Let us use some of the results of time-dependent perturbation theory to investigate the interaction of an atomic electron with classical (i.e., non-quantized) electromagnetic radiation.
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
where is the vector potential and is the scalar potential. Note that
This prescription also works in quantum mechanics. Thus, the Hamiltonian of an atomic electron placed in an electromagnetic field is
where and are real functions of the position operators. The above equation can be written
provided that we adopt the gauge . Hence,
Suppose that the perturbation corresponds to a monochromatic plane-wave, for which
where and are unit vectors that specify the direction of polarization and the direction of propagation, respectively. Note that . The Hamiltonian becomes
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
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
Now, the energy density of a radiation field is
where and are the peak electric and magnetic field-strengths, respectively. Hence,
and expressions (878) and (879) become
respectively. Finally, if we imagine that the incident radiation has a range of different frequencies, so that
where is the energy density of radiation whose frequency lies in the range to , then we can integrate our transition rates over to give
for absorption, and
for stimulated emission. Here, and . Furthermore, we are assuming that the radiation is incoherent, so that intensities can be added.
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)