Hydrogen

The Hydrogen atom consists of an electron bound to a proton by the **Coulomb potential**.

We can generalize the potential to a nucleus of charge without complication of the problem.

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.

We have already studied the spherical harmonics.

The **radial wavefunction** satisfies the differential equation that depends on the angular
momentum quantum number
,

where is the reduced mass of the nucleus and electron.

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 principle quantum number is actually the sum of the radial quantum number plus plus 1.

and therefore, the total angular momentum quantum number must be less than .

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**.