Absorption and Stimulated Emission of Radiation

(8.122) |

where is the atomic potential energy, and and are the vector and scalar potentials associated with the incident radiation. (See Section 3.6.) The previous equation can be written

(8.123) |

Now,

(8.124) |

provided that we adopt the Coulomb gauge, . (See Exercise 6.) Hence,

Suppose that the perturbation corresponds to a monochromatic plane-wave of angular frequency , for which [49]

(8.126) | ||

(8.127) |

where

(8.128) |

with

and

(8.130) |

where the term involving , which is second order in , has been neglected.

The perturbing Hamiltonian can be written

This has the same form as Equation (8.113), provided that

(8.132) |

It is clear, by analogy with the analysis of Section 8.8, that the first term on the right-hand side of Equation (8.134) describes a process by which the atom absorbs energy from the electromagnetic field. On the other hand, the second term describes a process by which the atom emits energy to the electromagnetic field. We can interpret the former and latter processes as the absorption and stimulated emission, respectively, of a photon of energy by the atom.

It is convenient to define

which has the dimensions of length. Note that

It follows, from Equations (8.122) and (8.123), that the rates of absorption and stimulated emission are

and

respectively. (See Exercise 19.) It can be seen that absorption involves the absorption by the atom of a photon of angular frequency

(8.137) |

and energy , causing a transition from an atomic state with an initial energy to a state with a final energy . On the other hand, stimulated emission involves the emission by the atom of a photon of angular frequency

(8.138) |

and energy , causing a transition from an atomic state with an initial energy to a state with a final energy . In both cases, and

Now, the energy density of an electromagnetic radiation field is [49]

(8.139) |

where and are the peak electric and magnetic field-strengths, respectively. Hence,

(8.140) |

and expressions (8.138) and (8.139) become

and

respectively, where is the fine structure constant.

Let us suppose that the incident radiation has a range of different angular frequencies, so that

(8.143) |

where is the energy density of radiation whose angular frequency lies in the range to . Equations (8.144) and (8.145) imply that

(8.144) |

and

(8.145) |

Here, is the rate of absorption associated with radiation whose angular frequency lies in the range to , et cetera. Incidentally, we are assuming that the radiation is incoherent, so that intensities can be added [64]. Of course,

The rate at which the atom gains energy from the electromagnetic field as a consequence of absorption can be written

(8.148) |

where is the so-called

Similarly, the

(8.150) |

Finally, comparing Equations (8.149) and (8.150) with the previous two equations, we deduce that

(8.151) | ||

(8.152) |

Here, we have written , where is the number density of photons whose angular frequencies lie in the range to .