Meshless local Petrov‐Galerkin (MLPG) methods in quantum mechanics

Abstract

Purpose

The purpose of this paper is to solve both eigenvalue and boundary value problems coming from the field of quantum mechanics through the application of meshless methods, particularly the one known as meshless local Petrov‐Galerkin (MLPG).

Design/methodology/approach

Regarding eigenvalue problems, the authors show how to apply MLPG to the time‐independent Schrödinger equation stated in three dimensions. Through a special procedure, the numerical integration of weak forms is carried out only for internal nodes. The boundary conditions are enforced through a collocation method. The final result is a generalized eigenvalue problem, which is readily solved for the energy levels. An example of boundary value problem is described by the time‐dependent nonlinear Schrödinger equation. The weak forms are again stated only for internal nodes, whereas the same collocation scheme is employed to enforce the boundary conditions. The nonlinearity is dealt with by a predictor‐corrector scheme.

Findings

Results show that the combination of MLPG and a collocation scheme works very well. The numerical results are compared to those provided by analytical solutions, exhibiting good agreement.

Originality/value

The flexibility of MLPG is made explicit. There are different ways to obtain the weak forms, and the boundary conditions can be enforced through a number of ways, the collocation scheme being just one of them. The shape functions used to approximate the solution can incorporate modifications that reflect some feature of the problem, like periodic boundary conditions. The value of this work resides in the fact that problems from other areas of electromagnetism can be attacked by the very same ideas herein described.

Citation

Nicomedes, W.L., Mesquita, R.C. and Moreira, F.J.S. (2011), "Meshless local Petrov‐Galerkin (MLPG) methods in quantum mechanics", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 30 No. 6, pp. 1763-1776. https://doi.org/10.1108/03321641111168084