Editing Atomic electron transition (section)

== Theory ==
An atom interacts with the oscillating electric field:
{{NumBlk|:|<math>  E(t) = |\textbf{E}_0| Re( e^{-i{\omega}t} \hat{\textbf{e}}_\mathrm{rad} )</math>|{{EquationRef|1}}}}
with amplitude <math>|\textbf{E}_0|</math>, angular frequency <math>\omega</math>, and polarization vector <math>\hat{\textbf{e}}_\mathrm{rad}</math>.<ref>{{Cite book|title=Atomic Physics|author=Foot, CJ|year=2004|
    publisher=Oxford University Press|isbn=978-0-19-850696-6}}</ref> Note that the actual phase is <math> (\omega t - \textbf{k} \cdot \textbf{r}) </math>. However, in many cases, the variation of <math> \textbf{k}  \cdot \textbf{r} </math> is small over the atom (or equivalently, the radiation wavelength is much greater than the size of an atom) and this term can be ignored. This is called the dipole approximation. The atom can also interact with the oscillating magnetic field produced by the radiation, although much more weakly.

The Hamiltonian for this interaction, analogous to the energy of a classical dipole in an electric field, is <math> H_I = e \textbf{r} \cdot \textbf{E}(t) </math>. The stimulated transition rate can be calculated using [[time-dependent perturbation theory]]; however, the result can be summarized using [[Fermi's golden rule]]:
<math display="block">
 Rate \propto |eE_0|^2 \times | \lang 2 | 
 \textbf{r} \cdot \hat{\textbf{e}}_\mathrm{rad} |1 \rang |^2
</math>
The dipole matrix element can be decomposed into the product of the radial integral and the angular integral. The angular integral is zero unless the [[selection rules]] for the atomic transition are satisfied.