$$\require{cancel}$$
Now, we saw earlier, in Section [s7.2], that the operators, $$p_i$$, which represent the Cartesian components of linear momentum in quantum mechanics, can be represented as the spatial differential operators $$-{\rm i}\,\hbar\,\partial/\partial x_i$$. Let us now investigate whether angular momentum operators can similarly be represented as spatial differential operators.
It is most convenient to perform our investigation using conventional spherical polar coordinates: that is, $$r$$, $$\theta$$, and $$\phi$$. These are defined with respect to our usual Cartesian coordinates as follows: \begin{aligned} \label{e8.21} x &= r\,\sin\theta\,\cos\phi,\\[0.5ex] y&= r\,\sin\theta\,\sin\phi,\\[0.5ex] z&= r\,\cos\theta.\label{e8.23}\end{aligned} We deduce, after some tedious analysis, that \begin{aligned} \frac{\partial}{\partial x} &= \sin\theta\,\cos\phi\,\frac{\partial}{\partial r} + \frac{\cos\theta\,\cos\phi}{r}\,\frac{\partial}{\partial\theta} - \frac{\sin\phi}{r\,\sin\theta}\,\frac{\partial}{\partial\phi},\label{e8xx}\\[0.5ex] \frac{\partial}{\partial y} &= \sin\theta\,\sin\phi\,\frac{\partial}{\partial r} + \frac{\cos\theta\,\sin\phi}{r}\,\frac{\partial}{\partial\theta} + \frac{\cos\phi}{r\,\sin\theta}\,\frac{\partial}{\partial\phi},\label{e8yy}\\[0.5ex] \frac{\partial}{\partial z} &= \cos\theta\,\frac{\partial}{\partial r} -\frac{\sin\theta}{r}\,\frac{\partial}{\partial \theta}.\label{e8zz}\end{aligned} Making use of the definitions ([e8.1])–([e8.3]), ([e8.9]), and ([e8.13]), the fundamental representation ([e6.12])–([e6.14]) of the $$p_i$$ operators as spatial differential operators, Equations ([e8.21])–([e8zz]), and a great deal of tedious analysis, we finally obtain \begin{aligned} L_x &= - {\rm i}\,\hbar\left(-\sin\phi\,\frac{\partial}{\partial\theta} -\cos\phi\,\cot\theta\,\frac{\partial}{\partial\phi}\right),\\[0.5ex] L_y &= - {\rm i}\,\hbar\left(\cos\phi\,\frac{\partial}{\partial\theta} -\sin\phi\,\cot\theta\,\frac{\partial}{\partial\phi}\right),\\[0.5ex] L_z &= -{\rm i}\,\hbar\,\frac{\partial}{\partial\phi},\label{e8.26}\end{aligned} as well as $L^2 = -\hbar^{\,2}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left( \sin\theta\,\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^{\,2}}{\partial\phi^{\,2}}\right],$ and $\label{e8.28} L_\pm = \hbar\,{\rm e}^{\pm{\rm i}\,\phi}\left(\pm\frac{\partial}{\partial\theta} +{\rm i}\,\cot\theta\,\frac{\partial}{\partial\phi}\right).$ We, thus, conclude that all of our angular momentum operators can be represented as differential operators involving the angular spherical coordinates, $$\theta$$ and $$\phi$$, but not involving the radial coordinate, $$r$$.