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{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2023-12-14T16:29:59","documenter_version":"1.2.1"}}
{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2023-12-14T23:52:04","documenter_version":"1.2.1"}}
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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Harmonic Oscillator · Antique.jl</title><meta name="title" content="Harmonic Oscillator · Antique.jl"/><meta property="og:title" content="Harmonic Oscillator · Antique.jl"/><meta property="twitter:title" content="Harmonic Oscillator · Antique.jl"/><meta name="description" content="Documentation for Antique.jl."/><meta property="og:description" content="Documentation for Antique.jl."/><meta property="twitter:description" content="Documentation for Antique.jl."/><meta property="og:url" content="https://ohno.github.io/Antique.jl/HarmonicOscillator/"/><meta property="twitter:url" content="https://ohno.github.io/Antique.jl/HarmonicOscillator/"/><link rel="canonical" href="https://ohno.github.io/Antique.jl/HarmonicOscillator/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/logo.ico" rel="icon" type="image/x-icon"/><link href="../assets/css/catalog.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img src="../assets/logo.svg" alt="Antique.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">Antique.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><a class="tocitem" href="../InfinitePotentialWell/">Infinite Potential Well</a></li><li class="is-active"><a class="tocitem" href>Harmonic Oscillator</a><ul class="internal"><li><a class="tocitem" href="#Definitions"><span>Definitions</span></a></li><li><a class="tocitem" href="#Usage-and-Examples"><span>Usage &amp; Examples</span></a></li><li><a class="tocitem" href="#Testing"><span>Testing</span></a></li></ul></li><li><a class="tocitem" href="../MorsePotential/">Morse Potential</a></li><li><a class="tocitem" href="../HydrogenAtom/">Hydrogen Atom</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Harmonic Oscillator</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Harmonic Oscillator</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/ohno/Antique.jl/blob/main/docs/src/HarmonicOscillator.md#" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Harmonic-Oscillator"><a class="docs-heading-anchor" href="#Harmonic-Oscillator">Harmonic Oscillator</a><a id="Harmonic-Oscillator-1"></a><a class="docs-heading-anchor-permalink" href="#Harmonic-Oscillator" title="Permalink"></a></h1><p>The harmonic oscillator is the most frequently used model in quantum physics.</p><h2 id="Definitions"><a class="docs-heading-anchor" href="#Definitions">Definitions</a><a id="Definitions-1"></a><a class="docs-heading-anchor-permalink" href="#Definitions" title="Permalink"></a></h2><p><span>$k$</span> is the spring constant. And <span>$\omega = \sqrt{k/m}$</span>, <span>$\xi = \sqrt{\frac{m\omega}{\hbar}}x$</span>, <span>$A_n = \sqrt{\frac{1}{n! 2^n} \sqrt{\frac{m\omega}{\pi\hbar}}}$</span>, <span>$H_n(x) = (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2}$</span> are used.</p><h4 id="Schrödinger-Equation"><a class="docs-heading-anchor" href="#Schrödinger-Equation">Schrödinger Equation</a><a id="Schrödinger-Equation-1"></a><a class="docs-heading-anchor-permalink" href="#Schrödinger-Equation" title="Permalink"></a></h4><p class="math-container">\[ \hat{H}\psi(x) = E \psi(x)\]</p><h4 id="Hamiltonian"><a class="docs-heading-anchor" href="#Hamiltonian">Hamiltonian</a><a id="Hamiltonian-1"></a><a class="docs-heading-anchor-permalink" href="#Hamiltonian" title="Permalink"></a></h4><p class="math-container">\[ \begin{aligned}
