Numerical investigation of entangled two-particle quantum systems

Using a grid-based relaxation algorithm, we find numerical solutions to the time-independent Schrödinger equation for some interacting two-particle systems. The first system is two particles trapped in an infinite square well with a contact interaction that varies in strength. The results show the ‘fermionization’ effect, in which the two particles are in a symmetric state but as they interact more strongly the energies become closer to those of an antisymmetric state. The second system is the helium atom, where we focus on states with spherical symmetry and neglect angular correlations. We obtain the helium ground-state energy to an accuracy of less than one percent, and obtain the 1s2s excited-state energies to even higher accuracy. This numerical method is easy to understand and is accessible to undergraduate physics students. It provides a useful tool for developing intuition in quantum mechanics and for investigating non-separable multidimensional systems.