Skip to main content
Chemistry LibreTexts

9.19: Numerical Solutions for the Two-Dimensional Harmonic Oscillator

  • Page ID
    137742
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Reduced mass: \( \mu\) = 1

    Angular momentum: L = 2

    Integration limit: rmax = 5

    Force constant: k = 1

    Energy guess: E = 3

    Solve Schrödinger's equation numerically. Use Mathcad's ODE solve block:

    Given

    \[ \frac{-1}{2 \mu} \frac{d^2}{dr^2} \psi (r) - \frac{1}{2 \mu} \frac{d}{dr} \psi (r) + \left( \frac{L^2}{2 \mu r^2} + \frac{1}{2} kr^2\right) \psi (r) = E \psi (r)~~~ \psi (.001) = 1~~~ \psi '(.001) = 0.1 \nonumber \]

    \[ \psi = Odesolve (r, r_{max}, .001) \nonumber \]

    \[ \psi (r) = \left( \int_{0}^{r_{max}} \psi (r)^2 4 \pi r^2 dr \right) ^{ \frac{-1}{2}} \psi (r) \nonumber \]

    Screen Shot 2019-02-19 at 11.44.13 AM.png


    This page titled 9.19: Numerical Solutions for the Two-Dimensional Harmonic Oscillator is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.