8.9: Absorption and Stimulated Emission of Radiation
- Page ID
- 1234
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 \( q\) in the presence of an electromagnetic field is
where \( {\bf A}(\bf r)\) is the vector potential and \( \phi({\bf r})\) 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 \( {\bf A}\) and \( \phi\) are real functions of the position operators. The above equation can be written
Now,
provided that we adopt the gauge \( \nabla\cdot{\bf A} = 0\) . Hence,
Suppose that the perturbation corresponds to a monochromatic plane-wave, for which
where \( \epsilon\) and \( {\bf n}\) are unit vectors that specify the direction of polarization and the direction of propagation, respectively. Note that \( \epsilon\) \( \cdot{\bf n} = 0\) . The Hamiltonian becomes
with
and
where the \( A^2\) term, which is second order in \( A_0\) , has been neglected.
The perturbing Hamiltonian can be written
This has the same form as Equation \ref{850}, provided that
It is clear, by analogy with the previous analysis, that the first term on the right-hand side of Equation \ref{876} describes the absorption of a photon of energy \( \hbar\,\omega\) , whereas the second term describes the stimulated emission of a photon of energy \( \hbar\,\omega\) . It follows from Equations \ref{859} and \ref{860} that the rates of absorption and stimulated emission are
and
respectively.
Now, the energy density of a radiation field is
where \( E_0\) and \( B_0=E_0/c= 2\,A_0\,\omega/c\) are the peak electric and magnetic field-strengths, respectively. Hence,
and expressions \ref{878} and \ref{879} become
and
respectively. Finally, if we imagine that the incident radiation has a range of different frequencies, so that
where \( d\omega\,u(\omega)\) is the energy density of radiation whose frequency lies in the range \( \omega\) to \( \omega+d\omega\) , then we can integrate our transition rates over \( \omega\) to give
for absorption, and
for stimulated emission. Here, \( \omega_{ni} = (E_n-E_i)/\hbar>0\) and \( \omega_{in} = (E_i-E_n)/\hbar>0\) . Furthermore, we are assuming that the radiation is incoherent, so that intensities can be added.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
\( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)