A COMPUTATIONAL DEMONSTRATION OF THE DISTRIBUTED APPROXIMATING FUNCTION-APPROACH TO REAL-TIME QUANTUM DYNAMICS

We report computational applications of the newly developed distributed approximating function (DAF) approach to real time quantal wavepacket propagation for several one-dimensional model problems. The DAF is constructed to fit all wavepackets accurately which can be represented, to the same accuracy, by a polynomial of degree M, or less, within the envelope of the DAF . (This defines the "DAF class" of functions.) By expressing the DAF (and thus the wavepacket to be propagated) in terms of Hermite functions (each a product of a Hermite polynomial and its Gaussian generating function), the DAF approximation to the wavepacket is propagated freely and exactly for a short time tau. The Hermite functions are the natural basis states for describing the free evolution of a localized particle and yield a highly banded representation for the free particle propagator. Combining the DAF class free propagation scheme with any of several short time approximations to the full propagator enables one to propagate the wavepacket through a potential. The DAF results for the propagated wavepacket and various scattering amplitudes are shown to be in good agreement with those obtained by more standard methods.