DISTRIBUTED APPROXIMATING FUNCTION-THEORY - A GENERAL, FULLY QUANTAL APPROACH TO WAVE-PROPAGATION

A new procedure for developing a coarse-grained representation of the free particle propagator in Cartesian, cylindrical, and spherical polar coordinates is presented. The approach departs from a standard basis representation of the propagator and the state function to which it is applied. Instead, distributed approximating functions (DAFs), developed recently in the context of propagating wave packets in 1-D on an infinite line, are used to create a coarse-grained, highly banded matrix which produces arbitrarily accurate results for the free propagation of wave packets. The new DAF formalism can be used with nonuniform grid spacings. The banded, discretized matrix DAF representation of <x\exp(-iK-tau/h)\x'> can be employed in any wave packet propagation scheme which makes use of the free propagator. A major feature of the DAF expression for the effective free propagator is that the modulus of the x(j),x(j), element is proportional to the Gaussian exp(-sigma(2)(0)(x(j) - x(f)2/2(sigma(4)(0) + h2-tau(2)/m2)). The occurrence of a tau-dependent width is a manifestation of the fundamental spreading of a wave packet as it evolves through time, and it is the minimum possible because the DAF representation of the free propagator is based on evolving the Gaussian generator of the Hermite polynomials. This suggests that the DAFs yield the most highly banded effective free propagator possible. The second major feature of the DAF representation of the free propagator is that it can be used for real time dynamics based on Feynman path integrals. This holds the possibility that the real time dynamics for multidimensional systems could be done by Monte Carlo methods with a Gaussian as the importance sampling function.