Hydrogen Atom: Its Spectrum and Degeneracy Importance of the Laplace-Runge-Lenz Vector

Consider the problem: why does the bound state spectrum $E(n)<0$, of hydrogen atom Hamiltonian $H$ have more degenerate eigenstates than those required by rotational symmetry? The answer is well known and was demonstrated by Pauli. It is due to an additional conserved vector, $\vec{A}$, of $H$, called the Laplace-Runge-Lenz vector, that was first discovered for planetary orbits. However, surprisingly, a direct link between degenerate eigenstates of $H$ and the physical labels that describe them is missing. To provide such a link requires, as we show, solving a subtle problem of self adjoint operators. In our discussions we address a number of conceptual historical aspects regarding hydrogen atom that also include a careful discussion of both the classical as well as the quantum vector $\vec{A}$.