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Chemistry LibreTexts

396: Particle in an Infinite Potential Well

  • Page ID
    135864
  • Numerical Solutions for Schrödinger's Equation

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

    Potential energy: V(x) := 0

    Numerical integration of Schrödinger's equation:

    Given: \( \frac{1}{2 \mu} \frac{d^{2}}{dx^{2}} \Psi (x) + V(x) \Psi (x) = E \Psi (x)\) \( \Psi (0) = 0\) \( \Psi '(0) = 0.1\)

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

    Enter energy guess: E = 4.934

    Screen Shot 2019-02-06 at 12.35.11 PM.png

    Fourier transform coordinate wave function into momentum space:

    p := -20, -19.5 .. 20

    \( \Phi (p) := \frac{1}{2 \mu} \int_{0}^{x_{max}} exp(-i \cdot p \cdot x) \cdot \Psi (x) dx\)

    Screen Shot 2019-02-06 at 12.35.18 PM.png