coordinates is
1 1 1
2 2 2 2 2 2 2 2
h = p(cid:23) + p(cid:22) (cid:0)(cid:15)((cid:23) +(cid:22) )+ (cid:23) (cid:22) ((cid:23) +(cid:22) ) (cid:17) 2; (8)
2 2 8
and the scaled energy (cid:15) enters as a parameter here.
The structure of the dynamics of the Hamiltonian (8) depends on the value of
the scaled energy (cid:15) alone. The E{(cid:13) parameter plane can be partitioned into 5
distinct regions [8, 30], see also Fig. 1:
4
0.02
0.015
(V)
0.01
0.005 (IV)
E 0
(III)
-0.005
(II)
-0.01
(I)
-0.015
-0.02
0 0.002 0.004 0.006 0.008 0.01
γ
Figure 1: The 5 main dynamical regions in the E{(cid:13) parameter space reaching from
bound and almostintegrable dynamics (I) to unbound completely chaotic dynamics
(V). See also the explanation in the text.
(I) (cid:15) < (cid:0)0:5: bounded almost integrable motion;
(II) (cid:0)0:5 < (cid:15) < (cid:0)0:13: bounded motion with mixed chaotic and regular motion;
(III) (cid:0)0:13 < (cid:15) < 0:0: thelastlargestableislanddisappear; thedynamicsismostly
chaotic;
(IV) 0:0 < (cid:15) < 0:328782:::: unbounded mostly chaotic motion; the symbolic
dynamic is not complete;
(V) 0:328782::: = (cid:15)c < (cid:15): unbounded chaotic motion with a complete symbolic
description.
In the following, we will always work in the region (cid:15) > (cid:15)c = 0:32878:::, where
a simple and complete symbolic description can be assigned to (cid:13)ow in phase space
[30]. Furthermore, all periodic orbits are unstable in this parameter regime. For (cid:15)
values below the critical energy (cid:15)c, the symbolic dynamics become incomplete and
stable islands appear. Stable classical motion implies additional complications in
the semiclassical description in section 3, which will not be included in the work
presented here.
1.1 Symbolic dynamics and periodic orbits
We can motivate the structures of the classical motion and the underlying sym-
bolic description by (cid:12)rst studying the dynamics of a symmetric four disk scattering
5
billiard [33]. The similarity between the DKP and the four disk system becomes
obvious, when looking at the shape of the potential in the regularized (cid:22);(cid:23) { coor-
dinates, see Fig. 2 (b). In the four disk billiard, a trajectory may after a bounce
with one disk either hit one of the other three disk or escape from the system. A
trajectory bouncing two times after leaving the (cid:12)rst disk may bounce in 9 di(cid:11)erent
n
ways. We (cid:12)nd that there are 3 possible sequences of disks for n bounces. This set
of non-escaping starting points for n ! 1 is called a Cantor set. We label each
disk with a number st 2 f1;2;3;4g and a string s1(cid:1)(cid:1)(cid:1)sn assigned to a trajectory
n
represents one of the 3 possible sequences. A particle cannot bounce twice o(cid:11) the
same disk, which is the only restriction in the 4{letter symbolic dynamics as long
as the distance between the disks is su(cid:14)ciently large compared to the disk radius.
The orbits remaining in the system both forward and backward in(cid:12)nitely long in
time form a two dimensional Cantor set and are uniquely described by bi{in(cid:12)nite
symbol strings (cid:1)(cid:1)(cid:1)s(cid:0)2s(cid:0)1s0s1s2(cid:1)(cid:1)(cid:1).
Since the Cantor set is ternary we can reduce the number of symbols to three
with no restrictions on the sequence of symbols. The reduction can be done in
several possible ways [31, 33]. In Appendix A, we construct a ternary alphabet
closely related to the C4v { symmetry of our system, which will be used throughout
the paper. A trajectory is now characterized by a symbol string (cid:1)(cid:1)(cid:1)g(cid:0)1g0g1(cid:1)(cid:1)(cid:1) with
gi 2 f0;1;2g de(cid:12)ned in Appendix A.
InAppendixB,weintroduceasocalledwellorderedsymbolicdynamics(cid:1)(cid:1)(cid:1)w(cid:0)1w0w1(cid:1)(cid:1)(cid:1)
with wi 2 f0;1;2g, which is in particular useful for (cid:12)nding periodic orbits numeri-
cally [34].
The structure of the Cantor set is similar for the DKP and for the four-disk
scattering system. We can associate a 4{letter symbol string to every trajectory
alsointheDKP,andthesymbolscanbeunderstood asasmooth\bounce"withone
of the steep hills in the potential (8). At this bounces, a trajectory has a caustic,
and we can determine the symbolic dynamics either by determining the number of
caustics [31] or by making a suitable partition curve in a Poincar(cid:19)e plane [35]. A
detailed discussion of this method will be given elsewhere [36].
