next up previous contents
Next: Eigenstates of angular momentum Up: Orbital angular momentum Previous: Angular momentum operators   Contents


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:

$\displaystyle x$ $\textstyle =$ $\displaystyle r\,\sin\theta\,\cos\phi,$ (527)
$\displaystyle y$ $\textstyle =$ $\displaystyle r\,\sin\theta\,\sin\phi,$ (528)
$\displaystyle z$ $\textstyle =$ $\displaystyle r\,\cos\theta.$ (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
$\displaystyle L_x$ $\textstyle =$ $\displaystyle - {\rm i}\,\hbar\left(-\sin\phi\,\frac{\partial}{\partial\theta}
-\cos\phi\,\cot\theta\,\frac{\partial}{\partial\phi}\right),$ (530)
$\displaystyle L_y$ $\textstyle =$ $\displaystyle - {\rm i}\,\hbar\left(\cos\phi\,\frac{\partial}{\partial\theta}
-\sin\phi\,\cot\theta\,\frac{\partial}{\partial\phi}\right),$ (531)
$\displaystyle L_z$ $\textstyle =$ $\displaystyle -{\rm i}\,\hbar\,\frac{\partial}{\partial\phi},$ (532)

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} (533)

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} (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, $r$.


next up previous contents
Next: Eigenstates of angular momentum Up: Orbital angular momentum Previous: Angular momentum operators   Contents
Richard Fitzpatrick 2006-12-12