Quantum Mechanics of the Kicked Harmonic Oscillator

*Züricher Lokalaberglauben.*

WOLFGANG PAULI commenting on

ERWIN SCHRÖDINGER's
interpretation

of quantum mechanics
[HvMW79].

In chapter 1
the kicked harmonic oscillator has been
introduced, and typical properties of this model system have been
discussed within the framework of *classical mechanics*.
On the other hand,
it is the main objective of this study to investigate the
*quantum mechanics*
of the kicked harmonic oscillator and to compare
that system's differing dynamical properties with respect to these two
dynamical theories.
To this end
a quantum description of the model,
consistent with the classical formulation of the previous chapter, is
supplied
in the present chapter.

In section 2.1
the Hamiltonian, scaled in a similar way as its
classical counterpart, is given and a quantum map for the quantum states
that takes the role of the classical POINCARÉ map is derived.
For this derivation,
based on
the periodicity of the Hamiltonian in time,
I employ FLOQUET theory,
some results of which
are also
an important
prerequisite for the discussion of the theory of ANDERSON-like localization
in chapter 5.
The derivation of the quantum map explicitly relies on
the -shaped time-dependence of the kicks,
as chosen in equation (1.12) for the excitation of
the harmonic oscillator. A different approach has been chosen, for example,
in [VT99] by replacing the -functions with
rectangularly peaked functions of finite width. While this simplifies the
resulting
time evolution operator
in the sense that the dynamics becomes piecewise autonomous,
at the same time this approach rules out the
existence of stochastic webs which are in the focus of attention here --
cf. the discussion in subsection 1.1.4,
in particular with respect to [Vec95].
Therefore, in order to study the quantum counterparts of stochastic webs,
keeping the -shaped kicks is essential.
What is more,
although they make the dynamics

nonautonomous in a nontrivial way,
the -kicks have the advantage of allowing to compute the
quantum kick dynamics in a comparatively simple way.

Another way of quantizing the problem is to consider the equation of motion not for the quantum states, but for the relevant operators, i.e. for and , or for the annihilation and creation operators , . This leads, in section 2.2, to two more versions of the quantum map, this time in the HEISENBERG picture rather than the SCHRÖDINGER picture.

In section 2.3 I then argue that it is difficult to discuss the classical limit of the quantum dynamics and its semiclassical approximation based on the analytic formulae for the quantum maps alone. This provides the motivation for the numerical analysis of the system in the following chapters.

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

- The Classical Limit
- Discussion of the Parameter Dependence