The unperturbed Hamiltonian is

The standard classical prescription for obtaining the Hamiltonian of a particle of charge in the presence of an electromagnetic field is

(863) | ||

(864) |

where is the vector potential and is the scalar potential. Note that

(865) | ||

(866) |

This prescription also works in quantum mechanics. Thus, the Hamiltonian of an atomic electron placed in an electromagnetic field is

(867) |

where and are real functions of the position operators. The above equation can be written

(868) |

Now,

(869) |

provided that we adopt the gauge . Hence,

Suppose that the perturbation corresponds to a monochromatic plane-wave, for which

(871) | ||

(872) |

where

(873) |

with

(874) |

and

(875) |

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

The perturbing Hamiltonian can be written

This has the same form as Equation (850), provided that

(877) |

It is clear, by analogy with the previous analysis, that the first term on the right-hand side of Equation (876) describes the absorption of a photon of energy , whereas the second term describes the stimulated emission of a photon of energy . It follows from Equations (859) and (860) that the rates of absorption and stimulated emission are

and

respectively.

Now, the energy density of a radiation field is

(880) |

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

(881) |

and expressions (878) and (879) become

and

respectively. Finally, if we imagine that the incident radiation has a range of different frequencies, so that

(884) |

where is the energy density of radiation whose frequency lies in the range to , then we can integrate our transition rates over to give

(885) |

for absorption, and

(886) |

for stimulated emission. Here, and . Furthermore, we are assuming that the radiation is incoherent, so that intensities can be added.