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Sakurai page 196: Orbital angular momentum as rotation generator

  1. May 9, 2013 #1
    From "Modern Quantum Mechanics, revised edition" by J.J. Sakurai, page 196.

    Equation (3.6.4),[tex]
    1-i \left( \frac{\delta \phi}{\hbar} \right) L_z = 1 - i \left( \frac{\delta \phi}{\hbar} \right) (x p_y - y p_x )
    [/tex]Making this act on an arbitrary position eigenket [itex]\mid x', y', z' \rangle[/itex],
    Equation (3.6.5),[tex]
    \begin{eqnarray}
    \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 \\
    & = & \mid x' - y' \delta \phi, y' + x \delta \phi, z' \rangle
    \end{eqnarray}
    [/tex]
    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 [itex]\mid x', y', z' \rangle[/itex] should be operated on by the momentum operators first.
     
  2. jcsd
  3. May 9, 2013 #2
    It wouldn't matter, since [itex][\hat p_{x},\hat y]=[\hat p_{y},\hat x]=0[/itex]. Remember [itex][\hat p_{i},\hat x_{j}]=i\hbar \delta_{ij}[/itex]?
     
  4. May 9, 2013 #3
    Oh, OK. Got it. Thanks.
     
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