We have solved the problem of a non-relativistic, spinless electron in a coulomb potential exactly. Real Hydrogen atoms have several small corrections to this simple solution. If we say that electron spin is a relativistic effect, they can all be called relativistic corrections which are off order compared to the Hydrogen energies we have calculated.

- The relativistic correction to the electron's kinetic energy.
- The Spin-Orbit correction.
- The ``Darwin Term'' correction to s states from Dirac equation.

Calculating these
fine structure effects
separately
and summing them we find that we get a nice cancellation yielding a simple formula.

The correction depends only on the total angular quantum number and does not depend on so the states of different total angular momentum split in energy but there is still a good deal of degeneracy. It makes sense, for a

We also compute the **Zeeman effect** in which an external magnetic field is applied to Hydrogen.
The external field is very important since it breaks the spherical symmetry
and splits degenerate states allowing us to understand Hydrogen through spectroscopy.

The correction due to a **weak magnetic field** is found to be

The factor is known as the

Thus, in a weak field, the

In the strong field limit we could use states of definite and and calculate the effects of the fine structure, , as a perturbation. In an intermediate strength field, on the order of 500 Gauss, the combination of the Hydrogen fine structure Hamiltonian and the term to the B field must be diagonalized on the set of states with the same principal quantum number .

Jim Branson 2013-04-22