Forbidden Transitions

We saw in Section 8.10 that a spontaneous electromagnetic transition between some initial atomic state, $i$ , and some final state, $f$ , is mediated by the matrix element

$$\mbox{\boldmath$\epsilon$} \cdot{\bf d}_{if} = \frac{-{\rm i}}{m_e\,\omega_{if}}\left\langle... ...\bf x}\right]\mbox{\boldmath$\epsilon$}\cdot{\bf p} \right\vert f\right\rangle.$$ (8.183)

Now,

$$\exp\left[\,{\rm i}\left(\frac{\omega}{c}\right){\bf n}\cdot{\bf x}\right] = 1 + {\rm i}\,\frac{\omega}{c} \,{\bf n}\cdot{\bf x} + \cdots.$$ (8.184)

However, as explained in the previous section, the fact that the wavelength of the radiation that is emitted during spontaneous transition is generally much larger than the typical size of the atom allows us to truncated the previous expansion. Retaining the first two terms, we obtain

$$\mbox{\boldmath$\epsilon$} \cdot{\bf d}_{if} = \langle i\vert\, \mbox{\boldmath$\epsilon$} \cdot{\bf x}\,\vert f\rangle + \frac{1}{m_e\,c}\, \langle i\vert\,({\bf n}\cdot{\bf x})\,( \mbox{\boldmath$\epsilon$} \cdot{\bf p})\,\vert f\rangle,$$ (8.185)

where use has been made of Equation (8.170). Moreover, we have assumed that $\omega=\omega_{if}$ (i.e., the angular frequency of the electromagnetic radiation matches that associated with the atomic transition.) Suppose, however, that the transition from state $i$ to state $f$ is forbidden according to the selection rules for electric dipole transitions. This implies that

$$\langle i\vert\, \mbox{\boldmath$\epsilon$} \cdot{\bf x}\,\vert f\rangle =0.$$ (8.186)

In this case, Equation (8.188) reduces to

$$\mbox{\boldmath$\epsilon$} \cdot{\bf d}_{if} = \frac{1}{m_e\,c}\, \langle i\vert\,({\bf n}\cdot{\bf x})\,( \mbox{\boldmath$\epsilon$} \cdot{\bf p})\,\vert f\rangle.$$ (8.187)

We deduce that a ``forbidden'' transition is not, strictly speaking, forbidden [i.e., Equation (8.189) does not necessarily mean that $\epsilon$ $\cdot{\bf d}_{if} =0$ ], but rather takes place at a significantly lower rate than an electric dipole transition [because, according to the previous expression, $\vert$$\epsilon$ $\cdot{\bf d}_{if}\vert\sim a_0\,\hbar/(m_e\,c\,a_0)\sim \alpha\,a_0\ll a_0$ , whereas $\vert$$\epsilon$ $\cdot{\bf d}_{if}\vert\sim a_0$ for an electric dipole transition (see Section 8.11)].

According to classical electromagnetic theory, the polarization direction of the magnetic component of an electromagnetic wave propagating in the direction ${\bf n}$ is given by ${\bf b}={\bf n}\times$$\epsilon$ , where $\epsilon$ specifies the direction of the wave's electric component [49]. Of course, ${\bf L} = {\bf x}\times{\bf p}$ represents orbital angular momentum. However,

$${\bf b}\cdot{\bf L} = ({\bf n}\times \mbox{\boldmath$\epsilon$} )\cdot ({\bf x}\times {\bf p}) = ({\bf n}\cdot{\bf x})\,( \mbox{\boldmath$\epsilon$} \cdot{\bf p})-( \mbox{\boldmath$\epsilon$} \cdot{\bf x})\,({\bf n}\cdot {\bf p}).$$ (8.188)

Furthermore, if

$$S = \frac{{\rm i}\,m_e}{\hbar}\,[H,\,( \mbox{\boldmath$\epsilon$} \cdot{\bf x})\,({\bf n}\cdot{\bf x})],$$ (8.189)

then

$$S = \frac{{\rm i}\,m_e}{\hbar}\,\epsilon_i\,n_j\,[H,\,x_i\,x_j] = \... ...{\rm i}\,m_e}{\hbar}\,\epsilon_i\,n_j\left(x_i\,[H,\,x_j]+[H,\,x_i]\,x_j\right)$$

$$= \epsilon_i\,n_j\left(x_i\,p_j+p_i\,x_j\right) = \epsilon_i\,n_j\left(x_i\,p_j+x_j\,p_i-{\rm i}\,\hbar\,\delta_{ij}\right)$$

$$= ( \mbox{\boldmath$\epsilon$} \cdot{\bf x})\,({\bf n}\cdot{\bf p}) + ({\bf n}\cdot{\bf x})\,( \mbox{\boldmath$\epsilon$} \cdot{\bf p}).$$ (8.190)

Here, use has been made of Equations (3.32) and (3.33), as well as the fact that $\epsilon$ $\cdot{\bf n} = 0$ . It follows, from the previous three equations, that

$$({\bf n}\cdot{\bf x})\,( \mbox{\boldmath$\epsilon$} \cdot{\bf p}) = \frac{1}{2}\,{\bf b}\,\cdot{\bf L} + \frac{{\rm i}\,m_e}{2\,\hbar}\,[H,\,( \mbox{\boldmath$\epsilon$} \cdot{\bf x})\,({\bf n}\cdot{\bf x})].$$ (8.191)

Hence, Equation (8.190) yields

$$\mbox{\boldmath$\epsilon$} \cdot{\bf d}_{if} = \frac{1}{2\,m_e\,c}\,{\bf b}\,\cdot\langle i\vert\,{\bf L}\,\vert f\rangle +\frac{{\rm i}\,\omega_{if}}{2\,c}\, \mbox{\boldmath$\epsilon$} \cdot{\bf Q}_{if}\cdot{\bf n},$$ (8.192)

where

$$({\bf Q}_{if})_{jk} = \langle i\vert\,x_j\,x_k - r^{\,2}\,\delta_{jk}/3\,\vert f\rangle.$$ (8.193)

Here, $r^{\,2}=x_j\,x_j$ . Moreover, we have made use of the fact that $\epsilon$ $\cdot{\bf n} = 0$ to write ${\bf Q}_{if}$ as a traceless tensor. In the following, we shall treat the two terms on the right-hand side of Equation (8.195) separately, because they give rise to completely different selection rules. The first term governs so-called magnetic dipole transitions, whereas the second governs so-called electric quadrupole transitions.