Non-dispersive wave packets in periodically driven quantum systems
Non-dispersive wave packets in periodically driven quantum systems
Non-dispersive wave packets in periodically driven quantum systems
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412 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
| 〉 keeps track of the classical time evolution of z and p, and z and p rema<strong>in</strong> small as time<br />
proceeds. After all, also classical bodies follow their center of mass trajectory even if they have a<br />
nite volume. Quantum states which exhibit these properties at least on a nite time-scale are called<br />
“<strong>wave</strong> <strong>packets</strong>”, simply due to their localization properties <strong>in</strong> phase space [2].<br />
More formally, a localized solution of a <strong>wave</strong> equation like the Schrod<strong>in</strong>ger equation can be<br />
conceived as a l<strong>in</strong>ear superposition of plane <strong>wave</strong>s (eigenstates of the momentum operator) or of<br />
any other suitable basis states. From a purely technical po<strong>in</strong>t of view, such a superposition may<br />
be seen as a packet of <strong>wave</strong>s, hence, a <strong>wave</strong> packet. Note, however, that any strongly localized<br />
object is a <strong>wave</strong> packet <strong>in</strong> this formal sense, though not all superpositions of plane <strong>wave</strong>s qualify<br />
as localized objects. In addition, this formal de nition quite obviously depends on the basis used<br />
for the decomposition. Therefore, the only sensible de nition of a <strong>wave</strong> packet can be through its<br />
localization properties <strong>in</strong> phase space, as outl<strong>in</strong>ed above.<br />
What can we say about the localization properties of a <strong>quantum</strong> state | 〉 as time evolves? For<br />
simplicity, let us assume that the Hamiltonian describ<strong>in</strong>g the dynamics has the time-<strong>in</strong>dependent<br />
form<br />
H = ˜p2<br />
+ V (˜r) (2)<br />
2m<br />
with V (˜r) some potential. The time evolution of | 〉 is then described by the Schrod<strong>in</strong>ger equation<br />
<br />
− ˝2<br />
<br />
9 (˜r;t)<br />
+ V (˜r) (˜r;t)=i˝ : (3)<br />
2m 9t<br />
The expectation values of position and momentum <strong>in</strong> this state are given by<br />
〈˜r(t)〉 = 〈 (t)|˜r| (t)〉 ; (4)<br />
〈˜p(t)〉 = 〈 (t)|˜p| (t)〉 (5)<br />
with time evolution<br />
d〈˜r〉<br />
dt<br />
d〈˜p〉<br />
dt<br />
= 1<br />
i˝<br />
= 1<br />
i˝<br />
〈[˜r;H]〉 = 〈˜p〉<br />
m<br />
; (6)<br />
〈[˜p; H]〉 = −〈∇V (˜r)〉 (7)<br />
and [:;:] the commutator. These are almost the classical equations of motion generated by H, apart<br />
from the right-hand side of Eq. (7), and apply for any | 〉, irrespective of its localization properties.<br />
If we additionally assume | 〉 to be localized with<strong>in</strong> a spatial region where ∇V (˜r) is essentially<br />
constant, we have 〈∇V (˜r)〉 ∇V (〈˜r〉), and therefore<br />
d〈˜r〉<br />
dt<br />
= 〈˜p〉<br />
m<br />
; (8)<br />
d〈˜p〉<br />
−∇V (〈˜r〉) ; (9)<br />
dt<br />
precisely identical to the classical equations of motion. This is noth<strong>in</strong>g but Ehrenfest’s theorem and<br />
tells us that the <strong>quantum</strong> expectation values of ˜r and ˜p of an <strong>in</strong>itially localized <strong>wave</strong> packet evolve