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PaulWAyers committed Jan 14, 2025
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16 changes: 9 additions & 7 deletions SchrodingerEq.html
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Expand Up @@ -569,7 +569,8 @@ <h2>Motivation for the Schrödinger Equation<a class="headerlink" href="#motivat
<div class="math notranslate nohighlight">
\[ E = h \nu = \hbar \omega \]</div>
<div class="math notranslate nohighlight">
\[ p = \tfrac{h}{\lambda} = \tfrac{h \nu}{c} = \hbar k = h \tilde{\nu} \]</div>
\[ p = \tfrac{h}{\lambda} = \tfrac{h \nu}{c} = \hbar k
= h \tilde{\nu} \]</div>
<section id="angular-frequency-and-wavenumber">
<h3>Angular Frequency and Wavenumber<a class="headerlink" href="#angular-frequency-and-wavenumber" title="Link to this heading">#</a></h3>
<p>In these equations I have introduced several new symbols, mostly related to the fact it is often convenient to use angular frequency,</p>
Expand All @@ -583,7 +584,7 @@ <h3>Angular Frequency and Wavenumber<a class="headerlink" href="#angular-frequen
\[ \tilde{\nu} = \tfrac{1}{\lambda} = \lambda^{-1} \]</div>
<p>or its angular analogue</p>
<div class="math notranslate nohighlight">
\[ k = \tfrac{2 \pi}{\lambda} = 2 \pi \tilde{nu} \]</div>
\[ k = \tfrac{2 \pi}{\lambda} = 2 \pi \tilde{\nu} \]</div>
<p>Sometimes it is also useful to consider the period of the wave,</p>
<div class="math notranslate nohighlight">
\[ \text{T} = \nu^{-1} \]</div>
Expand Down Expand Up @@ -716,13 +717,14 @@ <h4>📝 Exercise: Ground-State Energy of the Morse Potential<a class="headerlin
\[
\left(-\frac{d^{2}}{dx^{2}}+\lambda ^{2}\left(e^{-2x}-2e^{-x}\right)\right)\psi _{n}\left(x\right)=E_{n}\psi _{n}\left(x\right)
\]</div>
<p>The first two eigenfunctions of the Morse oscillator are given by the following expressions (which are not normalized)
$<span class="math notranslate nohighlight">\(
\begin{align}
<p>The first two eigenfunctions of the Morse oscillator are given by the following expressions (which are not normalized)</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{split}
\psi _{0}\left(x\right)&amp;=\exp \left(-\left(\lambda -\tfrac{1}{2}\right)x-\lambda e^{-x}\right)\\
\psi _{1}\left(x\right)&amp;=\exp \left(-\left(\lambda -\tfrac{3}{2}\right)x-\lambda e^{-x}\right)\left(2\lambda -2-2\lambda e^{-x}\right)
\end{align}
\)</span>$</p>
\end{split}
\end{split}\]</div>
<p><strong>What is the expression for the ground-state energy for the Morse oscillator?</strong></p>
<p><a class="reference internal" href="SchrodingerExercise3.html"><span class="std std-doc">Answer</span></a></p>
</section>
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45 changes: 26 additions & 19 deletions SchrodingerExercise3.html
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Expand Up @@ -8,7 +8,7 @@
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />

<title>&lt;no title&gt; &#8212; Quantum Chemistry</title>
<title>📝 Exercise &#8212; Quantum Chemistry</title>



Expand Down Expand Up @@ -449,7 +449,7 @@


<div id="jb-print-docs-body" class="onlyprint">
<h1><no title></h1>
<h1>📝 Exercise</h1>
<!-- Table of contents -->
<div id="print-main-content">
<div id="jb-print-toc">
Expand All @@ -458,9 +458,9 @@ <h1><no title></h1>
<h2> Contents </h2>
</div>
<nav aria-label="Page">
<ul class="simple visible nav section-nav flex-column">
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ground-state-energy-of-the-morse-potential">Ground-State Energy of the Morse Potential</a></li>
</ul>

</nav>
</div>
</div>
Expand All @@ -471,23 +471,27 @@ <h2> Contents </h2>
<div id="searchbox"></div>
<article class="bd-article">

