12.13: Forbidden Transitions

Atomic transitions which are forbidden by the electric dipole selection rules \ref{e13.133} and \ref{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

\[ \langle f|\epsilon \!\cdot\!{\bf p}\,\exp(\,{\rm i}\,{\bf k}\!\cdot\!{\bf r})|i\rangle \simeq {\rm i}\,\langle f|(\epsilon \!\cdot\!{\bf p})\,({\bf k}\!\cdot\!{\bf r})|i\rangle. \label{e13.146} \]

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

\[w_{i\rightarrow f} \sim \alpha^{\,3}\,\omega_{if}. \nonumber \]

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 \ref{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

\[w_{i\rightarrow f} \sim \alpha^{\,5}\,\omega_{if} \nonumber \]

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.

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