Sakurai page 196: Orbital angular momentum as rotation generator

[omoplata]

From "Modern Quantum Mechanics, revised edition" by J.J. Sakurai, page 196.

Equation (3.6.4),$$1-i \left( \frac{\delta \phi}{\hbar} \right) L_z = 1 - i \left( \frac{\delta \phi}{\hbar} \right) (x p_y - y p_x )$$Making this act on an arbitrary position eigenket $\mid x', y', z' \rangle$,
Equation (3.6.5),$$\begin{eqnarray}<br /> \left[ 1-i \left( \frac{\delta \phi}{\hbar} \right) L_z \right] \mid x', y', z' \rangle & = & \left[ 1 - i \left( \frac{p_y}{\hbar} \right) ( \delta \phi x' ) + i \left( \frac{p_x}{\hbar} \right) ( \delta \phi y' ) \right] \mid x', y', z' \rangle \\<br /> & = & \mid x' - y' \delta \phi, y' + x \delta \phi, z' \rangle<br /> \end{eqnarray}$$
What I don't understand is, in equation (3.6.5), why did they operate by the position operators first, and not the momentum operators. Looking at equation (3.6.4), it looks like the ket $\mid x', y', z' \rangle$ should be operated on by the momentum operators first.

 

[unchained1978]

It wouldn't matter, since $[\hat p_{x},\hat y]=[\hat p_{y},\hat x]=0$. Remember $[\hat p_{i},\hat x_{j}]=i\hbar \delta_{ij}$?

 

[omoplata]

Oh, OK. Got it. Thanks.