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 $\delta$-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 $\delta$-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 $\delta$-shaped kicks is essential. What is more, although they make the dynamics
nonautonomous in a nontrivial way, the $\delta$-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 ${\hat{x}}$ and ${\hat{p}}$, or for the annihilation and creation operators ${\hat{a}}$, ${\hat{a}}^\dagger$. 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.