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:

$$x = r \sin\theta \cos\phi,$$ (545)

$$y = r \sin\theta \sin\phi,$$ (546)

$$z = r \cos\theta.$$ (547)

It follows, after some tedious analysis, that

$$\frac{\partial}{\partial x} = \sin\theta \cos\phi \frac{\partial}{\partial r} + \frac{\cos\th... ...partial\theta} - \frac{\sin\phi}{r \sin\theta} \frac{\partial}{\partial\phi},$$ (548)

$$\frac{\partial}{\partial y} = \sin\theta \sin\phi \frac{\partial}{\partial r} + \frac{\cos\th... ...partial\theta} + \frac{\cos\phi}{r \sin\theta} \frac{\partial}{\partial\phi},$$ (549)

$$\frac{\partial}{\partial z} = \cos\theta \frac{\partial}{\partial r} -\frac{\sin\theta}{r} \frac{\partial}{\partial \theta}.$$ (550)

Making use of the definitions (527)-(529), (534), and (538), the fundamental representation (478)-(480) of the $p_i$ operators as spatial differential operators, the Eqs. (545)-(550), and a great deal of tedious algebra, we finally obtain

$$L_x = - {\rm i} \hbar\left(-\sin\phi \frac{\partial}{\partial\theta} -\cos\phi \cot\theta \frac{\partial}{\partial\phi}\right),$$ (551)

$$L_y = - {\rm i} \hbar\left(\cos\phi \frac{\partial}{\partial\theta} -\sin\phi \cot\theta \frac{\partial}{\partial\phi}\right),$$ (552)

$$L_z = -{\rm i} \hbar \frac{\partial}{\partial\phi},$$ (553)

as well as

$$L^2 = -\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\par... ...frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right],$$ (554)

and

$$L_\pm = \hbar {\rm e}^{\pm{\rm i} \phi}\left(\pm\frac{\par... ...a} +{\rm i} \cot\theta \frac{\partial}{\partial\phi}\right).$$ (555)

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, $r$.