Quantum roll: A study of the long-time behavior of the finite-element method
Fred Cooper, Kimball A. Milton, L. M. Simmons · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · 1985
Using the method of finite elements we investigate the quantum behavior of a particle starting in an unstable equilibrium at the top of a potential hill and rolling down. In order to study the numerical accuracy of the method for large times we consider the exactly solvable model with V(q)=-1/2${q}^{2}$ and an initial Gaussian wave function at t=0, \ensuremath{\psi}(q)\ensuremath{\propto}exp(-1/2${q}^{2}$) so that the initial state \ensuremath{\Vert}0〉 has 〈0\ensuremath{\Vert}${q}^{2}$\ensuremath{\Vert}0〉=〈0\ensuremath{\Vert}${p}^{2}$\ensuremath{\Vert}0〉=1/2. We study the accuracy of the large-time approximations to 〈0\ensuremath{\Vert}${q}^{2}$(t)\ensuremath{\Vert}0〉 based upon single finite elements of degree n. The Taylor series is exact up to ${t}^{2n}$ and even the coefficient of ${t}^{2n+2}$ is a very accurate representation of the exact coefficient. We then consider the convergence properties of the (N,n) approximations consisting of N iterations of the finite element of degree n. For these approximations the corrections to the Taylor-series coefficients in higher orders vanish as ${N}^{\mathrm{\ensuremath{-}}2n}$.