\ell^{\prime}=\ell+1
From Eq. 11.76, \left\langle n^{\prime} \ell^{\prime} m^{\prime}|z| n \ell m\right\rangle=0 \text { unless } m^{\prime}=m \text {, } so the only nonzero z term is
\left\{\begin{array}{l} \text { if } m^{\prime}=m, \quad \text { then }\left\langle n^{\prime} \ell^{\prime} m^{\prime}|x| n \ell m\right\rangle=\left\langle n^{\prime} \ell^{\prime} m^{\prime}|y| n \ell m\right\rangle=0 \\ \text { if } m^{\prime}=m \pm 1, \text { then }\left\langle n^{\prime} \ell^{\prime} m^{\prime}|x| n \ell m\right\rangle=\pm i\left(n^{\prime} \ell^{\prime} m^{\prime}|y| n \ell m\right\rangle \\ \text { and }\left\langle n^{\prime} \ell^{\prime} m^{\prime}|z| n \ell m\right\rangle=0 \end{array}\right. (11.76).
\left\langle n^{\prime} \ell^{\prime} m|z| n \ell m\right\rangle=\int R_{n^{\prime} \ell^{\prime}}\left(Y_{\ell^{\prime}}^{m}\right)^{*} r \cos \theta R_{n \ell} Y_{\ell}^{m} r^{2} d r \sin \theta d \theta d \phi .
=I \sqrt{\frac{\left(2 \ell^{\prime}+1\right)}{4 \pi} \frac{\left(\ell^{\prime}-|m|\right) !}{\left(\ell^{\prime}+|m|\right) !}} \sqrt{\frac{(2 \ell+1)}{4 \pi} \frac{(\ell-|m|) !}{(\ell+|m|) !}} 2 \pi \int_{0}^{\pi} P_{\ell^{\prime}}^{m} P_{\ell}^{m} \cos \theta \sin \theta d \theta . [ \star 1] .
This is independent of the sign of m, so we might as well assume m \geq 0 . The integral (changing variables to x \equiv \cos \theta ) is (using Eq. 11.134)
\operatorname{Int}_{\theta}=\int_{-1}^{1} x P_{\ell+1}^{m}(x) P_{\ell}^{m}(x) d x=\frac{1}{(2 \ell+1)}\left[(\ell+m) \int_{-1}^{1} P_{\ell+1}^{m} P_{\ell-1}^{m} d x+(\ell-m+1) \int_{-1}^{1} P_{\ell+1}^{m} P_{\ell+1}^{m} d x\right] .
Now, it follows from Eq. 4.33 that the associated Legendre functions satisfy the orthogonality relation
\int_{0}^{\pi} \int_{0}^{2 \pi}\left[Y_{\ell}^{m}(\theta, \phi)\right]^{*}\left[Y_{\ell^{\prime}}^{m^{\prime}}(\theta, \phi)\right] \sin \theta d \theta d \phi=\delta_{\ell \ell^{\prime}} \delta_{m m^{\prime}} . (4.33).
\int_{-1}^{1} P_{\ell^{\prime}}^{m}(x) P_{\ell}^{m}(x) d x=\frac{2}{(2 \ell+1)} \frac{(\ell+|m|) !}{(\ell-|m|) !} \delta_{\ell \ell^{\prime}} , [ \star 2] .
so
\operatorname{Int}_{\theta}=\frac{(\ell-m+1)}{(2 \ell+1)} \frac{2}{(2 \ell+3)} \frac{(\ell+1+m) !}{(\ell+1-m) !}=\frac{2}{(2 \ell+1)(2 \ell+3)} \frac{(\ell+1+m) !}{(\ell-m) !} .
Putting this into Eq. [ \star 1] :
\left\langle n^{\prime}(\ell+1) m|z| n \ell m\right\rangle=\frac{I}{2} \sqrt{(2 \ell+3) \frac{(\ell+1-m) !}{(\ell+1+m) !}} \sqrt{(2 \ell+1) \frac{(\ell-m) !}{(\ell+m) !}} \frac{2}{(2 \ell+1)(2 \ell+3)} \frac{(\ell+1+m) !}{(\ell-m) !} .
