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(*restored equation*)

$\begin{displaymath}= L_x L_x+ \imath L_x L_y - L_x L_x- \imath L_y L_x = L_x L_x... ...L_x L_y - L_y L_x) = + \imath (\imath \hbar L_z) = - \hbar L_z \end{displaymath}$

Continuing

$\begin{displaymath}[L_x,L^+]= -\hbar L_z \end{displaymath}$

(40)

$\begin{displaymath}[L_y,L^+]= L_y (L_x+\imath L_y) - (L_x+\imath L_y) L_y \end{displaymath}$

which is, expanding:

$\begin{displaymath}= L_y L_x+ \imath L_y L_y - L_x L_y- \imath L_y L_y = (L_y L_x - L_x L_y) = - \hbar L_z \end{displaymath}$

$\begin{displaymath}[L_y,L^+]= - \hbar L_x \end{displaymath}$

(41)

Our last ladder operator commutator with a component of $\vec{L}$ gives

$\begin{displaymath}[L_z,L^+]= L_z (L_x+\imath L_y) - (L_x+\imath L_y) L_z \end{displaymath}$

i.e.,

$\begin{displaymath}= L_z L_x + \imath L_z L_y - L_x L_z - \imath L_y L_K \end{displaymath}$

2001-12-26