$$\require{cancel}$$

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.
