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Chemistry LibreTexts

398: Particle in a One-dimensional Egg Carton

  • Page ID
    135866
  • Numerical Solutions for Schrödinger's Equation

    Integration limit: xmax = 10 Effective mass: \( \mu\) = 1

    Potential energy: Vo = 2 atoms = 2 \( V(x) = V_{o} ( \cos (atoms 2 \pi \frac{x}{x_{max}}) +1) \)

    Numerical integration of Schrödinger's equation:

    Given \( \frac{-1}{2 \mu} \psi (x) + V(x) \psi (x) = E \psi (x)\) \( \psi (0) = 0\) \( \psi '=0.1\)

    \( \psi = Odesolve (x, x_{max})\) Normalize wave function: \( \psi (x) = \frac{ \psi (x)}{ \sqrt{ \int_{0}^{x_{max}} \psi (x)^{2}dx}} \)

    Enter energy guess: E = 0.83583

    Screen Shot 2019-02-06 at 1.15.46 PM.png

    Fourier transform coordinate wave function into momentum space.

    p = -10, -9.9 .. 10 \( \Phi (p) = \frac{1}{ \sqrt{2 \pi}} \int_{0}^{x_{max}} exp(-i p x) \psi (x)~dx\)

    Screen Shot 2019-02-06 at 1.15.56 PM.png