Define an operator by its spectrum: Υ Υ l m | r ⟩ ⟩ = r l Y l m ( θ θ , ϕ ϕ ) | r ⟩ ⟩ = Υ Υ l m ( r → → ) | r ⟩ ⟩ Since for spherical harmonics under rotation: Y ℓ ℓ = k m = q ( n ) = ⟨ ⟨ n | k , q ⟩ ⟩ → → U ( R ) † † Y ℓ ℓ = k m = q ( n ) U ( R ) = Y ℓ ℓ = k m = q ( R n ) = ⟨ ⟨ n | D ( R ) † † | k , q ⟩ ⟩ = ∑ ∑ q ′ D q ′ , q ( k ) ( R − − 1 ) Y ℓ ℓ = k m = q ′ ( n ) It can also been shown that: Υ Υ l m ( r → → ) → → U ( R ) † † Υ Υ l m ( r → → ) U ( R ) = ∑ ∑ m ′ D m ′ , m ( l ) ( R − − 1 ) Υ Υ l m ′ ( r → → ) Then Υ Υ l m ( V → → ) , where V → → is a vector operator, also transforms in the same manner ie, is a spherical tensor operator. The process involves expressing Υ Υ l m ( r → → ) = r l Y l m ( θ θ , ϕ ϕ ) = Υ Υ l m ( x , y , z ) in terms of x, y and z and replacing x, y and z with operators V x V y and V z which from vector operator. The resultant operator is hence a spherical tensor operator T ^ ^ m ( l ) . ^ This may include constant due to normalization from spherical harmonics which is meaningless in context of operators.