They comprise the spherical basis, are complete, and satisfy the following eigenvalue equations, j 2 | j m ⟩ ⟩ = ℏ ℏ 2 j ( j + 1 ) | j m ⟩ ⟩ , j ∈ ∈ { 0 , 1 2 , 1 , 3 2 , … … } j z | j m ⟩ ⟩ = ℏ ℏ m | j m ⟩ ⟩ , m ∈ ∈ { − − j , − − j + 1 , … … , j } .