Ladder Operator for Angular Momentum

Defining

$$L^+ = L_x + \imath L_y$$

and

$$L^- = L_x - \imath L_y$$

and knowing

L_x L_y - L_y L_z = (yp_z - z p_y)(zp_x - xp_z) -(zp_x - xp_z) (yp_z - z p_y)

which is

= yp_z zp_x - yp_z xp_z - z p_y zp_x + z p_y xp_z -zp_x yp_z +zp_x z p_y + xp_z yp_z - xp_z z p_y

which is, reordering:

= yp_z zp_x -zp_x yp_z - yp_z xp_z + xp_z yp_z - z p_y zp_x +zp_x z p_y + z p_y xp_z - xp_z z p_y

which is, cancelling:

$$\begin{eqnarray*}= yp_z zp_x -zp_x yp_z + z p_y xp_z - xp_z z p_y \\ = y p_x (... ... = \imath \hbar ( x p_y -K p_x ) = \imath \hbar L_z = [L_x,L_y] \end{eqnarray*}$$