.

(*restored equation*)

$$L_x +\imath L_y = \hbar\left ( \imath \sin \phi + \cos \phi ... ...th\sin \phi \right )\cot \theta \frac{\partial}{\partial \phi}$$

or

$$L_x +\imath L_y = \hbar\left ( \imath \sin \phi + \cos \phi ... ...th\sin \phi \right )\cot \theta \frac{\partial}{\partial \phi}$$

which is, using DeMoivre's Theorem

$$L^+ = \hbar e^{\imath \phi} \left ( \frac{\partial}{\partial... ...} + \imath \cot \theta \frac{\partial}{\partial \phi} \right )$$

which is correct, as we see by applying L₊ to the angular part of a p_z orbital, which is the $m_\ell = 0$ orbital then $\ell = 0$:

$$L^+ \vert> \rightarrow \vert 1>$$ \rightarrow \vert 1> \end{displaymath}">

i.e.,

$$\hbar e^{\imath \phi} \left ( \frac{\partial}{\partial \theta... ...al \phi} \right ) \cos \theta = -sin \theta \hbar e^{\imath K}$$