=I \sqrt{\frac{(\ell+1)^{2}-m^{2}}{(2 \ell+1)(2 \ell+3)}} [ \star 3] .
From Eq. 11.76, \left\langle n^{\prime} \ell^{\prime} m^{\prime}|x| n \ell m\right\rangle=0 \text { unless } m^{\prime}=m \pm 1 \text {; let's start with } m+1 \text { : }
\left\{\begin{array}{l} \text { if } m^{\prime}=m, \quad \text { then }\left\langle n^{\prime} \ell^{\prime} m^{\prime}|x| n \ell m\right\rangle=\left\langle n^{\prime} \ell^{\prime} m^{\prime}|y| n \ell m\right\rangle=0 \\ \text { if } m^{\prime}=m \pm 1, \text { then }\left\langle n^{\prime} \ell^{\prime} m^{\prime}|x| n \ell m\right\rangle=\pm i\left(n^{\prime} \ell^{\prime} m^{\prime}|y| n \ell m\right\rangle \\ \text { and }\left\langle n^{\prime} \ell^{\prime} m^{\prime}|z| n \ell m\right\rangle=0 \end{array}\right. (11.76).
\left\langle n^{\prime} \ell^{\prime}(m+1)|x| n \ell m\right\rangle=\int R_{n^{\prime} \ell^{\prime}}\left(Y_{\ell^{\prime}}^{m+1}\right)^{*} r \sin \theta \cos \phi R_{n \ell} Y_{\ell}^{m} r^{2} d r \sin \theta d \theta d \phi .
=I \sqrt{\frac{\left(2 \ell^{\prime}+1\right)}{4 \pi} \frac{\left(\ell^{\prime}-m-1\right) !}{\left(\ell^{\prime}+m+1\right) !}} \sqrt{\frac{(2 \ell+1)}{4 \pi} \frac{(\ell-m) !}{(\ell+m) !}} \int_{0}^{\pi} P_{\ell^{\prime}}^{m+1} P_{\ell}^{m} \sin ^{2} \theta d \theta \int_{0}^{2 \pi} \cos \phi e^{-i(m+1) \phi} e^{i m \phi} d \phi [ \star 4] .
\operatorname{Int}_{\phi}=\frac{1}{2} \int_{0}^{2 \pi}\left(e^{i \phi}+e^{-i \phi}\right) e^{-i \phi} d \phi=\frac{1}{2} \int_{0}^{2 \pi}\left(1+e^{-2 i \phi}\right) d \phi=\pi [ \star 5] .
Changing variables (x \equiv \cos \theta) , and using Eqs. 11.135 and [ \star 2] :
\operatorname{Int}_{\theta}=\int_{-1}^{1} \sqrt{1-x^{2}} P_{\ell+1}^{m+1}(x) P_{\ell}^{m}(x) d x=\frac{1}{(2 \ell+1)}\left[\int_{-1}^{1} P_{\ell+1}^{m+1} P_{\ell+1}^{m+1} d x-\int_{-1}^{1} P_{\ell+1}^{m+1} P_{\ell-1}^{m+1} d x\right] .
=\frac{2}{(2 \ell+1)(2 \ell+3)} \frac{(\ell+m+2) !}{(\ell-m) !} .
\text { Thus }[\star 4] \text { becomes }
\left\langle n^{\prime}(\ell+1)(m+1)|x| n \ell m\right\rangle=\frac{I}{2} \sqrt{(2 \ell+3) \frac{(\ell-m) !}{(\ell+m+2) !}} \sqrt{(2 \ell+1) \frac{(\ell-m) !}{(\ell+m) !}} \frac{1}{(2 \ell+1)(2 \ell+3)} \frac{(\ell+m+2) !}{(\ell-m) !} .
=\frac{I}{2} \sqrt{\frac{(\ell+m+2)(\ell+m+1)}{(2 \ell+1)(2 \ell+3)}} [ \star 6] .
