# Electric dipole transition

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 $H_0$, interacting with a plane electromagnetic wave

${\mathbf E}({\mathbf r},t)=E_0 {\hat{\mathbf z}} \cos(ky-\omega t), \ \ \ {\mathbf B}({\mathbf r},t)=B_0{\hat{\mathbf x}} \cos(ky-\omega t).$

Write the Hamiltonian of the electron in this electromagnetic field as

$H(t) \ = \ H_0 + W(t).$

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 $W(t)$ written as

$W_{DE}(t) = \frac{q E_0}{m\omega} p_z \sin \omega t. \,$

Electric dipole transitions are the transitions between energy levels in the system with the Hamiltonian $H_0 + W_{DE}(t)$.

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 $W(t)$ is written as

$W_{DM}(t) = \frac{q}{2m} (L_x + 2S_x) B_0 \cos \omega t \,$

and describes magnetic dipole transitions.

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