Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 419
directions of phase space. This was Schrodinger’s original concern, without reference to the actual
value of z p [2].
Finally, for future applications, let us de ne the so-called Husimi representation of the quantum
wave function [21]. It is the squared projection of a given quantum state over a set of coherent states.
Let us denote the gaussian wavefunction of Eq. (24) (with given by Eq. (29)) as |Coh(z0;p0)〉.
Then the Husimi representation of (z) is de ned as
Hus(z; p)= 1 |〈Coh(z; p)| 〉| 2 ; (30)
where the factor 1= is due to the resolution of unity in the coherent states basis [19] and is often
omitted (to confuse the reader). Alternatively, the Husimi function may be looked upon as a Wigner
function convoluted with a Gaussian [12].
1.3. A simple example: the one-dimensional hydrogen atom
We now illustrate the ideas discussed in the preceding sections, using the speci c example of a
one-dimensional hydrogen atom. This object is both, representative of generic systems, and useful
for atomic systems to be discussed later in this paper. We choose the simplest hydrogen atom: we
neglect all relativistic, spin and QED e ects, and assume that the nucleus is in nitely massive. The
Hamiltonian reads:
H = p2 e2
−
2m z
; (31)
where m is the mass of the electron, e 2 =q 2 =4 0, with q the elementary charge, and z is restricted to
the positive real axis. The validity of this model as compared to the real 3D atom will be discussed
in Section 3.3.
Here and in the rest of this paper, we will use atomic units, de ned by m, e 2 and ˝. The unit of
length is the Bohr radius a0=˝ 2 =me 2 =5:2917×10 −11 m, the unit of time is ˝ 3 =me 4 =2:4189×10 −17 s,
the unit of energy is the Hartree me 4 =˝ 2 =27:2eV,twice the ionization energy of the hydrogen atom,
and the unit of frequency is me 4 =2 ˝ 3 =6:5796 × 10 16 Hz [22].
With these premises, the energy levels are 4
En = − 1
2n 2 for n ¿ 1 : (32)
Clearly, the levels are not equally spaced, and therefore (see Eq. (15)) any wave packet will spread.
Fig. 3 shows the evolution of a wave packet built from a linear combination of eigenstates of
H, using a Gaussian distribution of the coe cients cn in Eq. (13). The distribution is centered at
n0 = 60, with a width n =1:8 for the |cn| 2 . The calculation is done numerically, but is simple
in the hydrogen atom since all ingredients—energy levels and eigenstates—are known analytically.
At time t = 0, the wave packet is localized at the outer turning point (roughly at a distance 2n 2 0
from the origin), and has zero initial momentum; its shape is roughly Gaussian. After a quarter of
a classical Kepler period Trecurrence, it is signi cantly closer to the nucleus, with negative velocity,
4 The present analysis is restricted to bound states of the atom. Continuum (i.e., scattering) states also exist but usually
do not signi cantly contribute to the wave-packet dynamics. If needed, they can be incorporated without any fundamental
di culty [23,24].