The Hydrogen atom consists of an electron bound to a proton by the Coulomb potential.
Since the potential is spherically symmetric, the problem separates and the solutions
will be a product of a radial wavefunction and one of the spherical harmonics.
The radial wavefunction satisfies the differential equation that depends on the angular
momentum quantum number
The differential equation can be solved using techniques similar to those used to solve the 1D harmonic oscillator equation. We find the eigen-energies
The principle quantum number
is an integer from 1 to infinity.
This unusual way of labeling the states comes about because a radial excitation has the same energy as an angular excitation for Hydrogen. This is often referred to as an accidental degeneracy.