(8.212) |

where

is the

Here, is the solid angle associated with the direction of the emitted photon's normalized wavevector, . Moreover,

The selection rules for electric quadrupole transitions in a hydrogen-like atom are [23]

(8.215) | ||

(8.216) | ||

(8.217) |

where is the azimuthal quantum number, the magnetic quantum number, and the spin quantum number. A more general form of these selection rules is

(8.218) | ||

(8.219) |

where and are the standard quantum numbers associated with the total angular momentum of the system. Note, however, that electric quadrupole transitions between two states, or between two states, or between a state and a state, are forbidden [23].

Let us estimate the typical spontaneous emission rate for an electric quadrupole transition in a hydrogen atom. We expect the matrix element , defined in Equation (8.216), to be of order , where is the Bohr radius. We also expect to be of order , where is the hydrogen ground-state energy. It thus follows from Equation (8.217) that

(8.220) |

which is of order times smaller than a typical electric dipole transition rate. [See Equations (8.180).]