The hardwall method of solving the radial Schrodinger equation and unmasking hidden symmetries

Solving for the bound state eigenvalues of the Schrodinger equation is a tedious iterative process using the conventional shooting or matching method. In this paper, we bypass a eigenvalue's dependence on the eigenfunction by trying all eigenvalues to a desired accuracy. When the eigenvalue is known, the integration for the eigenfunction is then trivial. By outputting the radial distance at which the wave function crosses zero (the hardwall radius) for a given energy, the hardwall method automatically determines the entire spectrum of eigenvalues of the radial Schrodinger equation without iterative adjustments. Moreover, such a spherically symmetric hardwall can unmask the accidental degeneracy of eigenvalues due to hidden symmetries. We illustrate the method for the Coulomb, harmonic, Coulomb plus harmonic, and the Woods-Saxon potentials. (C) 2019 American Association of Physics Teachers.