Electric Dipole Transitions

is well approximated by its first term, unity. This approximation is known as the

(8.165) |

It is readily demonstrated from Equations (3.33) and (8.132) that

(8.166) |

so

which implies that

Here, is termed the

for absorption, and

for stimulated emission, and with

for spontaneous emission.

We have already seen, from Section 7.4, that
for a hydrogen-like atom,
unless the initial and final states satisfy
and
.
Here,
is the azimuthal quantum number, and
the magnetic quantum number. Moreover,
is the difference between the azimuthal quantum numbers of the initial and final
states, et cetera.
It is easily demonstrated that
and
are only non-zero if
,
and
.
(See Exercise 10.)
It follows that the *electric dipole matrix elements*,
, which control the rates
of so-called *electric dipole transitions*, via Equations (8.172)-(8.174), are only non-zero if

(8.172) | ||

(8.173) | ||

(8.174) |

Here, is the

(8.175) | ||

(8.176) |

where and are the standard quantum numbers associated with the total angular momentum, and the projection of the total angular momentum along the -axis, respectively [23]. (These new rules are evident from a perusal of the results of Exercise 15.) Note, however, that an electric dipole transition between two states is forbidden [23].

Let us estimate the typical spontaneous emission rate for an electric dipole transition in a hydrogen atom. We expect the matrix element , defined in Equation (8.171), 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 (8.174) that

where is the fine structure constant. This is an important result, because our perturbation expansion is based on the assumption that the transition rate between different energy eigenstates is much less than the frequency of phase oscillation of these states. In other words, . This is indeed the case.

According to Equation (8.152), the absorption cross-section associated with atomic transitions between an initial state of energy and a final state of energy can be written

where . Suppose, for the sake of argument, that the electromagnetic radiation is polarized in the -direction, so that

where the dimensionless parameter

(8.180) |

is termed the

(8.181) |

where the sum is over all atomic states. (See Exercise 11.) Thus, it follows that, provided state is the ground state,

(8.182) |

where is the