Single-particle and two-particle quantum numbers, brief review on angular momenta etc

We have that the operators for the orbital momentum are given by $$ L_x=-i\hbar(y\frac{\partial }{\partial z}-z\frac{\partial }{\partial y})= yp_z-zp_y, $$ $$ L_y=-i\hbar(z\frac{\partial }{\partial x}-x\frac{\partial }{\partial z})= zp_x-xp_z, $$ $$ L_z=-i\hbar(x\frac{\partial }{\partial y}-y\frac{\partial }{\partial x})=xp_y-yp_x. $$