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omoplata
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From "Modern Quantum Mechanics, revised edition" by J.J. Sakurai, page 196.
Equation (3.6.4),<br /> 1-i \left( \frac{\delta \phi}{\hbar} \right) L_z = 1 - i \left( \frac{\delta \phi}{\hbar} \right) (x p_y - y p_x )<br />Making this act on an arbitrary position eigenket \mid x', y', z' \rangle,
Equation (3.6.5),<br /> \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}<br />
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.
Equation (3.6.4),<br /> 1-i \left( \frac{\delta \phi}{\hbar} \right) L_z = 1 - i \left( \frac{\delta \phi}{\hbar} \right) (x p_y - y p_x )<br />Making this act on an arbitrary position eigenket \mid x', y', z' \rangle,
Equation (3.6.5),<br /> \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}<br />
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.