Bound states of one-dimensional helium on the half-line

We report a non-relativistic solution of the one-dimensional helium atom confined to the half-line r > 0. Using a variational product basis of Laguerre functions with dimension up to N = 45, we obtained convergence of the first four bound-state energies to six significant digits: -2.10839, -2.05037, -2.02901, and -2.01883 atomic units. Despite the theoretical spin and exchange-symmetry degeneracy, we observed a numerical anomaly in which the symmetric (singlet) states converge drastically more slowly than the antisymmetric (triplet) states, potentially requiring basis sizes exceeding N = 1000 to reach the same triplet energies. With N = 45, the symmetric energies converged only to the first four significant digits. The energy spectrum and wave function plots revealed a clear Rydberg-like ladder of two-electron configurations, characterized by a (1s, ns) nodal structure.