(8.183) |

Now,

(8.184) |

However, as explained in the previous section, the fact that the wavelength of the radiation that is emitted during spontaneous transition is generally much larger than the typical size of the atom allows us to truncated the previous expansion. Retaining the first two terms, we obtain

where use has been made of Equation (8.170). Moreover, we have assumed that (i.e., the angular frequency of the electromagnetic radiation matches that associated with the atomic transition.) Suppose, however, that the transition from state to state is forbidden according to the selection rules for electric dipole transitions. This implies that

In this case, Equation (8.188) reduces to

We deduce that a ``forbidden'' transition is not, strictly speaking, forbidden [i.e., Equation (8.189) does not necessarily mean that

According to classical electromagnetic theory, the polarization direction of the magnetic component of an electromagnetic wave propagating in the
direction
is given by
**
**
, where
**
**
specifies the direction of the
wave's electric component [49]. Of course,
represents orbital
angular momentum. However,

(8.188) |

Furthermore, if

(8.189) |

then

(8.190) |

Here, use has been made of Equations (3.32) and (3.33), as well as the fact that

(8.191) |

Hence, Equation (8.190) yields

where

(8.193) |

Here, . Moreover, we have made use of the fact that