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Solving for the bound state eigenvalues of the Schr\"odinger equation is atedious iterative process when the conventional shooting or matching method isused. In this work, we bypass the eigenvalue's dependence on the eigenfunctionby simply trying out all eigenvalues to a desired accuracy. When the eigenvalueis known, the integration for the eigenfunction is then trivial. At a givenenergy, by outputting the radial distance at which the wave function crosseszero (the hardwall radius), this method automatically determines the entirespectrum of eigenvalues of the radial Schr\"odinger equation without iterativeadjustments. Moreover, such a spherically symmetric "hardwall" can unmask"accidental degeneracy" of eigenvalues due to hidden symmetries. We illustratethe method on the Coulomb, harmonic, Coulomb+harmonic, and the Woods-Saxonpotentials.

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