Thesymbolicdynamicsiscomplete,i.e.eachpossiblesymbolstring(cid:1)(cid:1)(cid:1)g(cid:0)1g0g1(cid:1)(cid:1)(cid:1)
corresponds to a non-escaping orbit in the symmetric four disk system, if and only
if the gap between two disks are larger than 0.205 times the disk radius[37]. If
the disks come closer to each other certain paths between the disks are forbidden.
These paths lie now in the \shadow" of a disk in between. The corresponding sym-
bol strings cannot be related to a physical orbit and the symbolic dynamics is said
to be pruned [38, 39, 37].
We (cid:12)nd an analogous situation in the DKP. Here, each possible symbol string
corresponds to an orbit if the scaled energy is larger than (cid:15)c = 0:32878::: [30]. For
energies below the critical value, the hills of the potential will shadow some of the
trajectories which exist for larger (cid:15) and the symbolic dynamics becomes pruned.
The value of (cid:15) thus plays in the DKP the same role as the distance between the
disks in the 4-disk system. Note that pruning in the DKP is introduced through
bifurcationsandstableislandsappearanddisappearwithvarying(cid:15) < (cid:15)c. Inthedisk
system, however, periodic orbits are always unstable and disappear immediately at
6
3 (a) 4 (b)
2 4 3
1
2 2
1
ρ 0 ν 0
10
-1
-2
-2 20 1 2
-4
-3
-3 -2 -1 0 1 2 3 -4 -2 0 2 4
z µ
Figure 2: Periodic orbits in the (a) (z;(cid:26)) plane and (b) ((cid:23);(cid:22)) plane. The full line
denotes the boundary of the potential for (cid:15) = 0:5.
a bifurcation point.
The periodicorbitsare a subset of allnon-escaping orbitsexisting inthe system,
(i.e. of therepellor). A periodicorbitisdescribed by an in(cid:12)niterepetitionof a (cid:12)nite
symbol string G = g1g2(cid:1)(cid:1)(cid:1)gn. In the following, we focus mainly on periodic orbits,
which we characterize by the shortest non repeating symbol string directly.
FourshortperiodicorbitsoftheDKParedrawn inFig.2, bothinthecoordinate
space (z;(cid:26)), and in the semi parabolic coordinates ((cid:23);(cid:22)). Note that the two orbits
labelled G=2 and G=20 in Fig. 2 (a) collide with the nucleus. In Fig. 2 (b) we
have indicated the labelling of the four hills used to construct the 4{letter symbolic
dynamics. A translation from the 4 { letter to the 3 { letter alphabet can now be
read o(cid:11) directly with the help of Appendix A. The orbits labelled 1, 2, 10, and
20 in Fig. 2 (a) are given the symbolic descriptions S = 1234, 13, 1232, and 1324,
respectively, in the 4 { letter code, Fig. 2 (b).
1.2 Asymptotic in the classical Landau channel
Inthissection, wewillstudythedynamicsoftheHamiltonian(2)forpositiveenergy
E in the limit jzj ! 1. The coupling between the z and (cid:26) coordinates vanishes in
that limit and the Hamiltonian (2) has the asymptotic form
2
1 1 ! 1
2 2 2 2 3
H = p(cid:26) + pz + (cid:26) (cid:0) +O((cid:26) =z ) = E > 0; (9)
2 2 2 jzj
with ! = (cid:13)=2. The motion separates for large jzj values in a pure Coulomb part
along the z { axis and an harmonic oscillator perpendicular to the magnetic (cid:12)eld.
The energies
2
1 ! 1 1
2 2 2
Eh = p(cid:26) + (cid:26) and Ec = pz + (10)
2 2 2 jzj
7
become adiabatic constants of motion for jzj=(cid:26) ! 1. The regular region far from
the nucleus is always coupled to the chaotic motion near the core due to the at-
tractive Coulomb force in z { direction. The classical motion of the electron in
the DKP thus alternates between strong chaotic motion for (cid:26) (cid:25) jzj and regular
time intervals out in the channel jzj (cid:29) (cid:26). Almost all trajectories escape (cid:12)nally
to in(cid:12)nity with nonzero momentum in z { direction. Orbits which go further and
further out in the regular channel but return to the origin, approach a marginally
stable periodic orbit jzj (cid:17) 1, pz (cid:17) 0. In the symbolic description explained in the
appendix A, this marginally stable orbit is labelled 0. Escaping trajectories which
continue travelling along the z { axis can be separated from returning once by a
plane, that is part of the stable manifold of the marginal stable orbit. This plane
2
is in the asymptotic limit given by the condition Ec = pz=2(cid:0)1=jzj = 0. Returning
trajectories can be associated with bound motion in the z { direction and Ec < 0.
