12.13: Forbidden Transitions

Atomic transitions which are forbidden by the electric dipole selection rules ([e13.133]) and ([e13.134]) are unsurprisingly known as forbidden transitions. It is clear from the analysis in Section 1.8 that a forbidden transition is one for which the matrix element \(\langle f|\epsilon\!\cdot\!{\bf p}|i\rangle\) is zero. However, this matrix element is only an approximation to the true matrix element for radiative transitions, which takes the form \(\langle f|\epsilon\!\cdot\!{\bf p}\,\exp(\,{\rm i}\,{\bf k}\!\cdot\!{\bf r})|i\rangle\). Expanding \(\exp(\,{\rm i}\,{\bf k}\!\cdot\!{\bf r})\), and keeping the first two terms, the matrix element for a forbidden transition becomes \[\label{e13.146} \langle f|\epsilon\!\cdot\!{\bf p}\,\exp(\,{\rm i}\,{\bf k}\!\cdot\!{\bf r})|i\rangle \simeq {\rm i}\,\langle f|(\epsilon\!\cdot\!{\bf p})\,({\bf k}\!\cdot\!{\bf r})|i\rangle.\] Hence, if the residual matrix element on the right-hand side of the previous expression is non-zero then a “forbidden” transition can take place, albeit at a much reduced rate. In fact, in Section 1.9, we calculated that the typical rate of an electric dipole transition is \[w_{i\rightarrow f} \sim \alpha^{\,3}\,\omega_{if}.\] Because the transition rate is proportional to the square of the radiative matrix element, it is clear that the transition rate for a forbidden transition enabled by the residual matrix element ([e13.146]) is smaller than that of an electric dipole transition by a factor \((k\,r)^{\,2}\). Estimating \(r\) as the Bohr radius, and \(k\) as the wavenumber of a typical spectral line of hydrogen, it is easily demonstrated that \[w_{i\rightarrow f} \sim \alpha^{\,5}\,\omega_{if}\] for such a transition. Of course, there are some transitions (in particular, the \(2S\rightarrow 1S\) transition) for which the true radiative matrix element \(\langle f|\epsilon\!\cdot\!{\bf p}\,\exp(\,{\rm i}\,{\bf k}\!\cdot\!{\bf r})|i\rangle\) is zero. Such transitions are absolutely forbidden.

Finally, it is fairly obvious that excited states which decay via forbidden transitions have much longer life-times than those which decay via electric dipole transitions. Because the natural width of a spectral line is inversely proportional to the life-time of the associated decaying state, it follows that spectral lines associated with forbidden transitions are generally much sharper than those associated with electric dipole transitions.

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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