Now we do the same for m^{\prime}=m-1 :
\left\langle n^{\prime} \ell^{\prime}(m-1)|x| n \ell m\right\rangle=\int R_{n^{\prime} \ell^{\prime}}\left(Y_{\ell^{\prime}}^{m-1}\right)^{*} r \sin \theta \cos \phi R_{n \ell} Y_{\ell}^{m} r^{2} d r \sin \theta d \theta d \phi
=I \sqrt{\frac{\left(2 \ell^{\prime}+1\right)}{4 \pi} \frac{\left(\ell^{\prime}-m+1\right) !}{\left(\ell^{\prime}+m-1\right) !}} \sqrt{\frac{(2 \ell+1)}{4 \pi} \frac{(\ell-m) !}{(\ell+m) !}} \int_{0}^{\pi} P_{\ell^{\prime}}^{m-1} P_{\ell}^{m} \sin ^{2} \theta d \theta \int_{0}^{2 \pi} \cos \phi e^{-i(m-1) \phi} e^{i m \phi} d \phi \cdot[ \star 7] .
\operatorname{Int}_{\phi}=\frac{1}{2} \int_{0}^{2 \pi}\left(e^{i \phi}+e^{-i \phi}\right) e^{i \phi} d \phi=\frac{1}{2} \int_{0}^{2 \pi}\left(e^{2 i \phi}+1\right) d \phi=\pi . [\star 8] .
Changing variables (x \equiv \cos \theta) , and using Eqs. 11.135 and [\star 2] :
\operatorname{Int}_{\theta}=\int_{-1}^{1} \sqrt{1-x^{2}} P_{\ell+1}^{m-1}(x) P_{\ell}^{m}(x) d x=\frac{1}{(2 \ell+3)}\left[\int_{-1}^{1} P_{\ell+2}^{m} P_{\ell}^{m} d x-\int_{-1}^{1} P_{\ell}^{m} P_{\ell}^{m} d x\right] .
=-\frac{2}{(2 \ell+1)(2 \ell+3)} \frac{(\ell+m) !}{(\ell-m) !}.
\text { and }[\star 7] \text { becomes }
\left\langle n^{\prime}(\ell+1)(m-1)|x| n \ell m\right\rangle=-\frac{I}{4} \sqrt{(2 \ell+3) \frac{(\ell-m+2) !}{(\ell+m) !}} \sqrt{(2 \ell+1) \frac{(\ell-m) !}{(\ell+m) !}} \frac{2}{(2 \ell+1)(2 \ell+3)} \frac{(\ell+m) !}{(\ell-m) !} .
=-\frac{I}{2} \sqrt{\frac{(\ell-m+2)(\ell-m+1)}{(2 \ell+1)(2 \ell+3)}} . [\star 9] .
Meanwhile, Eq. 11.76 says \left|\left\langle n^{\prime} \ell^{\prime} m^{\prime}|y| n \ell m\right\rangle\right|^{2}=\left|\left\langle n^{\prime} \ell^{\prime} m^{\prime}|y| n \ell m\right\rangle\right|^{2} , so
\left\{\begin{array}{l} \text { if } m^{\prime}=m, \quad \text { then }\left\langle n^{\prime} \ell^{\prime} m^{\prime}|x| n \ell m\right\rangle=\left\langle n^{\prime} \ell^{\prime} m^{\prime}|y| n \ell m\right\rangle=0 \\ \text { if } m^{\prime}=m \pm 1, \text { then }\left\langle n^{\prime} \ell^{\prime} m^{\prime}|x| n \ell m\right\rangle=\pm i\left(n^{\prime} \ell^{\prime} m^{\prime}|y| n \ell m\right\rangle \\ \text { and }\left\langle n^{\prime} \ell^{\prime} m^{\prime}|z| n \ell m\right\rangle=0 \end{array}\right. (11.76).
