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2 changes: 1 addition & 1 deletion dev/.documenter-siteinfo.json
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{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2023-11-29T15:34:48","documenter_version":"1.2.0"}}
{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2023-11-29T21:33:28","documenter_version":"1.2.0"}}
54 changes: 27 additions & 27 deletions dev/HarmonicOscillator/index.html
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&= n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m}{m! (n-2m)!}(2 x)^{n-2m}.
\end{aligned}\]</p><p><span>$n=0:$</span> ✔</p><p class="math-container">\[\begin{aligned}
H_{0}(x)
= e^{x^{2}} e^{ - x^{2}}
= e^{ - x^{2}} e^{x^{2}}
&amp;= 1 \\
&amp;= 1
\end{aligned}\]</p><p><span>$n=1:$</span> ✔</p><p class="math-container">\[\begin{aligned}
Expand All @@ -67,46 +67,46 @@
&amp;= 2 x
\end{aligned}\]</p><p><span>$n=2:$</span> ✔</p><p class="math-container">\[\begin{aligned}
H_{2}(x)
= e^{x^{2}} \frac{\mathrm{d}^{2} e^{ - x^{2}}}{\mathrm{d}x^{2}}
= e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
&amp;= -2 + 4 x^{2} \\
&amp;= -2 + 4 x^{2}
\end{aligned}\]</p><p><span>$n=3:$</span> ✔</p><p class="math-container">\[\begin{aligned}
H_{3}(x)
= - e^{x^{2}} \frac{\mathrm{d}^{3} e^{ - x^{2}}}{\mathrm{d}x^{3}}
&amp;= 8 x^{3} - 12 x \\
&amp;= 8 x^{3} - 12 x
= - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} e^{x^{2}}
&amp;= - 12 x + 8 x^{3} \\
&amp;= - 12 x + 8 x^{3}
\end{aligned}\]</p><p><span>$n=4:$</span> ✔</p><p class="math-container">\[\begin{aligned}
H_{4}(x)
= e^{x^{2}} \frac{\mathrm{d}^{4} e^{ - x^{2}}}{\mathrm{d}x^{4}}
= e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
&amp;= 12 - 48 x^{2} + 16 x^{4} \\
&amp;= 12 - 48 x^{2} + 16 x^{4}
\end{aligned}\]</p><p><span>$n=5:$</span> ✔</p><p class="math-container">\[\begin{aligned}
H_{5}(x)
= - e^{x^{2}} \frac{\mathrm{d}^{5} e^{ - x^{2}}}{\mathrm{d}x^{5}}
&amp;= 32 x^{5} + 120 x - 160 x^{3} \\
&amp;= 32 x^{5} + 120 x - 160 x^{3}
= - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x} e^{x^{2}}
&amp;= 120 x - 160 x^{3} + 32 x^{5} \\
&amp;= 120 x - 160 x^{3} + 32 x^{5}
\end{aligned}\]</p><p><span>$n=6:$</span> ✔</p><p class="math-container">\[\begin{aligned}
H_{6}(x)
= e^{x^{2}} \frac{\mathrm{d}^{6} e^{ - x^{2}}}{\mathrm{d}x^{6}}
&amp;= -120 + 720 x^{2} + 64 x^{6} - 480 x^{4} \\
&amp;= -120 + 720 x^{2} + 64 x^{6} - 480 x^{4}
= e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
&amp;= -120 + 720 x^{2} - 480 x^{4} + 64 x^{6} \\
&amp;= -120 + 720 x^{2} - 480 x^{4} + 64 x^{6}
\end{aligned}\]</p><p><span>$n=7:$</span> ✔</p><p class="math-container">\[\begin{aligned}
H_{7}(x)
= - e^{x^{2}} \frac{\mathrm{d}^{7} e^{ - x^{2}}}{\mathrm{d}x^{7}}
&amp;= 128 x^{7} - 1344 x^{5} - 1680 x + 3360 x^{3} \\
&amp;= 128 x^{7} - 1344 x^{5} - 1680 x + 3360 x^{3}
= - e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
&amp;= - 1680 x + 3360 x^{3} - 1344 x^{5} + 128 x^{7} \\
&amp;= - 1680 x + 3360 x^{3} - 1344 x^{5} + 128 x^{7}
\end{aligned}\]</p><p><span>$n=8:$</span> ✔</p><p class="math-container">\[\begin{aligned}
H_{8}(x)
= e^{x^{2}} \frac{\mathrm{d}^{8} e^{ - x^{2}}}{\mathrm{d}x^{8}}
&amp;= 1680 - 13440 x^{2} - 3584 x^{6} + 13440 x^{4} + 256 x^{8} \\
&amp;= 1680 - 13440 x^{2} - 3584 x^{6} + 13440 x^{4} + 256 x^{8}
= e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
&amp;= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8} \\
&amp;= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8}
\end{aligned}\]</p><p><span>$n=9:$</span> ✔</p><p class="math-container">\[\begin{aligned}
