U ( R ) † † V ^ ^ i U ( R ) = ∑ ∑ j R i j V ^ ^ j Any observable vector quantity of a quantum mechanical system should be invariant of the choice of frame of reference. The transformation of expectation value vector which applies for any wavefunction, ensures the above equality. In Dirac notation: ⟨ ⟨ ψ ψ ¯ ¯ | V ^ ^ a | ψ ψ ¯ ¯ ⟩ ⟩ = ⟨ ⟨ ψ ψ | U ( R ) † † V ^ ^ a U ( R ) | ψ ψ ⟩ ⟩ = ∑ ∑ b R a b ⟨ ⟨ ψ ψ | V ^ ^ b | ψ ψ ⟩ ⟩ where the RHS is due to the rotation transformation acting on the vector formed by expectation values. Since | Ψ ⟩ is any quantum state, the same result follows: U ( R ) † † V ^ ^ a U ( R ) = ∑ ∑ b R a b V ^ ^ b Note that here, the term "vector" is used two different ways: kets such as | ψ ⟩ are elements of abstract Hilbert spaces, while the vector operator is defined as a quantity whose components transform in a certain way under rotations.