Suppose | ψ ψ ⟩ ⟩ is a simultaneous eigenstate of J 2 and J z (i.e., a state with a definite value for J 2 and a definite value for J z ). Then using the commutation relations for the components of J , one can prove that each of the states J + | ψ ψ ⟩ ⟩ and J − − | ψ ψ ⟩ ⟩ is either zero or a simultaneous eigenstate of J 2 and J z , with the same value as | ψ ψ ⟩ ⟩ for J 2 but with values for J z that are increased or decreased by ℏ ℏ respectively. The result is zero when the use of a ladder operator would otherwise result in a state with a value for J z that is outside the allowable range. Using the ladder operators in this way, the possible values and quantum numbers for J 2 and J z can be found.