The spin raising and lowering operators acting on these eigenvectors give S ^ ^ ± ± | s , m s ⟩ ⟩ = ℏ ℏ s ( s + 1 ) − − m s ( m s ± ± 1 ) | s , m s ± ± 1 ⟩ ⟩ , where S ^ ^ ± ± = S ^ ^ x ± ± i S ^ ^ y .