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- Express the angular momentum operators \begin{eqnarray} \hat{L}_x &=& -i\hbar{\hat{y} \frac{\partial}{\partial z} - \hat{z} \frac{\partial}{\partial y} }\label{EQ01}\\ \hat{L}_y &=& -i\hbar{\hat{z} \frac{\partial}{\partial x} - \hat{x} \frac{\partial}{\partial z} }\label{EQ02}\\ \hat{L}_z &=& -i\hbar{\hat{x} \frac{\partial}{\partial y} - \hat{y} \frac{\partial}{\partial x} }\label{EQ03} \end{eqnarray} in polar coordinates and show \begin{eqnarray} \hat{L}_x &=& i\hbar \left(\sin\phi \frac{\partial}{\partial \theta} +\cot\theta\cos\phi\frac{\partial}{\partial \phi} \right)\label{EQ04}\\ \hat{L}_x &=& i\hbar \left(\sin\phi \frac{\partial}{\partial \theta} +\cot\theta\cos\phi\frac{\partial}{\partial \phi} \right)\label{EQ05}\\ \hat{L}_z &= & i\hbar \frac{\partial }{\partial \phi}\label{EQ06} \end{eqnarray}
- Use the result of the previous part and show that The operator $\vec{L}^2$ in spherical polar coordinates is given by \begin{equation} \vec{L}^2 = \hat{L}_x^2 +\hat{L}_y^2 + \hat{L}_z^2 \label{EQ07} \end{equation} takes the form \begin{equation}\label{EQ08} \vec{L}^2 = -\hbar^2\left[ \frac{1}{\sin\theta} \frac{\partial}{\partial\theta}\left(\sin\theta \frac{\partial}{\partial\theta} \right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \right] \end{equation}
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