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Harmonic Oscillator · Antique.jl</title><meta name="title" content="Harmonic Oscillator · Antique.jl"/><meta property="og:title" content="Harmonic Oscillator · Antique.jl"/><meta property="twitter:title" content="Harmonic Oscillator · Antique.jl"/><meta name="description" content="Documentation for Antique.jl."/><meta property="og:description" content="Documentation for Antique.jl."/><meta property="twitter:description" content="Documentation for Antique.jl."/><meta property="og:url" content="https://ohno.github.io/Antique.jl/HarmonicOscillator/"/><meta property="twitter:url" content="https://ohno.github.io/Antique.jl/HarmonicOscillator/"/><link rel="canonical" href="https://ohno.github.io/Antique.jl/HarmonicOscillator/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/logo.ico" rel="icon" type="image/x-icon"/><link href="../assets/css/catalog.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img src="../assets/logo.svg" alt="Antique.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">Antique.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><a class="tocitem" href="../InfinitePotentialWell/">Infinite Potential Well</a></li><li class="is-active"><a class="tocitem" href>Harmonic Oscillator</a><ul class="internal"><li><a class="tocitem" href="#Definitions"><span>Definitions</span></a></li><li><a class="tocitem" href="#Usage-and-Examples"><span>Usage &amp; Examples</span></a></li><li><a class="tocitem" href="#Testing"><span>Testing</span></a></li></ul></li><li><a class="tocitem" href="../MorsePotential/">Morse Potential</a></li><li><a class="tocitem" href="../HydrogenAtom/">Hydrogen Atom</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Harmonic Oscillator</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Harmonic Oscillator</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/ohno/Antique.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/ohno/Antique.jl/blob/main/docs/src/HarmonicOscillator.md#" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Harmonic-Oscillator"><a class="docs-heading-anchor" href="#Harmonic-Oscillator">Harmonic Oscillator</a><a id="Harmonic-Oscillator-1"></a><a class="docs-heading-anchor-permalink" href="#Harmonic-Oscillator" title="Permalink"></a></h1><p>The harmonic oscillator is the most frequently used model in quantum physics.</p><h2 id="Definitions"><a class="docs-heading-anchor" href="#Definitions">Definitions</a><a id="Definitions-1"></a><a class="docs-heading-anchor-permalink" href="#Definitions" title="Permalink"></a></h2><p><span>$k$</span> is the spring constant. And <span>$\omega = \sqrt{k/m}$</span>, <span>$\xi = \sqrt{\frac{m\omega}{\hbar}}x$</span>, <span>$A_n = \sqrt{\frac{1}{n! 2^n} \sqrt{\frac{m\omega}{\pi\hbar}}}$</span>, <span>$H_n(x) = (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2}$</span> are used.</p><h4 id="Schrödinger-Equation"><a class="docs-heading-anchor" href="#Schrödinger-Equation">Schrödinger Equation</a><a id="Schrödinger-Equation-1"></a><a class="docs-heading-anchor-permalink" href="#Schrödinger-Equation" title="Permalink"></a></h4><p class="math-container">\[ \hat{H}\psi(x) = E \psi(x)\]</p><h4 id="Hamiltonian"><a class="docs-heading-anchor" href="#Hamiltonian">Hamiltonian</a><a id="Hamiltonian-1"></a><a class="docs-heading-anchor-permalink" href="#Hamiltonian" title="Permalink"></a></h4><p class="math-container">\[ \begin{aligned}
\hat{H}
&amp;= - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x) \\
&amp;= - \frac{1}{2} \hbar\omega \frac{\mathrm{d}^2}{\mathrm{d}\xi^2} + V(x)
Expand Down Expand Up @@ -443,4 +443,4 @@
5.0 | 9 | 21.242645786248 | 21.242617504750 ✔
Test Summary: | Pass Total
∫ψₙ*Hψₙdx = &lt;ψₙ|H|ψₙ&gt; = Eₙ | 40 40
</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../InfinitePotentialWell/">« Infinite Potential Well</a><a class="docs-footer-nextpage" href="../MorsePotential/">Morse Potential »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Thursday 14 December 2023 16:29">Thursday 14 December 2023</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../InfinitePotentialWell/">« Infinite Potential Well</a><a class="docs-footer-nextpage" href="../MorsePotential/">Morse Potential »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Thursday 14 December 2023 23:52">Thursday 14 December 2023</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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