The hardwall method of solving the radial Schr\"odinger equation and unmasking hidden symmetries

Solving for the bound state eigenvalues of the Schr\"odinger equation is a tedious iterative process when the conventional shooting or matching method is used. In this work, we bypass the eigenvalue's dependence on the eigenfunction by simply trying out all eigenvalues to a desired accuracy. When the eigenvalue is known, the integration for the eigenfunction is then trivial. At a given energy, by outputting the radial distance at which the wave function crosses zero (the hardwall radius), this method automatically determines the entire spectrum of eigenvalues of the radial Schr\"odinger equation without iterative adjustments. Moreover, such a spherically symmetric "hardwall" can unmask "accidental degeneracy" of eigenvalues due to hidden symmetries. We illustrate the method on the Coulomb, harmonic, Coulomb+harmonic, and the Woods-Saxon potentials.