Periodic orbits that go far out in the regular region jzj (cid:29) (cid:26) pick up regular
contributions to the actions and stability exponents, which can be given in analytic
form to leading order. In the ternary symbolic description the channel orbits are
characterized by a long string of consecutive symbols '0' in the symbol code. Of
n
particular interest are periodic orbits with a symbol code of the form G0 , i.e.
periodic orbits with a common head string G and a tail of n symbols '0'. We will
call these orbits a periodic orbit family. The orbits in a family have approximately
the same behavior before entering and after leaving the regular region. They di(cid:11)er
only in the '0' { tail string, i.e. in the number n of half { oscillations perpendicular
to the z { axis in the regular region. In Fig. 3, some members of the periodic orbit
families with head string G = 1 are shown explicitly.
The leading n { dependent behavior in the actions and stability exponents is
universal, i.e. it does not depend on the past or future of the periodic orbits before
entering and after leavingthe regular phase. Non universal contributionapproaches
a constant for orbits with the same head string G, but increasing number of oscil-
lations in the regular regime.
Toobtaintheleadingcontributionstotheactions,westartwiththeHamiltonian
(9) written in action{angle variables (Jh;'h;Jc;'c),
1
Hsep = 2!Jh (cid:0) (11)
2
2Jc
and omitting the coupling terms. The action variables are given as
I
1 1
Jh = p(cid:26)d(cid:26) = Eh (12)
2(cid:25) 2!
I
1 1
Jc = pzdz = p : (13)
2(cid:25) (cid:0)2Ec
The integration is taken over a single revolution of a trajectory in the potential of
the one dimensional Hamiltonians in (10). (Note, that (cid:26) (cid:21) 0 in (12)). The angle
variables propagate linearly in time with frequencies
@Hsep @Hsep (cid:0)3
!h = = 2!; !c = = Jc : (14)
@Jh @Jc
8
1.5
1
ν
0.5
0
0 2 4 6 8 10
µ
1.5
1
ν
0.5
0
0 2 4 6 8 10
µ
1.5
1
ν
0.5
0
0 2 4 6 8 10
µ
n
Figure 3: Three members of the periodic orbit family 10 with (a) n = 25, (b)
n = 50, and (c) n = 100 in the fundamental domain (cid:22) (cid:21) (cid:23) (cid:21) 0 of the regularized
(cid:22), (cid:23) coordinates.
9
The surface of constant Jh, Jc form a two dimensional torus in the 4{dimensional
phase space. The total action of a trajectory on a torus after the time Tc = 2(cid:25)=!c,
i.e. after one oscillation in z { direction, is
J((cid:11);E) = Jc((cid:11);E)+(cid:11)Jh((cid:11);E); (15)
where the winding number (cid:11) = !h=!c is given as
3
(cid:11) = 2!Jc: (16)
Inserting (12), (13) and (16) in (15), we can write the total action as function of
the energy E and (cid:11),
(cid:18) (cid:19)2=3!
(cid:11) 3 2!
J((cid:11);E) = E + : (17)
2! 2 (cid:11)
(cid:0)2=3
In terms of the scaled energy (cid:15) = E(cid:13) and (cid:13) = 2!, we obtain
(cid:18) (cid:19)
3
(cid:0)1=3 (cid:0)2=3
J((cid:11);(cid:15)) = (cid:11)(cid:13) (cid:15)+ (cid:11) : (18)
2
The symbol '0' corresponds to one oscillation perpendicular to the (cid:12)eld axis in the
(cid:26) coordinates or half an oscillation in the original (x;y) coordinates. A trajectory
with n consecutive zeros in the symbol code has an approximate winding number
(cid:11)(n) (cid:25) n+(cid:11)0 for n (cid:29) 1: (19)
Contributions due to the non { integrable part of the dynamics are collected here
in the shift term (cid:11)0, which approaches a constant in the limit n ! 1 for periodic
orbits in the same family. Inserting (19) in (18), the asymptotic n dependent
behavior of the (un{scaled) action of a channel trajectory can be written as
(cid:20) (cid:18) (cid:19) (cid:21)
3
(cid:0)1=3 (cid:0)2=3 (cid:0)2=3 (cid:0)1
S(n) = 2(cid:25)(cid:13) n (cid:15)+ n +S0 +S2=3n +O(n ) : (20)
2
The coe(cid:14)cients S0;S2=3;::: depend on (cid:15) and explicitly on the head string G or on
the past and future of the orbit before entering and after leaving the regular phase.
n n
Fig. 4 (a) shows results for the periodic orbit families 10 and 20 up to n = 100.
n
The constant family dependent term in (20) is S0 = (cid:0)0:6483 for the 10 family and
n
S0 = (cid:0)0:1440 for the 20 family, (see also Table 1).
The asymptotic behavior of the stability exponents for periodic orbits with long
sequences of symbols '0' can be extracted from the trace of the stability{or Jacobi{
matrix of the classical (cid:13)ow,
@(q(t);p(t))
M(t) = : (21)
@(q0;p0)
This matrix describes the evolution of the classical dynamics in the neighborhood
of a given trajectory (q(t),p(t)) in linear approximation.
10