A vector operator in the spherical basis is V = (V +1, V 0, V −1) where the components are: V + 1 = − − 1 2 ( V x + i V y ) V − − 1 = 1 2 ( V x − − i V y ) , V 0 = V z , using J ± ± = J x ± ± i J y , the various commutators with the rotation generators and ladder operators are: [ J z , V + 1 ] = + ℏ ℏ V + 1 [ J z , V 0 ] = 0 V 0 [ J z , V − − 1 ] = − − ℏ ℏ V − − 1 [ J + , V + 1 ] = 0 [ J + , V 0 ] = 2 ℏ ℏ V + 1 [ J + , V − − 1 ] = 2 ℏ ℏ V 0 [ J − − , V + 1 ] = 2 ℏ ℏ V 0 [ J − − , V 0 ] = 2 ℏ ℏ V − − 1 [ J − − , V − − 1 ] = 0