Interacting particles in two dimensions: numerical solution of the four-dimensional Schr\"odinger equation in a hypercube

• Main Authors: Vanyolos, Andras, Varga, Gabor
• Format: Journal Article
• Language: English
• Published: 28-08-2008
• Subjects: Physics - Quantum Physics
• DOI: 10.48550/arxiv.0808.3976

We study numerically the Coulomb interacting two-particle stationary states
of the Schr\"odinger equation, where the particles are confined in a
two-dimensional infinite square well. Inside the domain the particles are
subjected to a steeply increasing isotropic harmonic potential, resembling that
in a nucleus. For these circumstances we have developed a fully discretized
finite difference method of the Numerov-type that approximates the
four-dimensional Laplace operator, and thus the whole Schr\"odinger equation,
with a local truncation error of $\mathcal{O}(h^6)$, with $h$ being the uniform
step size. The method is built on a 89-point central difference scheme in the
four-dimensional grid. As expected from the general theorem by Keller [Num.\
Math. \textbf{7}, 412 (1965)], the error of eigenvalues so obtained are found
to be the same order of magnitude which we have proved analytically as well.