The quantum dynamics of the unscaled kicked harmonic oscillator
with the Hamiltonian function (1.12)
is
governed
by the SCHRÖDINGER equation

(2.2) |

combined with the position scaling (1.15a) -- which plays the same role classically and quantum mechanically -- yield the scaling

resulting in a dimensionless . The definition (2.3) of the momentum operator holds both before and after scaling; in particular, is retained after scaling, albeit in scaled form -- in contrast to the scaling used elsewhere [BRZ91]. Using the scaled PLANCK constant and the parameters and , scaled according to equations (1.16), I obtain the Hamiltonian operator in its dimensionless form:

This Hamiltonian is to be used in conjunction with the SCHRÖDINGER equation (2.1) where the scaled version of is employed.

By virtue of the scaling one is left with only the three dimensionless parameters , and . The first two of these describe the nature of the kick and have to be considered both in the classical and the quantum realms, whereas the third -- and only the third -- is a genuinely quantum mechanical parameter.

As discussed in the Introduction (pages ff), the main objective of the theory of quantum chaos is the investigation of the way in which the dynamics of the system changes when advancing from the quantum to the classical case, i.e. when passing from via the semiclassical to the limiting case .

For this purpose, comparison of the two dynamical theories of classical and quantum mechanics, the scaling used here is more appropriate than the one used in [BRZ91], for example: there, the oscillator length is used to scale lengths, which is a natural choice in the quantum context, but makes comparison with the classical case more difficult, as using this scale in classical mechanics does not make sense. This problem is avoided here by measuring lengths in units of , given by the kick function, which is present in both dynamical theories in exactly the same way. As a result, in this scaling the only parameter involving quantum effects is , and the other two remaining parameters and both play the same role classically and quantum mechanically.

- FLOQUET Theory
- The FLOQUET Operator of the Kicked Harmonic Oscillator
- Matrix Elements of the FLOQUET operator