<p>#📝 Exercise</p>
<p>##Ground-State Energy of the Morse Potential
The Morse potential is often used as an approximate model for the vibrations of diatomic molecules. In convenient units where <span class="math notranslate nohighlight">\(\frac{\hslash ^{2}}{2m}=1\)</span>, the time-independent Schrödinger equation for a Morse oscillator can be written as:</p>
<section class="tex2jax_ignore mathjax_ignore" id="exercise">
<h1>📝 Exercise<a class="headerlink" href="#exercise" title="Link to this heading">#</a></h1>
<section id="ground-state-energy-of-the-morse-potential">
<h2>Ground-State Energy of the Morse Potential<a class="headerlink" href="#ground-state-energy-of-the-morse-potential" title="Link to this heading">#</a></h2>
<p>The Morse potential is often used as an approximate model for the vibrations of diatomic molecules. In convenient units where <span class="math notranslate nohighlight">\(\frac{\hslash ^{2}}{2m}=1\)</span>, the time-independent Schrödinger equation for a Morse oscillator can be written as:</p>
<div class="math notranslate nohighlight">
\[
\left(-\frac{d^{2}}{dx^{2}}+\lambda ^{2}\left(e^{-2x}-2e^{-x}\right)\right)\psi _{n}\left(x\right)=E_{n}\psi _{n}\left(x\right)
\]</div>
<p>The first two eigenfunctions of the Morse oscillator are given by the following expressions (which are not normalized)
$<span class="math notranslate nohighlight">\(
<p>The first two eigenfunctions of the Morse oscillator are given by the following expressions (which are not normalized)</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{align}
\psi _{0}\left(x\right)&amp;=\exp \left(-\left(\lambda -\tfrac{1}{2}\right)x-\lambda e^{-x}\right)\\
\psi _{1}\left(x\right)&amp;=\exp \left(-\left(\lambda -\tfrac{3}{2}\right)x-\lambda e^{-x}\right)\left(2\lambda -2-2\lambda e^{-x}\right)
\end{align}
\)</span>$</p>
\end{split}\]</div>
<p><strong>What is the expression for the ground-state energy for the Morse oscillator?</strong></p>
<p>This requires applying the Hamiltonian to the ground-state wavefunction of the Morse oscillator and finding the eigenvalue. It’s a straightforward but tedious exercise:
$<span class="math notranslate nohighlight">\(
<p>This requires applying the Hamiltonian to the ground-state wavefunction of the Morse oscillator and finding the eigenvalue. It’s a straightforward but tedious exercise:</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{align}
&amp;\frac{d}{dx}\left[\exp \left(-\left(\lambda -\tfrac{1}{2}\right)x-\lambda e^{-x}\right)\right]=\exp \left(-\left(\lambda -\tfrac{1}{2}\right)x-\lambda e^{-x}\right)\cdot \frac{d}{dx}\left[-\left(\lambda -\frac{1}{2}\right)x-\lambda e^{-x}\right]\\
&amp;\qquad=\exp \left(-\left(\lambda -\frac{1}{2}\right)x-\lambda e^{-x}\right)\cdot \left[-\left(\lambda -\frac{1}{2}\right)+\lambda e^{-x}\right]\\
Expand All @@ -499,17 +503,20 @@ <h2> Contents </h2>
&amp;\qquad=\left(\exp \left(-\left(\lambda -\frac{1}{2}\right)x-\lambda e^{-x}\right)\right)\cdot \left(\left(\lambda -\frac{1}{2}\right)^{2}-2\lambda ^{2}e^{-x}+\lambda ^{2}e^{-2x}\right)\\
&amp;\qquad=\left(\exp \left(-\left(\lambda -\frac{1}{2}\right)x-\lambda e^{-x}\right)\right)\cdot \left(\left(\lambda -\frac{1}{2}\right)^{2}+\lambda ^{2}\left(e^{-2x}-2e^{-x}\right)\right)
\end{align}
\)</span><span class="math notranslate nohighlight">\(
Substituting into the Schr&amp;ouml;dinger equation gives:
\)</span><span class="math notranslate nohighlight">\(
\end{split}\]</div>
<p>Substituting into the Schrödinger equation gives:</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{align}
&amp;\left(-\frac{d^{2}}{dx^{2}}+\lambda ^{2}\left(e^{-2x}-2e^{-x}\right)\right)\left[\exp \left(-\left(\lambda -\frac{1}{2}\right)x-\lambda e^{-x}\right)\right]\\
&amp; \qquad =-\left(\exp \left(-\left(\lambda -\frac{1}{2}\right)x-\lambda e^{-x}\right)\right)\cdot \left(\left(\lambda -\frac{1}{2}\right)^{2}+\lambda ^{2}\left(e^{-2x}-2e^{-x}\right)\right)\\
&amp;\qquad \qquad+\left[\lambda ^{2}\left(e^{-2x}-2e^{-x}\right)\right]\left(\exp \left(-\left(\lambda -\frac{1}{2}\right)x-\lambda e^{-x}\right)\right)\\
&amp;\qquad =-\left(\lambda -\frac{1}{2}\right)^{2}\left(\exp \left(-\left(\lambda -\frac{1}{2}\right)x-\lambda e^{-x}\right)\right)
\end{align}
\)</span><span class="math notranslate nohighlight">\(
So the ground-state energy of the Morse oscillator is \)</span>E=-\left(\lambda -\frac{1}{2}\right)^{2}$.</p>
\end{split}\]</div>
<p>So the ground-state energy of the Morse oscillator is <span class="math notranslate nohighlight">\(E=-\left(\lambda -\frac{1}{2}\right)^{2}\)</span>.</p>
</section>
</section>

<script type="text/x-thebe-config">
{
Expand Down Expand Up @@ -557,9 +564,9 @@ <h2> Contents </h2>
<i class="fa-solid fa-list"></i> Contents
</div>
<nav class="bd-toc-nav page-toc">
<ul class="simple visible nav section-nav flex-column">
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ground-state-energy-of-the-morse-potential">Ground-State Energy of the Morse Potential</a></li>
</ul>

