Representation of angular momentum

Now, we saw earlier, in Sect. 7.2, 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: i.e., $r$, $\theta$, and $\phi$. These are defined with respect to our usual cartesian coordinates as follows:

$$x = r\,\sin\theta\,\cos\phi,$$ (527)

$$y = r\,\sin\theta\,\sin\phi,$$ (528)

$$z = r\,\cos\theta.$$ (529)

Making use of the definitions (509)-(511), (516), and (520), the fundamental representation (460)-(462) of the $p_i$ operators as spatial differential operators, the definitions (527)-(529), and a great deal of tedious analysis, we finally obtain

$$L_x = - {\rm i}\,\hbar\left(-\sin\phi\,\frac{\partial}{\partial\theta} -\cos\phi\,\cot\theta\,\frac{\partial}{\partial\phi}\right),$$ (530)

$$L_y = - {\rm i}\,\hbar\left(\cos\phi\,\frac{\partial}{\partial\theta} -\sin\phi\,\cot\theta\,\frac{\partial}{\partial\phi}\right),$$ (531)

$$L_z = -{\rm i}\,\hbar\,\frac{\partial}{\partial\phi},$$ (532)

as well as

$$L^2 = -\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\par... ...frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right],$$ (533)

and

$$L_\pm = \hbar\,{\rm e}^{\pm{\rm i}\,\phi}\left(\pm\frac{\par... ...a} +{\rm i}\,\cot\theta\,\frac{\partial}{\partial\phi}\right).$$ (534)

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, $r$.