On the application of the Heisenberg uncertainty relations to the hydrogen atom

Abstract The Heisenberg uncertainty relations are cornerstones of quantum theory. Yet care is required in applying them. While they determine the lowest energy level in some systems, such as the isotropic three-dimensional harmonic oscillator, they fail to do so in other systems, notably, the hydrogen atom. This failure has been overlooked in many textbooks and pedagogic articles. The failure arises due to the singularity of the Coulomb potential. To elucidate the point, we present four subfamilies of position probability distributions. By delineating the limitations in applying position-momentum uncertainty to the hydrogen atom, we hope to improve the teaching and learning of modern physics.