The phases of α and β are not physically significant, thus they can be chosen to be positive and real (Condon–Shortley phase convention). We then have J + | j , m ⟩ ⟩ = ℏ ℏ ( j − − m ) ( j + m + 1 ) | j , m + 1 ⟩ ⟩ = ℏ ℏ j ( j + 1 ) − − m ( m + 1 ) | j , m + 1 ⟩ ⟩ , J − − | j , m ⟩ ⟩ = ℏ ℏ ( j + m ) ( j − − m + 1 ) | j , m − − 1 ⟩ ⟩ = ℏ ℏ j ( j + 1 ) − − m ( m − − 1 ) | j , m − − 1 ⟩ ⟩ .