Suppose I want to solve the time-independent Schrödinger equation (ħ2/2m ∂2/∂x2 + V)ψ = Eψ using a numerical approach. I then discretize the equation on a lattice of N points such that x=(x1,x2,...,xN) etc. Finally I approximate the second order derivative with the well known central difference formula: ∂2/∂x2 ≈ 1/Δx2(ψi+1+ψi-1-2ψi) My question is now: How do you estimate the validity of this approximation? I have already talked to my teacher about it and he said the following: The discrete approximation is a tight-binding model with dispersion: E = ħ2/2m * 2/Δx2(1-cos(kΔx)) So for Δx<<1/k we can taylor expand this expression to give: E ≈ ħ2/2m * 2/Δx2(1-(1-1/2(kΔx)2))=ħ2k2/2m Which, according to my teacher, shows that the approximation holds provided that the lattice spacing is much shorter than the wavelength. What I don't get is how you can argue that because the dispersion is parabolic in k the finite difference approximation for the derivative ∂2/∂x2 is a good approximation. In short: What "connects" ħ2k2/2m with ħ2/2m ∂2/∂x2?