Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Finite difference method for Schrödinger equation

  1. Oct 23, 2016 #1
    Suppose I want to solve the time-independent Schrödinger equation
    2/2m ∂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/Δx2i+1i-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?
     
  2. jcsd
  3. Oct 24, 2016 #2

    DrClaude

    User Avatar

    Staff: Mentor

    You mean apart from the fact that ##p = \hbar k## and ##\hat{p}^2= -\hbar^2 \frac{d^2}{dx^2}##?
     
  4. Oct 24, 2016 #3
    But k is not an operator in this case. It is a wavenumber. What connects the number form with the operator form?
     
  5. Oct 24, 2016 #4

    DrClaude

    User Avatar

    Staff: Mentor

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Finite difference method for Schrödinger equation
  1. Schrödinger equation (Replies: 11)

Loading...