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Physics LibreTexts

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

    \( H_0 = \frac{p^2}{2 \,m_e} + V_0(r).\) \ref{862}

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

    \( {\bf p}\) \( \rightarrow {\bf p} - q\,{ \bf A},\) \ref{863} \( H\) \( \rightarrow H-q\,\phi,\) \ref{864}

    where \( {\bf A}(\bf r)\) is the vector potential and \( \phi({\bf r})\) is the scalar potential. Note that

    \( {\bf E}\) \( = - \nabla\phi - \frac{\partial {\bf A}}{\partial t},\) \ref{865} \( {\bf B}\) \( = \nabla\times {\bf A}.\) \ref{866}

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

    \( H = \frac{\left\vert{\bf p} + e\, {\bf A}\right\vert^{\,2} }{2\,m_e}- e \,\phi + V_0(r),\) \ref{867}

    where \( {\bf A}\) and \( \phi\) are real functions of the position operators. The above equation can be written

    \( H = \frac{ \left(p^2 +e \,{\bf A}\cdot {\bf p} +e \,{\bf p}\cdot{\bf A} + e^2 A^2\right)}{2\,m_e}- e \,\phi + V_0(r).\) \ref{868}

    Now,

    \( {\bf p}\cdot {\bf A} = {\bf A}\cdot {\bf p},\) \ref{869}

    provided that we adopt the gauge \( \nabla\cdot{\bf A} = 0\) . Hence,

    \( H = \frac{p^2}{2\,m_e} -\frac{e\,{\bf A}\cdot{\bf p}}{m_e} +\frac{ e^2 A^2}{2\,m_e}- e\, \phi + V_0(r).\) \ref{870}

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

    \( \phi\) \( = 0,\) \ref{871} \( {\bf A}\) \( = 2\, A_0 \,\)\( \mbox{\boldmath\)\( \,\cos\left (\frac{\omega}{c} \,{\bf n}\cdot{\bf x} - \omega\, t\right),\) \ref{872}

    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

    \( H = H_0 + H_1(t),\) \ref{873}

    with

    \( H_0 = \frac{p^2}{2\,m_e} + V(r),\) \ref{874}

    and

    \( H_1 \simeq \frac{e\,{\bf A}\cdot{\bf p}}{m_e},\) \ref{875}

    where the \( A^2\) term, which is second order in \( A_0\) , has been neglected.

    The perturbing Hamiltonian can be written

    $ H_1 = \frac{e \,A_0\, \mbox{\boldmath$\epsilon$}\cdot{\bf p} }{m_...
...\exp[-{\rm i}\,(\omega/c)\, {\bf n}\cdot{\bf x} + {\rm i}\, \omega\, t]\right).$ \ref{876}

    This has the same form as Equation \ref{850}, provided that

    \( V = - \frac{e \,A_0\, \mbox{\boldmath\) \ref{877}

    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

    $ w_{i\rightarrow n} = \frac{2\pi}{\hbar} \frac{e^2}{m_e^{\,2}}\, \...
...\,2}\, \vert\langle n\vert\, \exp[\,{\rm i}\,(\omega/c)\,{\bf n}\cdot{\bf x}]\,$ \( \mbox{\boldmath\)\( \cdot{\bf p} \,\vert i\rangle\vert^{\,2}\, \delta(E_n-E_i -\hbar\,\omega),\) \ref{878}

    and

    $ w_{i\rightarrow n} = \frac{2\pi}{\hbar} \frac{e^2}{m_e^{\,2}}\, \...
...{\,2}\, \vert\langle n\vert\, \exp[-{\rm i}\,(\omega/c)\,{\bf n}\cdot{\bf x}]\,$ \( \mbox{\boldmath\)\( \cdot{\bf p} \,\vert i\rangle\vert^{\,2}\, \delta(E_n-E_i +\hbar\,\omega),\) \ref{879}

    respectively.

    Now, the energy density of a radiation field is

    \( U = \frac{1}{2}\left(\frac{\epsilon_0\,E_0^{\,2}}{2}+ \frac{B_0^{\,2}}{2\,\mu_0} \right),\) \ref{880}

    where \( E_0\) and \( B_0=E_0/c= 2\,A_0\,\omega/c\) are the peak electric and magnetic field-strengths, respectively. Hence,

    \( U = 2\,\epsilon_0\,c\,\omega^2\,\vert A_0\vert^{\,2},\) \ref{881}

    and expressions \ref{878} and \ref{879} become

    $ w_{i\rightarrow n} = \frac{\pi}{\hbar} \frac{e^2}{\epsilon_0\,m_e...
...\, U\, \vert\langle n\vert\, \exp[\,{\rm i}\,(\omega/c)\,{\bf n}\cdot{\bf x}]\,$ \( \mbox{\boldmath\)\( \cdot{\bf p} \,\vert i\rangle\vert^{\,2}\, \delta(E_n-E_i -\hbar\,\omega),\) \ref{882}

    and

    $ w_{i\rightarrow n} = \frac{\pi}{\hbar} \frac{e^2}{\epsilon_0\,m_e...
...}\, U\, \vert\langle n\vert\, \exp[-{\rm i}\,(\omega/c)\,{\bf n}\cdot{\bf x}]\,$ \( \mbox{\boldmath\)\( \cdot{\bf p} \,\vert i\rangle\vert^{\,2}\, \delta(E_n-E_i +\hbar\,\omega),\) \ref{883}

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

    \( U = \int d\omega\,u(\omega),\) \ref{884}

    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

    $ w_{i\rightarrow n} = \frac{\pi}{\hbar^2} \frac{e^2}{\epsilon_0\,m...
..., \vert\langle n\vert\, \exp[\,{\rm i}\,(\omega_{ni}/c)\,{\bf n}\cdot{\bf x}]\,$ \( \mbox{\boldmath\)\( \cdot{\bf p} \,\vert i\rangle\vert^{\,2}\) \ref{885}

    for absorption, and

    $ w_{i\rightarrow n} = \frac{\pi}{\hbar^2} \frac{e^2}{\epsilon_0\,m...
...\, \vert\langle n\vert\, \exp[-{\rm i}\,(\omega_{in}/c)\,{\bf n}\cdot{\bf x}]\,$ \( \mbox{\boldmath\)\( \cdot{\bf p} \,\vert i\rangle\vert^{\,2}\) \ref{886}

    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)

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)