\left|\left\langle n^{\prime}(\ell+1)(m+1)| r | n \ell m\right\rangle\right|^{2}+\left|\left\langle n^{\prime}(\ell+1) m| r | n \ell m\right\rangle\right|^{2}+\left|\left\langle n^{\prime}(\ell+1)(m-1)| r | n \ell m\right\rangle\right|^{2} .
=2\left[\frac{I}{2} \sqrt{\frac{(\ell+m+2)(\ell+m+1)}{(2 \ell+1)(2 \ell+3)}}\right]^{2}+\left[I \sqrt{\frac{(\ell+1)^{2}-m^{2}}{(2 \ell+1)(2 \ell+3)}}\right]^{2}+2\left[-\frac{I}{2} \sqrt{\frac{(\ell-m+2)(\ell-m+1)}{(2 \ell+1)(2 \ell+3)}}\right]^{2} .
=\frac{I^{2}}{2}\left\{\frac{(\ell+m+2)(\ell+m+1)+2\left[(\ell+1)^{2}-m^{2}\right]+(\ell-m+2)(\ell-m+1)}{(2 \ell+1)(2 \ell+3)}\right\} .
=I^{2} \frac{\left(2 \ell^{2}+5 \ell+3\right)}{(2 \ell+1)(2 \ell+3)}=I^{2} \frac{(\ell+1)}{(2 \ell+1)} [\star 10] .
Therefore, | \wp |^{2} (summed over the three allowed transitions) is e^{2} I^{2}(\ell+1) /(2 \ell+1) , and the spontaneous emission rate (Eq. 11.63) is
A=\frac{\omega_{0}^{3}| S |^{2}}{3 \pi \epsilon_{0} \hbar c^{3}} (11.63).
A_{\ell \rightarrow \ell+1}=\frac{e^{2} \omega^{3} I^{2}}{3 \pi \epsilon_{0} \hbar c^{3}} \frac{(\ell+1)}{(2 \ell+1)} [\star 11] .
\ell^{\prime}=\ell-1 .
Return to Eq [\star 1] . This time the integral is
\operatorname{Int}_{\theta}=\int_{-1}^{1} x P_{\ell-1}^{m}(x) P_{\ell}^{m}(x) d x=\frac{1}{(2 \ell+1)}\left[(\ell+m) \int_{-1}^{1} P_{\ell-1}^{m} P_{\ell-1}^{m} d x+(\ell-m+1) \int_{-1}^{1} P_{\ell-1}^{m} P_{\ell+1}^{m} d x\right] .
=\frac{(\ell+m)}{(2 \ell+1)} \frac{2}{(2 \ell-1)} \frac{(\ell-1+m) !}{(\ell-1-m) !}=\frac{2}{(2 \ell-1)(2 \ell+1)} \frac{(\ell+m) !}{(\ell-m-1) !} .
Therefore
\left\langle n^{\prime}(\ell-1) m|z| n \ell m\right\rangle=\frac{I}{2} \sqrt{(2 \ell-1) \frac{(\ell-1-m) !}{(\ell-1+m) !}} \sqrt{(2 \ell+1) \frac{(\ell-m) !}{(\ell+m) !}} \frac{2}{(2 \ell-1)(2 \ell+1)} \frac{(\ell+m) !}{(\ell-m-1) !} .
=I \sqrt{\frac{\ell^{2}-m^{2}}{(2 \ell-1)(2 \ell+1)}} . [\star 12] .
\text { From }\left\langle n^{\prime} \ell^{\prime} m^{\prime}|x| n \ell m\right\rangle \text { with } m^{\prime}=m+1 \text {, Eqs. }[\star 4] \text { and }[ \star 5] are unchanged; this time
\operatorname{Int}_{\theta}=\int_{-1}^{1} \sqrt{1-x^{2}} P_{\ell-1}^{m+1}(x) P_{\ell}^{m}(x) d x=\frac{1}{(2 \ell+1)}\left[\int_{-1}^{1} P_{\ell-1}^{m+1} P_{\ell+1}^{m+1} d x-\int_{-1}^{1} P_{\ell-1}^{m+1} P_{\ell-1}^{m+1} d x\right] .