H_{9}(x)
= - e^{x^{2}} \frac{\mathrm{d}^{9} e^{ - x^{2}}}{\mathrm{d}x^{9}}
&amp;= 512 x^{9} - 9216 x^{7} + 30240 x + 48384 x^{5} - 80640 x^{3} \\
&amp;= 512 x^{9} - 9216 x^{7} + 30240 x + 48384 x^{5} - 80640 x^{3}
= - e^{x^{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} e^{ - x^{2}}}{\mathrm{d}x}
&amp;= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9} \\
&amp;= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9}
\end{aligned}\]</p><pre><code class="nohighlight hljs">Test Summary: | Pass Total Time
Hₙ(x) = (-1)ⁿ exp(x²) dⁿ/dxⁿ　exp(-x²) = ... | 10 10 34.9s</code></pre><h4 id="Normalization-and-Orthogonality-of-H_n(x)"><a class="docs-heading-anchor" href="#Normalization-and-Orthogonality-of-H_n(x)">Normalization &amp; Orthogonality of <span>$H_n(x)$</span></a><a id="Normalization-and-Orthogonality-of-H_n(x)-1"></a><a class="docs-heading-anchor-permalink" href="#Normalization-and-Orthogonality-of-H_n(x)" title="Permalink"></a></h4><p class="math-container">\[\int_{-\infty}^\infty H_j(x) H_i(x) \mathrm{e}^{-x^2} \mathrm{d}x = \sqrt{\pi} 2^j j! \delta_{ij}\]</p><pre><code class="nohighlight hljs"> n m numerical analytical |error|
Hₙ(x) = (-1)ⁿ exp(x²) dⁿ/dxⁿ exp(-x²) = ... | 10 10 35.0s</code></pre><h4 id="Normalization-and-Orthogonality-of-H_n(x)"><a class="docs-heading-anchor" href="#Normalization-and-Orthogonality-of-H_n(x)">Normalization &amp; Orthogonality of <span>$H_n(x)$</span></a><a id="Normalization-and-Orthogonality-of-H_n(x)-1"></a><a class="docs-heading-anchor-permalink" href="#Normalization-and-Orthogonality-of-H_n(x)" title="Permalink"></a></h4><p class="math-container">\[\int_{-\infty}^\infty H_j(x) H_i(x) \mathrm{e}^{-x^2} \mathrm{d}x = \sqrt{\pi} 2^j j! \delta_{ij}\]</p><pre><code class="nohighlight hljs"> n m numerical analytical |error|
0 0 1.7724538509055137 1.7724538509055159 0.0000000000001253% ✔
0 1 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔
0 2 0.0000000000000001 0.0000000000000000 0.0000000000000000% ✔
Expand Down Expand Up @@ -208,7 +208,7 @@
9 8 0.0000000016298145 0.0000000000000000 0.0000000000000000% ✔
9 9 329312283.3492956757545471 329312283.3492959141731262 0.0000000000000724% ✔
Test Summary: | Pass Total Time
∫Hⱼ(x)Hᵢ(x)exp(-x²)dx = √π2ʲj!δᵢⱼ | 100 100 1.0s</code></pre><h4 id="Normalization-and-Orthogonality-of-\\psi_n(x)"><a class="docs-heading-anchor" href="#Normalization-and-Orthogonality-of-\\psi_n(x)">Normalization &amp; Orthogonality of <span>$\psi_n(x)$</span></a><a id="Normalization-and-Orthogonality-of-\\psi_n(x)-1"></a><a class="docs-heading-anchor-permalink" href="#Normalization-and-Orthogonality-of-\\psi_n(x)" title="Permalink"></a></h4><p class="math-container">\[\int \psi_i^\ast(x) \psi_j(x) \mathrm{d}x = \delta_{ij}\]</p><pre><code class="nohighlight hljs"> i j numerical analytical |error|
∫Hⱼ(x)Hᵢ(x)exp(-x²)dx = √π2ʲj!δᵢⱼ | 100 100 1.2s</code></pre><h4 id="Normalization-and-Orthogonality-of-\\psi_n(x)"><a class="docs-heading-anchor" href="#Normalization-and-Orthogonality-of-\\psi_n(x)">Normalization &amp; Orthogonality of <span>$\psi_n(x)$</span></a><a id="Normalization-and-Orthogonality-of-\\psi_n(x)-1"></a><a class="docs-heading-anchor-permalink" href="#Normalization-and-Orthogonality-of-\\psi_n(x)" title="Permalink"></a></h4><p class="math-container">\[\int \psi_i^\ast(x) \psi_j(x) \mathrm{d}x = \delta_{ij}\]</p><pre><code class="nohighlight hljs"> i j numerical analytical |error|
0 0 0.9999999999999991 1.0000000000000000 0.0000000000000888% ✔
0 1 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔
0 2 0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔
Expand Down Expand Up @@ -310,7 +310,7 @@
9 8 -0.0000000000000000 0.0000000000000000 0.0000000000000000% ✔
9 9 0.9999999999999998 1.0000000000000000 0.0000000000000222% ✔
Test Summary: | Pass Total Time
&lt;ψᵢ|ψⱼ&gt; = δᵢⱼ | 100 100 0.7s</code></pre><h4 id="Virial-Theorem"><a class="docs-heading-anchor" href="#Virial-Theorem">Virial Theorem</a><a id="Virial-Theorem-1"></a><a class="docs-heading-anchor-permalink" href="#Virial-Theorem" title="Permalink"></a></h4><p>The virial theorem <span>$\langle T \rangle = \langle V \rangle$</span> and the definition of Hamiltonian <span>$\langle H \rangle = \langle T \rangle + \langle V \rangle$</span> derive <span>$\langle H \rangle = 2 \langle V \rangle = 2 \langle T \rangle$</span>.</p><p class="math-container">\[2 \int \psi_n^\ast(x) V(x) \psi_n(x) \mathrm{d}x = E_n\]</p><pre><code class="nohighlight hljs"> k n numerical analytical |error|
&lt;ψᵢ|ψⱼ&gt; = δᵢⱼ | 100 100 1.2s</code></pre><h4 id="Virial-Theorem"><a class="docs-heading-anchor" href="#Virial-Theorem">Virial Theorem</a><a id="Virial-Theorem-1"></a><a class="docs-heading-anchor-permalink" href="#Virial-Theorem" title="Permalink"></a></h4><p>The virial theorem <span>$\langle T \rangle = \langle V \rangle$</span> and the definition of Hamiltonian <span>$\langle H \rangle = \langle T \rangle + \langle V \rangle$</span> derive <span>$\langle H \rangle = 2 \langle V \rangle = 2 \langle T \rangle$</span>.</p><p class="math-container">\[2 \int \psi_n^\ast(x) V(x) \psi_n(x) \mathrm{d}x = E_n\]</p><pre><code class="nohighlight hljs"> k n numerical analytical |error|
0.1 0 0.4999999999999719 0.5000000000000000 0.0000000000056288% ✔
0.1 1 1.4999999999999998 1.5000000000000000 0.0000000000000148% ✔
0.1 2 2.5000000000000004 2.5000000000000000 0.0000000000000178% ✔
Expand Down Expand Up @@ -352,7 +352,7 @@
5.0 8 8.4999999999999929 8.5000000000000000 0.0000000000000836% ✔
5.0 9 9.5000000000000000 9.5000000000000000 0.0000000000000000% ✔
Test Summary: | Pass Total Time
2 × &lt;ψₙ|V|ψₙ&gt; = Eₙ | 40 40 0.6s</code></pre><h4 id="Eigen-Values-2"><a class="docs-heading-anchor" href="#Eigen-Values-2">Eigen Values</a><a class="docs-heading-anchor-permalink" href="#Eigen-Values-2" title="Permalink"></a></h4><p class="math-container">\[ \begin{aligned}
2 × &lt;ψₙ|V|ψₙ&gt; = Eₙ | 40 40 1.0s</code></pre><h4 id="Eigen-Values-2"><a class="docs-heading-anchor" href="#Eigen-Values-2">Eigen Values</a><a class="docs-heading-anchor-permalink" href="#Eigen-Values-2" title="Permalink"></a></h4><p class="math-container">\[ \begin{aligned}
E_n
&amp;= \int \psi^\ast_n(x) \hat{H} \psi_n(x) \mathrm{d}x \\
&amp;= \int \psi^\ast_n(x) \left[ \hat{V} + \hat{T} \right] \psi(x) \mathrm{d}x \\
Expand Down Expand Up @@ -438,5 +438,5 @@
5.0 8 19.0065551524155296 19.0065778087482116 0.0001192025882298% ✔
5.0 9 21.2426175047498660 21.2426457862480049 0.0001331354786194% ✔
Test Summary: | Pass Total Time
∫ψₙ*Hψₙdx = &lt;ψₙ|H|ψₙ&gt; = Eₙ | 40 40 0.6s
</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 1.2.0 on <span class="colophon-date" title="Wednesday 29 November 2023 15:34">Wednesday 29 November 2023</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
∫ψₙ*Hψₙdx = &lt;ψₙ|H|ψₙ&gt; = Eₙ | 40 40 1.4s
</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 1.2.0 on <span class="colophon-date" title="Wednesday 29 November 2023 21:33">Wednesday 29 November 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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