In Equation 9.31 I assumed that the atom is so small (in comparison to the wavelength of...

In Equation 9.31 I assumed that the atom is so small (in comparison to the wavelength of light) that spatial variations in the field...can be ignored. The true electric field would be

If the atom is centered at the origin, then k • r << 1 over the relevant volume , and that's why we could afford to drop this term. Suppose we keep the first-order correction:

The first term gives rise to the allowed (electric dipole) transitions we considered in the text; the second leads to so-called forbidden (magnetic dipole and electric quadrupole) transitions (higher powers of k • r lead to even more "forbidden" transitions, associated with higher multipole moments).21

(a) Obtain the spontaneous emission rate for forbidden transitions (don't bother to average over polarization and propagation directions, though this should really be done to complete the calculation). Answer: