Find all of the matrix elements of between the states with n=2, \ell=1 and n’=3 , \ell '=2:
\left\langle \begin{matrix}32\acute{m} \begin{vmatrix} r_{i} \\ \end{vmatrix} 21m\\ \end{matrix} \right\rangle
where m=-1,0,1 ,m’=-2,-1 ,0,1,2 and r_{i}=x,y,z.
With the vector operator \hat{V}=\hat{r} , our components are V_{z}=z , V_{+}=x+iy, and V_{-}=x-iy. We start by calculating one of the matrix elements,
\left\langle \begin{matrix}320 \begin{vmatrix} z \\ \end{vmatrix} 210\\ \end{matrix} \right \rangle=\int{\psi_{320}}(r)r\cos\theta \psi_{210}(r)d^{3}r
=\int R_{32} (r)^{*}r R_{21}(r)r^{2}dr\int Y^{0}_{2}(\theta ,\phi )^{*}\cos\theta Y^{0}_{1}(\theta ,\phi )\cos\theta d\Omega =\frac{2^{12}3^{3}\sqrt{3} }{5^{7}}a
From Equation 6.61 [\left\langle \acute{n} \acute{\ell}\acute{m} \begin{matrix}\begin{vmatrix} \hat{V}_{z} \\ \end{vmatrix} n\ell m\\ \end{matrix} \right\rangle =C^{\ell}_{m} {}^{l}_{0}{}^{\acute {\ell} }_{\acute{m} }\left\langle \begin{matrix} n \acute{\ell} \begin{Vmatrix} V \\ \end{Vmatrix} n\ell \\ \end{matrix} \right\rangle ] we can then determine the reduced matrix element
\left\langle \begin{matrix}320 \begin{vmatrix} z \\ \end{vmatrix} 210\\ \end{matrix}\right\rangle=C^{112}_{000} \left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right\rangle
=\frac{2^{12}3^{3}\sqrt{3} }{5^{7}}a =\sqrt{\frac{2}{3} } \left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right\rangle
Therefore
\left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right\rangle =\frac{2^{12}3^{4}}{5^{7}\sqrt{2} }a (6.63)
We can now find all of the remaining matrix elements from Equations 6.59–6.60 [\left\langle \acute{n} \acute{\ell}\acute{m} \begin{matrix} \begin{vmatrix} \hat{V}_{+} \\ \end{vmatrix} n\ell m\\ \end{matrix} \right\rangle =-\sqrt{2} C^{\ell} _{m} {}^{l}_{l}{}^{\acute{\ell} }_{\acute{m} }\left \langle \begin{matrix} n \acute{\ell} \begin{Vmatrix} V \\ \end{Vmatrix} n\ell \\ \end{matrix} \right\rangle] ,[\left\langle \acute{n} \acute{\ell}\acute{m} \begin{matrix}\begin{vmatrix} \hat{V}_{-} \\ \end{vmatrix} n\ell m\\ \end{matrix} \right\rangle =\sqrt{2}C^{\ell}_{m} {}^{l}_{-l}{}^{\acute{\ell} }_{\acute{m} }\left\langle \begin{matrix} n \acute{\ell} \begin{Vmatrix} V \\ \end{Vmatrix} n\ell \\ \end{matrix} \right\rangle ] with the help of the Clebsch–Gordan table. The relevant coefficients are shown in Figure 6.8. The nonzero matrix elements are
\left\langle \begin{matrix} 322\begin{vmatrix} \hat{V}_{+} \\ \end{vmatrix} 211 \\ \end{matrix} \right\rangle=-\sqrt{2}C^{112}_{112} \left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right\rangle=-\sqrt{2}\left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right\rangle
\left\langle \begin{matrix} 321\begin{vmatrix} \hat{V}_{+} \\ \end{vmatrix} 210 \\ \end{matrix} \right\rangle=-\sqrt{2}C^{112}_{011} \left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right\rangle=-\left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right\rangle
\left\langle \begin{matrix} 320\begin {vmatrix} \hat{V}_{+} \\ \end{vmatrix} 21-1 \\ \end{matrix} \right\rangle=-\sqrt{2} C^{112} _{-110} \left\langle \begin{matrix} 32\begin {Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right \rangle=-\frac{1}{\sqrt{3} } \left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end {Vmatrix} 21 \\ \end{matrix} \right\rangle
