where 1 is the identity operator. Also notice that R is an additive morphism : R ( n ^ ^ , ϕ ϕ 1 + ϕ ϕ 2 ) = R ( n ^ ^ , ϕ ϕ 1 ) R ( n ^ ^ , ϕ ϕ 2 ) ; as a consequence R ( n ^ ^ , ϕ ϕ ) = exp ⁡ ⁡ ( − − i ϕ ϕ J n ^ ^ ℏ ℏ ) where exp is matrix exponential. The existence of the generator is guaranteed by the Stone's theorem on one-parameter unitary groups.