Hydrogen Atom: Fine Structure

The fine structure is a result of relativistic corrections to the Schrödinger equation, derived from the relativistic Dirac equation for an electron of mass $m$ and charge $-e<0$ in an external electrical field $-{\bf\nabla}\Phi({\bf r})$. Performing an expansion in $v/c$, where $v$ is the velocity of the electron and $c$ is the speed of light, the result for the Hamiltonian $\hat{H}$ can be written as $\hat{H}=\hat{H}_0+ \hat{H}_1$, where

$$\hat{H}_0=- \frac{\hbar^2}{2m}\Delta - \frac{Ze^2}{4\pi \varepsilon_0 r}$$ (3.1)

is the non-relativistic Hydrogen atom, ( $Z=1$), cf. Eq. (II.1.7), and $\hat{H}_1$ is treated as a perturbation to $\hat{H}_0$, using perturbation theory. $\hat{H}_1$ consists of three terms: the kinetic energy correction, the Darwin term, and the Spin-Orbit coupling,

$$\hat{H}_1 = \hat{H}_{\rm KE} + \hat{H}_{\rm Darwin} + \hat{H}_{\rm SO}.$$ (3.2)

Literature: Gasiorowicz [3] cp. 12 (Kinetic Energy Correction, Spin-Orbit coupling); Weissbluth [4] (Dirac equation, Darwin term); Landau Lifshitz Vol IV chapter. 33,34.