91Ó°ÊÓ

Write equation 4.130.

${\mathbf{L}}_{\mathbf{Â\pm }}\mathbf{=}\mathbf{Â\pm }\sout{\mathbf{h}}{\mathbf{e}}^{\mathbf{Â\pm }\mathbf{¾\pm Ï•}}\left(\frac{∂}{∂\mathrm{θ}}Â\pm i\cot \mathrm{θ}\frac{∂}{∂\mathrm{Ï•}}\right)$

Write equation 4.129.

${\mathbf{L}}_{\mathbf{z}}\mathbf{=}\frac{\sout{\mathbf{h}}}{\mathbf{i}}\frac{\mathbf{∂}}{\mathbf{∂}\mathbf{ϕ}}$

Solve to derive equation 4.131

$$\begin{gathered} {\mathrm{L}}_{+}{\mathrm{L}}_{-}\mathrm{f}=-\sout{{\mathrm{h}}^2}{\mathrm{e}}^{\mathrm{¾\pm φ}}\left\{\left(\frac{∂}{∂\mathrm{θ}}+\mathrm{i}\cot \mathrm{θ}\frac{∂}{∂\mathrm{Ï•}}\right)\left[{\mathrm{e}}^{-\mathrm{¾\pm φ}}\left(\frac{∂\mathrm{f}}{∂\mathrm{θ}}-\mathrm{i}\cot \mathrm{θ}\frac{∂\mathrm{f}}{∂\mathrm{Ï•}}\right)\right]\right\} \\ =-{\mathrm{h}}^2{\mathrm{e}}^{\mathrm{¾\pm φ}}\left\{\begin{array}{c}{\mathrm{e}}^{\mathrm{¾\pm φ}}\left(\frac{{∂}^2\mathrm{f}}{∂{\mathrm{θ}}^2}+\mathrm{i}{\csc }^2\mathrm{θ}\frac{∂\mathrm{f}}{∂\mathrm{Ï•}}-\mathrm{i}\cot \mathrm{θ}\frac{{∂}^2}{∆\mathrm{Ï•}∂\mathrm{θ}}\right)\\ +\mathrm{i}\cot \mathrm{θ}\left[-{\mathrm{e}}^{-\mathrm{¾\pm φ}\left(\frac{∂\mathrm{f}}{∂\mathrm{θ}}\right)+{\mathrm{e}}^{-\mathrm{¾\pm φ}}}\left(\frac{{∂}^2\mathrm{f}}{∂\mathrm{θ}∂\mathrm{Ï•}}-\mathrm{i}\cot \mathrm{θ}\frac{{∂}^2\mathrm{f}}{∂{\mathrm{Ï•}}^2}\right)\right]\end{array}\right\} \\ =-{\mathrm{h}}^2{\mathrm{e}}^{\mathrm{¾\pm φ}}\left\{\begin{array}{c}{\mathrm{e}}^{\mathrm{¾\pm φ}}\left(\frac{{∂}^2\mathrm{f}}{∂{\mathrm{θ}}^2}+{\mathrm{ie}}^{-\mathrm{¾\pm φ}}{\csc }^2\mathrm{θ}\frac{∂\mathrm{f}}{∂\mathrm{Ï•}}-{\mathrm{ie}}^{-\mathrm{¾\pm φ}}\cot \mathrm{θ}\frac{{∂}^2}{∂∂∂\mathrm{Ï•}}+{\mathrm{e}}^{\mathrm{¾\pm φ}}\mathrm{³¦´Ç³Ùθ}\frac{∂\mathrm{f}}{∂\mathrm{θ}}-{\mathrm{ie}}^{-\mathrm{¾\pm φ}}{\cot }^2\mathrm{θ}\frac{∂\mathrm{f}}{∂\mathrm{Ï•}}\right)\\ +{\mathrm{ie}}^{-\mathrm{¾\pm φ}}\cot \mathrm{θ}\frac{{∂}^2\mathrm{f}}{∂\mathrm{Ï•}∂\mathrm{θ}}+{\mathrm{e}}^{\mathrm{¾\pm φ}}{\cot }^2\mathrm{θ}\frac{{∂}^2\mathrm{f}}{∂{\mathrm{Ï•}}^2}\end{array}\right\} \\ =-{\sout{\mathrm{h}}}^2\left\{\frac{{∂}^2}{∂{\mathrm{θ}}^2}+\mathrm{i}\left({\csc }^2\mathrm{θ}-{\cot }^2\mathrm{θ}\right)\frac{∂}{∂\mathrm{Ï•}}+\mathrm{³¦´Ç³Ùθ}\frac{∂}{∂\mathrm{θ}}+{\cot }^2\mathrm{θ}\frac{{∂}^2}{∂{\mathrm{Ï•}}^2}\right\}\mathrm{f} \\ {\mathrm{L}}_{+}{\mathrm{L}}_{-}=-{\sout{\mathrm{h}}}^2\left\{\frac{{∂}^2}{∂{\mathrm{θ}}^2}+\mathrm{i}\frac{∂}{∂\mathrm{Ï•}}+\mathrm{³¦´Ç³Ùθ}\frac{∂}{∂\mathrm{θ}}+{\cot }^2\mathrm{θ}\frac{{∂}^2}{∂{\mathrm{Ï•}}^2}\right\} \end{gathered}$$

Thus, equation 4.131 is derived.

Solve to derive equation 4.132.

$$\begin{gathered} {\mathrm{L}}^2=-{\sout{\mathrm{h}}}^2\frac{{∂}^2}{∂{\mathrm{θ}}^2}-\mathrm{i}{\sout{\mathrm{h}}}^2\frac{∂}{∂\mathrm{ϕ}}-{\sout{\mathrm{h}}}^2{\cot }^2\mathrm{θ}\frac{{∂}^2}{∂{\mathrm{ϕ}}^2}-{\sout{\mathrm{h}}}^2\frac{{∂}^2}{∂{\mathrm{ϕ}}^2}+\mathrm{i}{\sout{\mathrm{h}}}^2\frac{∂}{∂\mathrm{ϕ}} \\ =-{\sout{\mathrm{h}}}^2\left(\frac{{∂}^2}{∂{\mathrm{θ}}^2}+\cot \mathrm{θ}\frac{∂}{∂\mathrm{θ}}+\frac{1}{{\sin }^2\mathrm{θ}}\frac{{∂}^2}{∂{\mathrm{ϕ}}^2}\right) \\ =-{\sout{\mathrm{h}}}^2\left(\frac{1}{\sin \mathrm{θ}}\frac{∂}{∂\mathrm{θ}}\left(\sin \mathrm{θ}\frac{∂}{∂\mathrm{θ}}\right)+\frac{1}{{\sin }^2\mathrm{θ}}\frac{{∂}^2}{∂{\mathrm{ϕ}}^2}\right) \\ =-{\sout{\mathrm{h}}}^2\left(\frac{1}{\sin \mathrm{θ}}\frac{∂}{∂\mathrm{θ}}\left(\sin \mathrm{θ}\frac{∂}{∂\mathrm{θ}}\right)+\frac{1}{{\sin }^2\mathrm{θ}}\frac{{∂}^2}{∂{\mathrm{ϕ}}^2}\right) \end{gathered}$$

Thus, equation 4.132 is derived.