Having discussed the properties of the FLOQUET operator from a general point of view, I now proceed to the investigation of the specific defined by equations (2.8) and (2.14) with the Hamiltonian (2.5) of the kicked harmonic oscillator.
As its classical counterpart
in equations
(1.20),
can be decomposed into a contribution
, describing the kick, and the propagator of the free
harmonic
oscillator dynamics for time ,
:
The free propagator -- as many more expressions to follow -- is most
conveniently expressed in terms of the ladder operators
which can be used to write
The computation of the kick propagator requires slightly more effort
because of the explicit time-dependence of the kick part of the
Hamiltonian.
Dividing the SCHRÖDINGER equation in coordinate representation by the
corresponding wave function
, and
integrating over time at the -th kick I
obtain
(2.27) |
Summarizing,
this gives for the full FLOQUET operator
(2.28) |
Note that equation (2.37) cannot be simplified significantly by restricting the discussion to values of that satisfy a resonance condition (1.23) as in the classical case (cf. equations (1.25-1.29)). This statement also holds with respect to the explicit expressions for the matrix elements of that I derive in subsection 2.1.3; there, may take any value as well. For the comparison of classical and quantum results on stochastic webs, one has to concentrate on resonant values of . The way in which the choice of controls the existence of quantum mechanical periodic stochastic webs is discussed in chapter 4; for the discussion of ANDERSON localization in chapter 5, nonresonant values of are in the focus of attention.