# Fine Structure¶

## Fine Structure Constant¶

$$\large \alpha \ = \ \frac{e^2}{4\pi\epsilon\hbar c} \ \approx \ \frac{1}{137}$$

## Hydrogen Fine Structure¶

The small splitting of the spectral line is attributed to an interaction between the electron spin $S$ and the orbital angular momentum $L$.

This split can be handeled using the perturbation theory and gives an $O(\alpha^2)$ correction.

For this correction, we use eigenstates of total angular momentum $\hat{J}^2$ and $\hat{J}_z$.

Taking the fine structure into account always lowers the energy value with lower $j$ corresponding to lower energy.

$$\large E_{nj} \ = \ \frac{E_1}{n^2} \left[ 1 + \frac{\alpha^2}{n^2} \left( \frac{n}{j+1/2} - \frac{3}{4} \right) \right]$$

where

• $j = |l-1/2|, |l+1/2|$

Each energy level for $l > 0$ is split in two by the fine structure.