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# Numerical Evaluation of the Propagator in the Position Representation

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,

 (3.21)

expressing the time evolution over one period using the position representation propagator
 (3.22)

of the unkicked harmonic oscillator (2.29a) [FH65]. In contrast to the methods discussed in section 3.1 these expressions by construction take into account the FLOQUET nature of the system.

Using the -discretization as given by equations (3.10), the integral expression (3.25) can be approximated by

 (3.23)

which is once again denoted more concisely in vectorial form,
 (3.24)

with the vectors
 (3.25)

and the propagator matrix in the position representation, with the matrix elements
 (3.26)

takes the same role with respect to the position representation as the propagator of equation (2.42) does with respect to the eigenrepresentation of the harmonic oscillator. Note that , which describes the propagation over a full period of the excitation, is the same for all , such that it suffices to determine its matrix elements once and store them for repeated use.

In order to obtain a numerically stable algorithm, must be required to be unitary,

 (3.27)

whence with equations (3.26) and (3.30)

 (3.28)

can be concluded; the parameters , , of the numerical algorithm, on the left hand side of equation (3.32), have to be chosen as specified by the physical parameters , on the right hand side. The kick amplitude does not interfere with the unitarity of .

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 .

Next: The Propagator in the Up: Numerical Methods Previous: The GOLDBERG Algorithm   Contents
Martin Engel 2004-01-01