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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:

$\displaystyle x$ $\textstyle =$ $\displaystyle r \sin\theta \cos\phi,$ (545)
$\displaystyle y$ $\textstyle =$ $\displaystyle r \sin\theta \sin\phi,$ (546)
$\displaystyle z$ $\textstyle =$ $\displaystyle r \cos\theta.$ (547)

It follows, after some tedious analysis, that
$\displaystyle \frac{\partial}{\partial x}$ $\textstyle =$ $\displaystyle \sin\theta \cos\phi \frac{\partial}{\partial r} + \frac{\cos\th...
...partial\theta} - \frac{\sin\phi}{r \sin\theta} \frac{\partial}{\partial\phi},$ (548)
$\displaystyle \frac{\partial}{\partial y}$ $\textstyle =$ $\displaystyle \sin\theta \sin\phi \frac{\partial}{\partial r} + \frac{\cos\th...
...partial\theta} + \frac{\cos\phi}{r \sin\theta} \frac{\partial}{\partial\phi},$ (549)
$\displaystyle \frac{\partial}{\partial z}$ $\textstyle =$ $\displaystyle \cos\theta \frac{\partial}{\partial r} -\frac{\sin\theta}{r} \frac{\partial}{\partial \theta}.$ (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
$\displaystyle L_x$ $\textstyle =$ $\displaystyle - {\rm i} \hbar\left(-\sin\phi \frac{\partial}{\partial\theta}
-\cos\phi \cot\theta \frac{\partial}{\partial\phi}\right),$ (551)
$\displaystyle L_y$ $\textstyle =$ $\displaystyle - {\rm i} \hbar\left(\cos\phi \frac{\partial}{\partial\theta}
-\sin\phi \cot\theta \frac{\partial}{\partial\phi}\right),$ (552)
$\displaystyle L_z$ $\textstyle =$ $\displaystyle -{\rm i} \hbar \frac{\partial}{\partial\phi},$ (553)

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

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


next up previous
Next: Eigenstates of Angular Momentum Up: Orbital Angular Momentum Previous: Angular Momentum Operators
Richard Fitzpatrick 2010-07-20