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The orbital angular momentum commutation relations giveConsequentlyThe same commutation relations imply $[L^2,L_x]=[L^2,L_y]=0$, and henceThese are the defining identities for the angular momentum ladder operators.

Ifthen, whenever $L_{\pm }\varphi \ne 0$,Thus $L_{\pm }\varphi$ has quantum numbers $(\ell ,m\pm 1)$.