Atomic electron transition: Difference between revisions

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== Theory ==

Consider an atom interactinteracting with electromagnetic radiation which produces an oscillating electric field <ref>{{Citeproduced book|title=Atomicby Physics|author=Foot,electromagnetic CJ|year=2004|radiation:

publisher=Oxford University Press|isbn=978-0-19-850696-6}}</ref>:

{{NumBlk|:|<math> E(t) = |\textbf{E}_0| Re( e^{-i{\omega}t} \hat{\textbf{e}}_{rad} )</math>|{{EquationRef|1}}}}

with amplitude <math>|\textbf{E}_0|</math>, angular frequency <math>\omega</math> and polarization vector <math>\hat{\textbf{e}}_{rad}</math>. Note that the actual phase of wave should be <mathref> (\omega t - \textbf{k} \cdot \textbf{r})Cite </math>.book|title=Atomic HoweverPhysics|author=Foot, 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. This is called dipole approximation, and this approximation allows us to replace <math> E(r, t) </math> with <math> E(0, t) </math> in ({{EquationNoteCJ|year=2004|1}}). Atom can also interact with oscillating magnetic field produced in the radiaiton with the interaction being much weaker.

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 a electric field, is <math> H_I = e \textbf{r} \cdot \textbf{E}(t) </math>,. analogousThe tostimulated thetransition energyrate ofcan abe classicalcalculated dipole in a electric field.using [[Timetime-dependent perturbation theory]] is required for calculating the stimulate transition rate. However, the result can be summarized with Fermi's Golden rule:

<math display="block">

Rate \propto |eE_0|^2 \times | \lang 2 |

\textbf{r} \cdot \hat{\textbf{e}}_{rad} |1 \rang |^2

</math>

The dipole matrix element can be decompose into the product of the radial integral and the angular integral. The angular integral is zero unless certain condition is met, which are the [[selection rulerules]] for allowedthe atomic transitionstransition are satisfied.

== Recent discoveries ==