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# 8.9: Absorption and Stimulated Emission of Radiation

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

 (862)

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,

 (870)

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

 (871) (872)

where and are unit vectors that specify the direction of polarization and the direction of propagation, respectively. Note that . The Hamiltonian becomes

 (873)

with

 (874)

and

 (875)

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

The perturbing Hamiltonian can be written

 (876)

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

 (878)

and

 (879)

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

 (882)

and

 (883)

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.