Non-dispersive wave packets in periodically driven quantum systems

450 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
which, combined with Eqs. (121)–(123), allows for a complete expansion of the classical trajectories
in terms of action-angle coordinates.
The situation is somewhat simpler for the 2D model of the hydrogen atom. There, the angular
momentum ˜L is aligned along the z-axis, which means that L=M and =0. Also, the rotation around
˜L by an angle can be absorbed in a rotation by an angle around the z-axis. Therefore, one is
left with two pairs (I; ) and (M; ) of action-angle variables. The relation between the laboratory
and the local coordinates reads
x = x ′ cos − y ′ sin
y = x ′ sin + y ′ cos ; (127)
which is nothing but a rotation of angle in the plane of the trajectory. Formally, the 2D result,
Eq. (127), can be obtained from the 3D one, Eq. (126), by specializing to
Eq. (124), of the trajectory now reads
= =0. The eccentricity,
e = 1 −
M 2
I 2 : (128)
3.2.5. Scaling laws
It is well known that the Coulomb interaction exhibits particular scaling properties: for example,
all bounded trajectories are similar (ellipses), whatever the (negative) energy. Also, the classical
period scales in a well-de ned way with the size of the orbit (third Kepler law). This originates
from the fact that the Coulomb potential is a homogeneous function—of degree −1—of the radial
distance r. Similarly, the dipole operator responsible for the coupling between the Kepler electron
and the external driving eld is a homogenous function—of degree 1—of r. It follows that the
classical equations of motion of the hydrogen atom exposed to an electromagnetic eld are invariant
under the following scaling transformation:
˜r → −1 ˜r;
˜p → 1=2 ˜p;
H0 → H0 ;
t → −3=2 t;
F → 2 F;
! → 3=2 !;
V → V ; (129)
where is an arbitrary, positive real number. Accordingly, the action-angle variables transform as
I → −1=2 I;
L → −1=2 L;
M → −1=2 M;