=-\frac{2}{(2 \ell-1)(2 \ell+1)} \frac{(\ell+m) !}{(\ell-m-2) !} .
\text { and }[\star 4] \text { becomes}
\left\langle n^{\prime}(\ell-1)(m+1)|x| n \ell m\right\rangle=-\frac{I}{2} \sqrt{(2 \ell-1) \frac{(\ell-m-2) !}{(\ell+m) !}} \sqrt{(2 \ell+1) \frac{(\ell-m) !}{(\ell+m) !}} \frac{1}{(2 \ell-1)(2 \ell+1)} \frac{(\ell+m) !}{(\ell-m-2) !} .
=-\frac{I}{2} \sqrt{\frac{(\ell-m)(\ell-m-1)}{(2 \ell-1)(2 \ell+1)}} [\star 13] .
Now we do the same for m^{\prime}=m-1 . \text { Eqs. }[\star 7] \text { and }[\star 8] are unchanged, the θ integral is
\operatorname{Int}_{\theta}=\int_{-1}^{1} \sqrt{1-x^{2}} P_{\ell-1}^{m-1}(x) P_{\ell}^{m}(x) d x=\frac{1}{(2 \ell-1)}\left[\int_{-1}^{1} P_{\ell}^{m} P_{\ell}^{m} d x-\int_{-1}^{1} P_{\ell-2}^{m} P_{\ell}^{m} d x\right] .
=\frac{2}{(2 \ell-1)(2 \ell+1)} \frac{(\ell+m) !}{(\ell-m) !} .
\text { and }[\star 7] \text { becomes }
\left\langle n^{\prime}(\ell-1)(m-1)|x| n \ell m\right\rangle=\frac{I}{2} \sqrt{(2 \ell-1) \frac{(\ell-m) !}{(\ell+m-2) !}} \sqrt{(2 \ell+1) \frac{(\ell-m) !}{(\ell+m) !}} \frac{1}{(2 \ell-1)(2 \ell+1)} \frac{(\ell+m) !}{(\ell-m) !} .
=\frac{I}{2} \sqrt{\frac{(\ell+m)(\ell+m-1)}{(2 \ell-1)(2 \ell+1)}} [\star 14] .
Thus
\left|\left\langle n^{\prime}(\ell-1)(m+1)| r | n \ell m\right\rangle\right|^{2}+\left|\left\langle n^{\prime}(\ell-1) m| r | n \ell m\right\rangle\right|^{2}+\left|\left\langle n^{\prime}(\ell-1)(m-1)| r | n \ell m\right\rangle\right|^{2} .
=2\left[-\frac{I}{2} \sqrt{\frac{(\ell-m)(\ell-m-1)}{(2 \ell-1)(2 \ell+1)}}\right]^{2}+\left[I \sqrt{\frac{\ell^{2}-m^{2}}{(2 \ell-1)(2 \ell+1)}}\right]^{2}+2\left[\frac{I}{2} \sqrt{\frac{(\ell+m)(\ell+m-1)}{(2 \ell-1)(2 \ell+1)}}\right]^{2} .
=\frac{I^{2}}{2}\left\{\frac{(\ell-m)(\ell-m-1)+2\left(\ell^{2}-m^{2}\right)+(\ell+m)(\ell+m-1)}{(2 \ell-1)(2 \ell+1)}\right\} .
=I^{2} \frac{\left(2 \ell^{2}-\ell\right)}{(2 \ell-1)(2 \ell+1)}=I^{2} \frac{\ell}{(2 \ell+1)} [\star 15] .
and the emission rate is
A_{\ell \rightarrow \ell-1}=\frac{e^{2} \omega^{3} I^{2}}{3 \pi \epsilon_{0} \hbar c^{3}} \frac{\ell}{(2 \ell+1)} [\star 16] .
(Of course, I is different for the two cases \ell \rightarrow \ell \pm 1 ).