  The Pauli method of quantizing the Hydrogen system using the Runge-Lenz
vector is ingenious. It is well known that the energy spectrum is identical
with the one obtained from the Schr\"{o}dinger equation and the consistency
contributed significantly to the development of Quantum Mechanics in the early
days. Since the Runge-Lenz vector is a vector and it commutes with the
Hamiltonian, it is natural to use it to connect energy eigenstate
$|n,l,m\rangle$ with other degenerate states $|n, l\pm 1,m'\rangle$. Recursive
relations can be obtained and the wave functions of the whole spectrum can be
obtained easily. Note that the recursive relations are consistent with those
used in factorizing the Schr\"{o}dinger equation. Nevertheless, the present
analysis provide a better reasoning originated from the conserved vector, the
Runge-Lenz vector. As in the Pauli analysis, group theory or symmetry plays a
prominent role in the present analysis, while the rest of the derivations are
mostly elementary.
