9.8 Diamagnetic Hydrogen
329
where p and r are the relative momentum and displacement, respectively, of the
electron and proton, μ is the electron-proton reduced mass, E 0 is the total energy,
κ 0 = 1/4ππ 0 ( 0 is the permittivity constant), L z = xp y − yp x is the z-component
of angular momentum, and ω = eB/μ is the cyclotron frequency. If we now
change to atomic units, so L z = m ¯
h, and introduce cylindrical coordinates (ρ, φ, z),
Eq. (9.149) takes the form
H =
1
2
p
2
ρ +
m 2
ρ 2 + p
2
z
+
1
8
γ
2 ρ
2
−
1
2
γ m −
1
ρ 2 + z 2
= E,
(9.150)
where γ = ¯
hω/E B = B/B o and E = E 0 /E B . Here E B is twice the hydrogen
dissociation energy and B o = μE B /e ¯
h = 2.35×10 5 T.
Diamagnetic hydrogen exhibits scaling behavior. To see this, let us scale the
radial coordinates and momenta so that ˜
ρ = γ 2/3 ρ/2 and ˜
p ρ = γ −1/3 p ρ , with
similar scaling for the other coordinates and momenta. (Note that the time must
scale as ˜
t = γ t/2.) The scaled Hamiltonian can be written
1
2
˜
p
2
ρ + ˜
p z + ˜
ρ
2
−
1
˜
ρ 2 + ˜
z 2
+
(mγ 1/3 ) 2
8 ˜
ρ 2
= ,
(9.151)
where = γ −2/3 (E + γ m/2). In the absorption experiments that we discuss below,
the system can be excited into states of fixed m. We will only consider the case
m = 0. Then the dynamics of diamagnetic hydrogen depends only on the single
parameter .
Diamagnetic hydrogen is a nonintegrable system. In the limit → −∞, the
phase space is dominated by KAM surfaces and is quasi-integrable. This limit
actually describes two quasi-integrable regimes of diamagnetic hydrogen. In one
regime, the magnetic field is weak but we are looking at very tightly bound orbits, so
the motion is that of a slightly perturbed hydrogen system. The other regime is one
for which the magnetic field is very strong, so that the system is dominated by the
magnetic field and the Coulomb potential only slightly perturbs it. As we increase
, the phase space becomes increasingly chaotic. This is shown in Fig. 9.4, where
we plot surfaces of section for the classical phase space for = −1.0, = −0.4,
= −0.3, and = −0.1. A fairly abrupt transition to large-scale chaos occurs for
= −0.32. At about = −0.127, all stable elliptic islands have disappeared, and
for > −0.127, the phase space is completely chaotic. It is interesting to view the
orbits in configuration space. In Fig. 9.5 we show a plot of a chaotic orbit near the
ionization threshold.
When quantum effects are taken into account, diamagnetic hydrogen behaves
in an analogous manner. As is increased from large negative values, the energy
spectrum exhibits increasing level repulsion. The transition between the low-field
and high-field quasi-integrable regimes (with passage through a chaotic regime) for
the quantum case has been studied in some detail in Delande and Gay (1986a,b);
Wintgen and Friedrich (1986). (For other reviews of past work, see Wunner and
329
where p and r are the relative momentum and displacement, respectively, of the
electron and proton, μ is the electron-proton reduced mass, E 0 is the total energy,
κ 0 = 1/4ππ 0 ( 0 is the permittivity constant), L z = xp y − yp x is the z-component
of angular momentum, and ω = eB/μ is the cyclotron frequency. If we now
change to atomic units, so L z = m ¯
h, and introduce cylindrical coordinates (ρ, φ, z),
Eq. (9.149) takes the form
H =
1
2
p
2
ρ +
m 2
ρ 2 + p
2
z
+
1
8
γ
2 ρ
2
−
1
2
γ m −
1
ρ 2 + z 2
= E,
(9.150)
where γ = ¯
hω/E B = B/B o and E = E 0 /E B . Here E B is twice the hydrogen
dissociation energy and B o = μE B /e ¯
h = 2.35×10 5 T.
Diamagnetic hydrogen exhibits scaling behavior. To see this, let us scale the
radial coordinates and momenta so that ˜
ρ = γ 2/3 ρ/2 and ˜
p ρ = γ −1/3 p ρ , with
similar scaling for the other coordinates and momenta. (Note that the time must
scale as ˜
t = γ t/2.) The scaled Hamiltonian can be written
1
2
˜
p
2
ρ + ˜
p z + ˜
ρ
2
−
1
˜
ρ 2 + ˜
z 2
+
(mγ 1/3 ) 2
8 ˜
ρ 2
= ,
(9.151)
where = γ −2/3 (E + γ m/2). In the absorption experiments that we discuss below,
the system can be excited into states of fixed m. We will only consider the case
m = 0. Then the dynamics of diamagnetic hydrogen depends only on the single
parameter .
Diamagnetic hydrogen is a nonintegrable system. In the limit → −∞, the
phase space is dominated by KAM surfaces and is quasi-integrable. This limit
actually describes two quasi-integrable regimes of diamagnetic hydrogen. In one
regime, the magnetic field is weak but we are looking at very tightly bound orbits, so
the motion is that of a slightly perturbed hydrogen system. The other regime is one
for which the magnetic field is very strong, so that the system is dominated by the
magnetic field and the Coulomb potential only slightly perturbs it. As we increase
, the phase space becomes increasingly chaotic. This is shown in Fig. 9.4, where
we plot surfaces of section for the classical phase space for = −1.0, = −0.4,
= −0.3, and = −0.1. A fairly abrupt transition to large-scale chaos occurs for
= −0.32. At about = −0.127, all stable elliptic islands have disappeared, and
for > −0.127, the phase space is completely chaotic. It is interesting to view the
orbits in configuration space. In Fig. 9.5 we show a plot of a chaotic orbit near the
ionization threshold.
When quantum effects are taken into account, diamagnetic hydrogen behaves
in an analogous manner. As is increased from large negative values, the energy
spectrum exhibits increasing level repulsion. The transition between the low-field
and high-field quasi-integrable regimes (with passage through a chaotic regime) for
the quantum case has been studied in some detail in Delande and Gay (1986a,b);
Wintgen and Friedrich (1986). (For other reviews of past work, see Wunner and