In the Schroedinger representation, the z component of the orbital angular momentum operator can be expressed in spherical coordinates as, L z = − − i ℏ ℏ ∂ ∂ ∂ ∂ ϕ ϕ . For L z and eigenfunction ψ ψ with eigenvalue L z ′ , − − i ℏ ℏ ∂ ∂ ∂ ∂ ϕ ϕ ψ ψ = L z ′ ψ ψ . Solving for ψ ψ , ψ ψ = A e i L z ′ ϕ ϕ / ℏ ℏ , where A is independent of ϕ ϕ . Since ψ ψ is required to be single valued, and adding 2 π π to ϕ ϕ results in a coordinate for the same point in space, A e i L z ′ ( ϕ ϕ + 2 π π ) / ℏ ℏ = A e i L z ′ ϕ ϕ / ℏ ℏ , e i L z ′ 2 π π / ℏ ℏ = 1. Solving for the eigenvalue L z ′ , L z ′ = m l ℏ ℏ , where m l is an integer. From the above and the relation m ℓ ℓ = − − ℓ ℓ , ( − − ℓ ℓ + 1 ) , … … , ( ℓ ℓ − − 1 ) , ℓ ℓ , it follows that ℓ ℓ is also an integer. This shows that the quantum numbers m ℓ ℓ and ℓ ℓ for the orbital angular momentum L are restricted to integers, unlike the quantum numbers for the total angular momentum J and spin S , which can have half-integer values.