The tensor product of these spaces, V 3 ≡ V 1 ⊗ V 2, has a (2 j 1 + 1) (2 j 2 + 1) -dimensional uncoupled basis | j 1 m 1 j 2 m 2 ⟩ ⟩ ≡ ≡ | j 1 m 1 ⟩ ⟩ ⊗ ⊗ | j 2 m 2 ⟩ ⟩ , m 1 ∈ ∈ { − − j 1 , − − j 1 + 1 , … … , j 1 } , m 2 ∈ ∈ { − − j 2 , − − j 2 + 1 , … … , j 2 } . Angular momentum operators are defined to act on states in V 3 in the following manner: ( j ⊗ ⊗ 1 ) | j 1 m 1 j 2 m 2 ⟩ ⟩ ≡ ≡ j | j 1 m 1 ⟩ ⟩ ⊗ ⊗ | j 2 m 2 ⟩ ⟩ and ( 1 ⊗ ⊗ j ) | j 1 m 1 j 2 m 2 ⟩ ⟩ ≡ ≡ | j 1 m 1 ⟩ ⟩ ⊗ ⊗ j | j 2 m 2 ⟩ ⟩ , where 1 denotes the identity operator.