Hydrogen Atom: Its Spectrum and Degeneracy Importance of the Laplace-Runge-Lenz Vector
Abstract
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, 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 A.