Electric Dipole Approximation

can be approximated by its first term, unity (remember that ). This approximation is known as the

(888) |

It is readily demonstrated that

(889) |

so

(890) |

Thus, making use of the electric dipole approximation, we obtain

for absorption, and

for stimulated emission, where

(893) |

and is the fine structure constant.

Suppose that the radiation is polarized in the
-direction,
so that
**
**
. We have already seen, from Section 7.4, that
unless the initial and final states satisfy

(894) | ||

(895) |

Here, is the quantum number describing the total orbital angular momentum of the electron, and is the quantum number describing the projection of the orbital angular momentum along the -axis. It is easily demonstrated that and are only non-zero if

(896) | ||

(897) |

Thus, for generally directed radiation

(898) | ||

(899) |

These are termed the

Forbidden transitions are not strictly forbidden. Instead, they take
place at a far lower rate than transitions that are allowed
according to the electric
dipole approximation.
After electric dipole transitions, the next most likely type of transition
is a *magnetic dipole transition*, which is due to the interaction between
the electron spin and the oscillating magnetic field of the
incident electromagnetic
radiation. Magnetic dipole transitions are typically about
times
more unlikely than similar electric dipole transitions. The first-order term
in Equation (887) yields so-called *electric quadrupole transitions*.
These are typically about
times more unlikely than electric
dipole transitions. Magnetic dipole and electric quadrupole transitions
satisfy different selection rules than electric dipole transitions. For instance, the selection rules for electric quadrupole transitions
are
. Thus, transitions that are forbidden as
electric dipole transitions may well be allowed as magnetic dipole
or electric quadrupole transitions.