It is customary to set , and write . The separation between adjacent levels is , it increases linear with l . Later a measurement of L y is made. Show that if any operator commutes with two of the components of an angular momentum operator, it commutes with the third. The commutation relations for the Cartesian components of any angular momentum operator.
Commutation relations are what defines a vector operator as a angular momentum operator.
We define angular momentum through . Then . . They commute with H . However a state cannot be simultaneously an eigenstate of J z and J x . If one of the operators is non degenerate, then all of its eigenvectors are also eigenvectors of the other operator.