Finite difference method for Schrödinger equation

[aaaa202]

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,...,x_N) etc. Finally I approximate the second order derivative with the well known central difference formula:
∂²/∂x² ≈ 1/Δx²(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 = 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?

 

[DrClaude]

In short: What "connects" ħ²k²/2m with ħ²/2m ∂²/∂x²?

You mean apart from the fact that $p = \hbar k$ and $\hat{p}^2= -\hbar^2 \frac{d^2}{dx^2}$?

 

[aaaa202]

But k is not an operator in this case. It is a wavenumber. What connects the number form with the operator form?

 

[DrClaude]

But k is not an operator in this case. It is a wavenumber. What connects the number form with the operator form?

Plane waves. Eigenfunctions of the $\hat{p}^2$ operator are of the form $\exp(i k x)$ where $k = p/\hbar$, with $p$ the momentum of the corresponding plane wave.