  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 $\vec A$.
