The properties of the ladder operators can be determined by observing how they modify the action of the J z operator on a given state: J z J ± ± | j m ⟩ ⟩ = ( J ± ± J z + [ J z , J ± ± ] ) | j m ⟩ ⟩ = ( J ± ± J z ± ± ℏ ℏ J ± ± ) | j m ⟩ ⟩ = ℏ ℏ ( m ± ± 1 ) J ± ± | j m ⟩ ⟩ .