\left\langle \begin{matrix} 320\begin {vmatrix} \hat{V}_{-} \\ \end{vmatrix} 211 \\ \end{matrix} \right\rangle=\sqrt{2} C^{1} _{1}{}^{12}_{-10} \left\langle \begin{matrix} 32\begin {Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right \rangle=\frac{1}{\sqrt{3} } \left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end {Vmatrix} 21 \\ \end{matrix} \right\rangle
\left\langle \begin{matrix} 32-1\begin {vmatrix} \hat{V}_{-} \\ \end{vmatrix} 210 \\ \end{matrix} \right\rangle=\sqrt{2} C^{1} _{0}{}^{1}_{-1} {}^{2}_{-1} \left\langle \begin{matrix} 32\begin {Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right \rangle= \left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end {Vmatrix} 21 \\ \end{matrix} \right\rangle
\left\langle \begin{matrix} 32-2\begin {vmatrix} \hat{V}_{-} \\ \end{vmatrix} 21-1 \\ \end{matrix} \right\rangle=\sqrt{2} C^{1} _{-1}{}^{1}_{-1} {}^{2}_{-2} \left\langle \begin{matrix} 32\begin {Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right \rangle=\sqrt{2} \left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end {Vmatrix} 21 \\ \end{matrix} \right\rangle
\left\langle \begin{matrix} 321\begin {vmatrix} \hat{V}_{z} \\ \end{vmatrix} 211 \\ \end{matrix} \right\rangle= C^{1} _{1}{}^{1}_{0} {}^{2}_{1} \left\langle \begin{matrix} 32\begin {Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right \rangle=\frac{1}{\sqrt{2}} \left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end {Vmatrix} 21 \\ \end{matrix} \right\rangle
\left\langle \begin{matrix} 320\begin {vmatrix} \hat{V}_{z} \\ \end{vmatrix} 210 \\ \end{matrix} \right\rangle= C^{1} _{0}{}^{1}_{0} {}^{2}_{0} \left\langle \begin{matrix} 32\begin {Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right \rangle=\sqrt{\frac{2}{3}} \left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end {Vmatrix} 21 \\ \end{matrix} \right\rangle
\left\langle \begin{matrix} 32-1\begin {vmatrix} \hat{V}_{z} \\ \end{vmatrix} 21-1 \\ \end{matrix} \right\rangle= C^{1} _{-1}{}^{1}_{0} {}^{2}_{-1} \left\langle \begin{matrix} 32\begin {Vmatrix} V \\ \end{Vmatrix} 21 \\ \end{matrix} \right \rangle=\frac{1}{\sqrt{2}}\left\langle \begin{matrix} 32\begin{Vmatrix} V \\ \end {Vmatrix} 21 \\ \end{matrix} \right\rangle
with the reduced matrix element given by Equation 6.63 [\left\langle \begin {matrix} 32\begin{Vmatrix} V \\ \end {Vmatrix} 21 \\ \end{matrix} \right\rangle =\frac{2^{13}3^{4}}{5^{7}\sqrt{2} }a ]. The other thirty-six matrix elements vanish due to the selection rules (Equations 6.56–6.58 and 6.62 [\left\langle \acute{n} \acute{\ell}\acute{m} \begin{matrix}\begin{vmatrix} \hat{V}_{+} \\ \end{vmatrix} n\ell m\\ \end{matrix} \right\rangle =0 unless \acute{m}=m+1 ] [\left\langle \acute{n} \acute{\ell}\acute{m} \begin{matrix}\begin{vmatrix} \hat{V}_{-} \\ \end{vmatrix} n\ell m\\ \end{matrix} \right\rangle =0 unless \acute{m} =m-1 ] , [\Delta \ell=0,\pm 1, and \Delta m=0,\pm 1.]) . We have determined all forty-five matrix elements and have only needed to evaluate a single integral. I’ve left the matrix elements in terms of V_{+} and V_{-} but it’s straightforward to write them in terms of x and y using the expressions on page 259.
