Non-dispersive wave packets in periodically driven quantum systems

412 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
| 〉 keeps track of the classical time evolution of z and p, and z and p remain small as time
proceeds. After all, also classical bodies follow their center of mass trajectory even if they have a
nite volume. Quantum states which exhibit these properties at least on a nite time-scale are called
“wave packets”, simply due to their localization properties in phase space [2].
More formally, a localized solution of a wave equation like the Schrodinger equation can be
conceived as a linear superposition of plane waves (eigenstates of the momentum operator) or of
any other suitable basis states. From a purely technical point of view, such a superposition may
be seen as a packet of waves, hence, a wave packet. Note, however, that any strongly localized
object is a wave packet in this formal sense, though not all superpositions of plane waves qualify
as localized objects. In addition, this formal de nition quite obviously depends on the basis used
for the decomposition. Therefore, the only sensible de nition of a wave packet can be through its
localization properties in phase space, as outlined above.
What can we say about the localization properties of a quantum state | 〉 as time evolves? For
simplicity, let us assume that the Hamiltonian describing the dynamics has the time-independent
form
H = ˜p2
+ V (˜r) (2)
2m
with V (˜r) some potential. The time evolution of | 〉 is then described by the Schrodinger equation
− ˝2
9 (˜r;t)
+ V (˜r) (˜r;t)=i˝ : (3)
2m 9t
The expectation values of position and momentum in this state are given by
〈˜r(t)〉 = 〈 (t)|˜r| (t)〉 ; (4)
〈˜p(t)〉 = 〈 (t)|˜p| (t)〉 (5)
with time evolution
d〈˜r〉
dt
d〈˜p〉
dt
= 1
i˝
= 1
i˝
〈[˜r;H]〉 = 〈˜p〉
m
; (6)
〈[˜p; H]〉 = −〈∇V (˜r)〉 (7)
and [:;:] the commutator. These are almost the classical equations of motion generated by H, apart
from the right-hand side of Eq. (7), and apply for any | 〉, irrespective of its localization properties.
If we additionally assume | 〉 to be localized within a spatial region where ∇V (˜r) is essentially
constant, we have 〈∇V (˜r)〉 ∇V (〈˜r〉), and therefore
d〈˜r〉
dt
= 〈˜p〉
m
; (8)
d〈˜p〉
−∇V (〈˜r〉) ; (9)
dt
precisely identical to the classical equations of motion. This is nothing but Ehrenfest’s theorem and
tells us that the quantum expectation values of ˜r and ˜p of an initially localized wave packet evolve