</nav></div>

</div></div>
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10 changes: 6 additions & 4 deletions _sources/SchrodingerEq.ipynb
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Expand Up @@ -29,7 +29,8 @@
"\n",
"$$ E = h \\nu = \\hbar \\omega $$\n",
"\n",
"$$ p = \\tfrac{h}{\\lambda} = \\tfrac{h \\nu}{c} = \\hbar k = h \\tilde{\\nu} $$\n",
"$$ p = \\tfrac{h}{\\lambda} = \\tfrac{h \\nu}{c} = \\hbar k \n",
"= h \\tilde{\\nu} $$\n",
"\n",
"### Angular Frequency and Wavenumber\n",
"In these equations I have introduced several new symbols, mostly related to the fact it is often convenient to use angular frequency, \n",
Expand All @@ -46,7 +47,7 @@
"\n",
"or its angular analogue\n",
"\n",
"$$ k = \\tfrac{2 \\pi}{\\lambda} = 2 \\pi \\tilde{nu} $$\n",
"$$ k = \\tfrac{2 \\pi}{\\lambda} = 2 \\pi \\tilde{\\nu} $$\n",
"\n",
"Sometimes it is also useful to consider the period of the wave, \n",
"\n",
Expand Down Expand Up @@ -231,11 +232,12 @@
"\n",
"\n",
"The first two eigenfunctions of the Morse oscillator are given by the following expressions (which are not normalized)\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\begin{split}\n",
"\\psi _{0}\\left(x\\right)&=\\exp \\left(-\\left(\\lambda -\\tfrac{1}{2}\\right)x-\\lambda e^{-x}\\right)\\\\\n",
"\\psi _{1}\\left(x\\right)&=\\exp \\left(-\\left(\\lambda -\\tfrac{3}{2}\\right)x-\\lambda e^{-x}\\right)\\left(2\\lambda -2-2\\lambda e^{-x}\\right)\n",
"\\end{align}\n",
"\\end{split}\n",
"$$ \n",
"\n",
"**What is the expression for the ground-state energy for the Morse oscillator?**\n",
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13 changes: 9 additions & 4 deletions _sources/SchrodingerExercise3.md
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@@ -1,6 +1,6 @@
#&#x1f4dd; Exercise
# &#x1f4dd; Exercise

##Ground-State Energy of the Morse Potential
## Ground-State Energy of the Morse Potential
The Morse potential is often used as an approximate model for the vibrations of diatomic molecules. In convenient units where $\frac{\hslash ^{2}}{2m}=1$, the time-independent Schr&ouml;dinger equation for a Morse oscillator can be written as:

$$
Expand All @@ -9,16 +9,18 @@ $$


The first two eigenfunctions of the Morse oscillator are given by the following expressions (which are not normalized)

$$
\begin{align}
\psi _{0}\left(x\right)&=\exp \left(-\left(\lambda -\tfrac{1}{2}\right)x-\lambda e^{-x}\right)\\
\psi _{1}\left(x\right)&=\exp \left(-\left(\lambda -\tfrac{3}{2}\right)x-\lambda e^{-x}\right)\left(2\lambda -2-2\lambda e^{-x}\right)
\end{align}
$$
$$

**What is the expression for the ground-state energy for the Morse oscillator?**

This requires applying the Hamiltonian to the ground-state wavefunction of the Morse oscillator and finding the eigenvalue. It’s a straightforward but tedious exercise:

$$
\begin{align}
&\frac{d}{dx}\left[\exp \left(-\left(\lambda -\tfrac{1}{2}\right)x-\lambda e^{-x}\right)\right]=\exp \left(-\left(\lambda -\tfrac{1}{2}\right)x-\lambda e^{-x}\right)\cdot \frac{d}{dx}\left[-\left(\lambda -\frac{1}{2}\right)x-\lambda e^{-x}\right]\\
Expand All @@ -32,7 +34,9 @@ $$
&\qquad=\left(\exp \left(-\left(\lambda -\frac{1}{2}\right)x-\lambda e^{-x}\right)\right)\cdot \left(\left(\lambda -\frac{1}{2}\right)^{2}+\lambda ^{2}\left(e^{-2x}-2e^{-x}\right)\right)
\end{align}
$$

Substituting into the Schr&ouml;dinger equation gives:

$$
\begin{align}
&\left(-\frac{d^{2}}{dx^{2}}+\lambda ^{2}\left(e^{-2x}-2e^{-x}\right)\right)\left[\exp \left(-\left(\lambda -\frac{1}{2}\right)x-\lambda e^{-x}\right)\right]\\
Expand All @@ -41,4 +45,5 @@ $$
&\qquad =-\left(\lambda -\frac{1}{2}\right)^{2}\left(\exp \left(-\left(\lambda -\frac{1}{2}\right)x-\lambda e^{-x}\right)\right)
\end{align}
$$
So the ground-state energy of the Morse oscillator is $E=-\left(\lambda -\frac{1}{2}\right)^{2}$.

So the ground-state energy of the Morse oscillator is $E=-\left(\lambda -\frac{1}{2}\right)^{2}$.
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2 changes: 1 addition & 1 deletion searchindex.js
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