Non-dispersive wave packets in periodically driven quantum systems

448 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
by [128,129]
a
I =
2 ;
− sin ;
=
2 − + sin ;
p ¿ 0 ;
p ¡ 0 ;
= 2 sin −1
z
a ; a= −E−1 ; (117)
where a is the maximum distance. In celestial mechanics, and are known as the mean and the
eccentric anomaly, respectively [93]. The Hamilton function depends on the action through:
H0 = − 1
2I 2
and the classical Kepler frequency reads
= dH0
dI
(118)
1
= : (119)
I 3
Due to the Coulomb singularity at z = 0, the Maslov index of this system is = 0 instead of =2,
and the semiclassical energy spectrum, Eq. (39), 14 matches the exact quantum spectrum, Eq. (115).
The classical equations of motion can be solved exactly and it is easy to obtain the Fourier
components of the dipole operator [130]:
z( )=I 2
∞ 3 J
− 2
2 ′
m(m)
cos(m ) ;
m
(120)
m=1
where J ′ n(x) denotes the derivative of the usual Bessel function. The strongly non-linear character of
the Coulomb interaction is responsible for the slow decrease of the Fourier components at high m.
For the 2D and 3D hydrogen atom, the action-angle variables are similar, but more complicated
because of the existence of angular degrees of freedom. The classical trajectories are ellipses with
focus at the nucleus. The fact that all bounded trajectories are periodic manifests the degeneracy
of the classical dynamics. As a consequence, although phase space is six dimensional with three
angle and three action variables in 3D—four dimensional with two angle and two action variables
in 2D—the Hamilton function depends only on the total action I, precisely like the 1D hydrogen
atom, i.e., through Eq. (118). In 3D, the Maslov index is zero, so that the energy spectrum is again
given by Eq. (108), and the semiclassical approximation is exact. However, a di erent result holds
for the 2D hydrogen atom, where the Maslov index is = 2 (still yielding exact agreement between
the semiclassical and the quantum spectrum, compare Eqs. (46) and (116)).
The action-angle variables which parametrize a general Kepler ellipse are well known [128]. In
addition to the action-angle variables (I; ) which determine the total action and the angular position
of the electron along the Kepler ellipse, respectively, the orientation of the ellipse in space is de ned
by two angles: , canonically conjugate to the total angular momentum L, and the polar angle ,
14 As we are using atomic units, ˝ is unity, and the principal quantum number just coincides with the action.