Suppose I want to solve the time-independent Schrödinger equation(adsbygoogle = window.adsbygoogle || []).push({});

(ħ^{2}/2m ∂^{2}/∂x^{2}+ V)ψ = Eψ

using a numerical approach. I then discretize the equation on a lattice of N points such that x=(x_{1},x_{2},...,x_{N}) etc. Finally I approximate the second order derivative with the well known central difference formula:

∂^{2}/∂x^{2}≈ 1/Δx^{2}(ψ_{i+1}+ψ_{i-1}-2ψ_{i})

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 = ħ^{2}/2m * 2/Δx^{2}(1-cos(kΔx))

So for Δx<<1/k we can taylor expand this expression to give:

E ≈ ħ^{2}/2m * 2/Δx^{2}(1-(1-1/2(kΔx)^{2}))=ħ^{2}k^{2}/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}/∂x^{2}is a good approximation. In short: What "connects" ħ^{2}k^{2}/2m with ħ^{2}/2m ∂^{2}/∂x^{2}?

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# I Finite difference method for Schrödinger equation

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