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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 \bgroup\color{col1}$ m$\egroup and charge \bgroup\color{col1}$ -e<0$\egroup in an external electrical field \bgroup\color{col1}$ -{\bf\nabla}\Phi({\bf r})$\egroup. Performing an expansion in \bgroup\color{col1}$ v/c$\egroup, where \bgroup\color{col1}$ v$\egroup is the velocity of the electron and \bgroup\color{col1}$ c$\egroup is the speed of light, the result for the Hamiltonian \bgroup\color{col1}$ \hat{H}$\egroup can be written as \bgroup\color{col1}$ \hat{H}=\hat{H}_0+ \hat{H}_1$\egroup, where
$\displaystyle \hat{H}_0=- \frac{\hbar^2}{2m}\Delta - \frac{Ze^2}{4\pi \varepsilon_0 r}$     (3.1)

is the non-relativistic Hydrogen atom, ( \bgroup\color{col1}$ Z=1$\egroup), cf. Eq. (II.1.7), and \bgroup\color{col1}$ \hat{H}_1$\egroup is treated as a perturbation to \bgroup\color{col1}$ \hat{H}_0$\egroup, using perturbation theory. \bgroup\color{col1}$ \hat{H}_1$\egroup consists of three terms: the kinetic energy correction, the Darwin term, and the Spin-Orbit coupling,
$\displaystyle \hat{H}_1 = \hat{H}_{\rm KE} + \hat{H}_{\rm Darwin} + \hat{H}_{\rm SO}.$     (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.



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Tobias Brandes 2005-04-26