With the inclusion of the

The quantity is typically small for atomic transitions

Note that we have take the full binding energy as the energy difference between states so almost all transitions will have smaller than this estimate. This makes an excellent parameter in which to expand decay rate formulas.

The approximation that
is a very good one and is called the
**electric dipole or E1 approximation**.
We previously
derived the E1 selection rules.

The **general E1 decay result depends on photon direction and polarization**.
If information about angular distributions or polarization is needed, it can be pried out of this formula.

**Summing over polarization and integrating over photon direction**, we get a simpler formula that is quite useful
to compute the decay rate from one initial atomic state to one final atomic state.

Here is the matrix element of the coordinate vector between final and initial states.

For single electron atoms, we can **sum over the final states** with different
and get a formula only requires us to do a radial integral.

The decay rate does not depend on the of the initial state.