For a particle in one dimension on ℓ ℓ 2 ( Z ) , we have the Hamiltonian H = − − Δ Δ + V where − − Δ Δ ≡ ≡ 2 I − − S − − S ∗ ∗ is the discrete Laplacian, with S being the right shift operator on ℓ ℓ 2 ( Z ) . Then the probability current is defined as j ≡ ≡ 2 ℑ ℑ { Ψ Ψ ¯ ¯ i v Ψ Ψ } , with v the velocity operator, equal to v ≡ ≡ − − i [ X , H ] and X is the position operator on ℓ ℓ 2 ( Z ) . Since V is usually a multiplication operator on ℓ ℓ 2 ( Z ) , we get to safely write − − i [ X , H ] = − − i [ X , − − Δ Δ ] = − − i [ X , − − S − − S ∗ ∗ ] = i S − − i S ∗ ∗ .