From J 2 = J x 2 + J y 2 + J z 2 is obtained, J x 2 + J y 2 = J 2 − − J z 2 . Applying both sides of the above equation to ψ ψ ( J 2 ′ J z ′ ) , ( J x 2 + J y 2 ) ψ ψ ( J 2 ′ J z ′ ) = ( J 2 ′ − − J z ′ 2 ) ψ ψ ( J 2 ′ J z ′ ) . Since J x and J y are real observables, J 2 ′ − − J z ′ 2 is not negative and | J z ′ | ≤ ≤ J 2 ′ . Thus J z ′ has an upper and lower bound.