The Gaussian wave packet transform: Efficient computation of the semi-classical limit of the Schrödinger equation. Part 1 - Formulation and the one dimensional case

In this paper an efficient and accurate method for simulating the propagation of a localized solution of the Schrödinger equation near the semiclassical limit is presented. We are interested computing arbitrarily accurate solutions when the non dimensional Plank’s constant, ε, 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, satisfies a new Schrödinger equation with a time dependent potential which is a perturbation of the harmonic oscillator, the perturbation being O( √ ε), 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 ε 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 paper, the Gaussian Wave Packet transform is carefully outlined and applied to the Schrödinger equation in one dimension. Detailed numerical tests show the efficiency and accuracy of the approach. In the sequel of this paper, this approach has been extended to the multidimensional case.