# Electric dipole transition

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Electric dipole transition is the dominant effect of an interaction of an electron in an atom with the electromagnetic field.

Following,[1] consider an electron in an atom with quantum Hamiltonian ${\displaystyle H_{0}}$, interacting with a plane electromagnetic wave

${\displaystyle {\mathbf {E} }({\mathbf {r} },t)=E_{0}{\hat {\mathbf {z} }}\cos(ky-\omega t),\ \ \ {\mathbf {B} }({\mathbf {r} },t)=B_{0}{\hat {\mathbf {x} }}\cos(ky-\omega t).}$

Write the Hamiltonian of the electron in this electromagnetic field as

Treating this system by means of time-dependent perturbation theory, one finds that the most likely transitions of the electron from one state to the other occur due to the summand of ${\displaystyle W(t)}$ written as

${\displaystyle W_{DE}(t)={\frac {qE_{0}}{m\omega }}p_{z}\sin \omega t.\,}$

Electric dipole transitions are the transitions between energy levels in the system with the Hamiltonian ${\displaystyle H_{0}+W_{DE}(t)}$.

Between certain electron states the electric dipole transition rate may be zero due to one or more selection rules, particularly the angular momentum selection rule. In such a case, the transition is termed electric dipole forbidden, and the transitions between such levels must be approximated by higher-order transitions.

The next order summand in ${\displaystyle W(t)}$ is written as

${\displaystyle W_{DM}(t)={\frac {q}{2m}}(L_{x}+2S_{x})B_{0}\cos \omega t\,}$

and describes magnetic dipole transitions.

Even smaller contributions to transition rates are given by higher electric and magnetic multipole transitions.