The Quantum Map

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

$$\H\left\vert \psi(t) \right> \; = \; i\hbar \frac{\partial}{\partial t} \left\vert \psi(t) \right>,$$ (2.1)

where $H$ is the explicitly time-dependent Hamiltonian operator

$$\H \; = \; \H({\hat{x}},{\hat{p}},t) \; = \; \frac{1}{2m_0}... ...}}^2 + V_0T\cos k{\hat{x}}\sum_{n=-\infty}^\infty\delta(t-nT)$$ (2.2)

and $\left\vert \psi(t) \right>$ the quantum state at time $t$. As in the classical case I switch to dimensionless variables by means of the scaling transformation (1.15). In the quantum mechanical context the classical momentum scaling can be interpreted as the implicit definition of a scaled PLANCK constant $\hbar$, since equation (1.15b) and the usual definition of the momentum operator in the position representation,

$${\hat{p}}\; = \; \frac{\hbar}{i}\frac{\partial}{\partial x},$$ (2.3)

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

$$\frac{k^2}{m_0\omega_0}\hbar \; \longmapsto \; \hbar,$$ (2.4)

resulting in a dimensionless $\hbar$. The definition (2.3) of the momentum operator holds both before and after scaling; in particular, $\hbar$ is retained after scaling, albeit in scaled form -- in contrast to the scaling used elsewhere [BRZ91]. Using the scaled PLANCK constant and the parameters $V_0$ and $T$, scaled according to equations (1.16), I obtain the Hamiltonian operator in its dimensionless form:

$$\hspace*{-0.2cm} \fbox{$ \displaystyle \rule{0.0cm}{0.75cm... ...os x \sum_{n=-\infty}^\infty \delta(t-nT). \hspace*{0.1cm} $}$$ (2.5)

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

By virtue of the scaling one is left with only the three dimensionless parameters $V_0$, $T$ and $\hbar$. 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 $\hbar\neq 0$ via the semiclassical $\hbar\approx 0$ to the limiting case $\hbar=0$.

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 $\sqrt{\hbar/m_0\omega_0}$ 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 $k$, 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 $\hbar$, and the other two remaining parameters $V_0$ and $T$ both play the same role classically and quantum mechanically.