arXiv subjects

Ilan Degani

Publications and source records attributed to Ilan Degani.

5 recordsLinked to original sources

arXiv2007

Observations on Gaussian bases for Schrodinger's equation

One of the few methods for generating efficient function spaces for multi-D Schrodinger eigenproblems is given by Garashchuk and Light in J.Chem.Phys. 114 (2001) 3929. Their Gaussian basis functions are wider and sparser in high potential regions, and narrower and denser in low ones. We suggest a modification of their approach based on the following observation: In very steep potential regions, wide, sparse, Gaussians should be avoided even if their centers have high potential values. Our numerical results illustrate that a dramatic improvement in accuracy may be obtained in this way. We also compare the errors of collocation to those of a Galerkin approach, test a criterion for scaling Gaussian widths based on deviation from orthogonality of collocation eigenfunctions, and suggest a criterion for scaling Gaussian widths based on Hamiltonian trace minimization.

physics.comp-ph↗

arXiv2007

Semiclassical approximation with zero velocity trajectories

We present a new semiclassical method that yields an approximation to the quantum mechanical wavefunction at a fixed, predetermined position. In the approach, a hierarchy of ODEs are solved along a trajectory with zero velocity. The new approximation is local, both literally and from a quantum mechanical point of view, in the sense that neighboring trajectories do not communicate with each other. The approach is readily extended to imaginary time propagation and is particularly useful for the calculation of quantities where only local information is required. We present two applications: the calculation of tunneling probabilities and the calculation of low energy eigenvalues. In both applications we obtain excellent agrement with the exact quantum mechanics, with a single trajectory propagation.

quant-ph↗

arXiv2006

Bohmian Mechanics with Complex Action: A New Trajectory-Based Formulation of Quantum Mechanics

In recent years there has been a resurgence of interest in Bohmian mechanics as a numerical tool because of its local dynamics, which suggest the possibility of significant computational advantages for the simulation of large quantum systems. However, closer inspection of the Bohmian formulation reveals that the nonlocality of quantum mechanics has not disappeared -- it has simply been swept under the rug into the quantum force. In this paper we present a new formulation of Bohmian mechanics in which the quantum action, S, is taken to be complex. This leads to a single equation for complex S, and ultimately complex x and p but there is a reward for this complexification -- a significantly higher degree of localization. The quantum force in the new approach vanishes for Gaussian wavepacket dynamics, and its effect on barrier tunneling processes is orders of magnitude lower than that of the classical force. We demonstrate tunneling probabilities that are in virtually perfect agreement with the exact quantum mechanics down to 10^{-7} calculated from strictly localized quantum trajectories that do not communicate with their neighbors. The new formulation may have significant implications for fundamental quantum mechanics, ranging from the interpretation of non-locality to measures of quantum complexity.

quant-ph↗

arXiv2005

Gaussian Quadrature without Orthogonal Polynomials

A novel development is given of the theory of Gaussian quadrature, not relying on the theory of orthogonal polynomials. A method is given for computing the nodes and weights that is manifestly independent of choice of basis in the space of polynomials. This method can be extended to compute nodes and weights for Gaussian quadrature on the unit circle and Gauss type quadrature rules with some fixed nodes.

math.NA↗

arXiv2004

Commuting Extensions and Cubature Formulae

Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to the computation of d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A_1,...,A_d, related to the coordinate operators x_1,...,x_d, in R^d. We prove a correspondence between cubature formulae and "commuting extensions" of A_1,...,A_d, satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and describe our attempts at computing them, as well as examples of cubature formulae obtained using the new approach.

math.NA↗