For choices of n ^ ^ = x ^ ^ ± ± i y ^ ^ or n ^ ^ = z ^ ^ , we get: [ J ± ± , T ^ ^ m ( j ) ] = ℏ ℏ ( j ∓ ∓ m ) ( j ± ± m + 1 ) T ^ ^ m ± ± 1 ( j ) [ J z , T ^ ^ m ( j ) ] = ℏ ℏ m T ^ ^ m ( j ) Note the similarity of the above to: J ± ± | j , m ⟩ ⟩ = ℏ ℏ ( j ∓ ∓ m ) ( j ± ± m + 1 ) | j , m ± ± 1 ⟩ ⟩ J z | j , m ⟩ ⟩ = ℏ ℏ m | j , m ⟩ ⟩ Since J x and J y are linear combinations of J ± ± , they share the same similarity due to linearity.