The Gaussian Wave Packet Transform: Efficient Computation of the Semi-Classical Limit of the Schroedinger Equation.

Description
n this talk an efficient and accurate method for simulating the
propagation of a localized solution of the Schroedinger equation near the
semiclassical limit is presented.  We are interested in computing arbitrarily
accurate solutions when the non dimensional Plank's constant, epsilon, is small, but
not negligible. The method is based on a time dependent transformation of the
independent variables, closely related to Gaussian wave packets. A rescaled wave
function, w(x,t), satisfies a new Schroeodinger equation with a time dependent
potential which is a perturbation of the harmonic oscillator, the perturbation being
O(sqrt{epsilon}), so that all stiffness (in space and time) are removed. In fact,
for integration in a fixed time interval, the number of modes required to fully
resolve the problem decreases when epsilon is decreased. The original wave function
may be reconstructed by Fourier interpolation, although expectation values of the
observables can be computed directly from the function w itself. If the initial
condition is a Gaussian wave packet, very few modes are necessary to fully resolve
the w variable, so for short time very accurate solutions can be obtained at low
computational cost. Initial conditions other than Gaussians wave packets can also be
used. In this talk, the Gaussian Wave Packet transform is carefully outlined and
applied to the Schroedinger equation in one and two space dimensions.
Given that very few modes are needed to fully resolve the problem for each space
dimension, the approach should be successfully extended to higher dimensional
problems, in which direct solution of the Schroedinger equation is  impractical.
Organised by Prof. Giacomo Fonte
Support Email: giacomo.fonte@ct.infn.it