Interacting particles in two dimensions: numerical solution of the four-dimensional Schrödinger equation in a hypercube

Abstract

We study numerically the Coulomb interacting two-particle stationary states of the Schrödinger 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 Schrödinger equation, with a local truncation error of 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. 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. Based on this difference scheme we have obtained a generalized matrix Schrödinger equation by vectorization. In the course of its numerical solution group theoretical methods were applied extensively to classify energy eigenvalues and associated two-particle wave functions. This classification scheme inherently accounts for the symmetry property of the two-particle state under permutation, thereby making it very easy to fully explore the completely symmetric and antisymmetric subspaces of the full Hilbert space. We have obtained the invariance group of the interacting Hamiltonian and determined its irreducible representations. With the help of these and Wigner-Eckart theorem, we derived an equivalent block diagonal form of the eigenvalue equation. In the low-energy subspace of the full Hilbert space we numerically computed the ground state and many (≈ 200) excited states for the noninteracting as well as the interacting cases. Comparison with the noninteracting exact results, which we have found analytically too, reveals indeed that already at a modest resolution of h ≈ 1/15 our numerical data for the eigenvalues are accurate globally up to three or more digits of precision. Having obtained the energy eigenvalues and eigenstates we calculated some relevant physical quantities with and without interaction. These are the static two-particle densities labeled by irreducible representations, oneand two-particle density of states and some different measures of entanglement, including the reduced density matrix and the von Neumann entropy. All these quantities signal concordantly that the symmetric states (with respect to permutation of particles) as well as their energies are affected by the interaction more dramatically then their antisymmetric counterparts.