For any of these eigenvectors ψ ψ with L 2 ψ ψ = ℏ ℏ 2 ℓ ℓ ( ℓ ℓ + 1 ) ψ ψ L z ψ ψ = ℏ ℏ m ψ ψ H 1 ψ ψ = ℏ ℏ ( n 1 + 1 2 ) ψ ψ H 2 ψ ψ = ℏ ℏ ( n 2 + 1 2 ) ψ ψ for some integers n 1 , n 2 , we find m = ( n 1 + 1 2 ) − − ( n 2 + 1 2 ) = n 1 − − n 2 As a difference of two integers, m must be an integer, from which ℓ ℓ is also integral.