In [BR95] a numerical ``brute force approach'' to
determining the kicked harmonic oscillator dynamics is described.
Using the present terminology and scaling, this approach
can be formulated as a
discretization of the
integral form of the quantum map (2.37)
in the position representation,
Using the -discretization
as given by equations
(3.10), the integral expression (3.25)
can be approximated by
In order to obtain a numerically stable algorithm,
must be required to be unitary,
In addition to the condition (3.32), a boundary condition of the type (3.24a) also needs to be satisfied, i.e. the interval must be chosen large enough to allow the discrete mapping (3.27) to approximate the original (3.25) well enough. This is granted if the wave function decays rapidly enough with reaching out to the boundaries of the interval considered.
These two restrictions combined are the reason why the algorithm given by the mapping (3.27) is of limited practical use when the long-time dynamics of spreading states is to be studied: for delocalized states spreading widely along the -axis, very large values of are required by the condition (3.32), in particular when takes on small values. This means that the memory requirements on the computer grow rapidly, and the evaluation of each iteration of (3.27) quickly becomes more time-consuming. For example, the values , and lead to , making it practically impossible to store the matrix elements on a typical workstation and slowing down the speed of the computation considerably. It is important to keep in mind this limitation of the algorithm when working with it in practice.
Conversely, if the dynamics is followed for times not too large, such that the initial wave packets have not spread too much, then equation (3.27) provides a simple and efficient way to evaluate the quantum map (2.37), especially for larger values of .