Numerical Methods

Erst die natürlichen Betrachtungen gemacht,

ehe die subtilen kommen.

$\mathrm{K\acute{\ubepsilon}\ubrho\ubalpha\ubvarsigma}\,$ $\mathrm{'A\ubmu\ubalpha\ublambda\ubvartheta\ubepsilon \acute{\ubiota}\ubalpha\ubvarsigma}$ (276)

GEORG CHRISTOPH LICHTENBERG

In the previous chapter it has been shown that in order to study the quantum dynamics of the kicked harmonic oscillator the time-dependent SCHRÖDINGER equation (2.1) needs to be solved for the Hamiltonian (2.5). Equivalently, the quantum map (2.37) is to be iterated. I now discuss some numerical methods that may be used for this purpose.

In section 3.1 two finite differences methods for solving the partial differential equation (2.1) in the position representation are presented; in section 3.2 I discuss how the the FLOQUET operator can be evaluated numerically in the position representation, and in section 3.3 some technical aspects of iterating the quantum map in the eigenrepresentation of the harmonic oscillator as outlined in subsection 2.1.3 are addressed.

The main result of this chapter is that from a numerical point of view the method best suited for studying the dynamics (including long-time effects) of the quantum kicked harmonic oscillator is the method described in subsection 2.1.3 and section 3.3. This is discussed in section 3.4.