<head>
<title>Testing MathJax Example 2 (autonumbering)</title>
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script>
MathJax = {
tex: {
tags: 'ams' // should be 'ams', 'none', or 'all'
}
};
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
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<body>
This is the test of equation \ref{num1} (but ( eqref(num1) )\eqref{num1} also works)
\begin{equation}
\sqrt{b^2} = \pm b
\label{num1}
\end{equation}
In equation \eqref{eq:sample}, we find the value of an
interesting integral:
\begin{equation}
\int_0^\infty \frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15}
\label{eq:sample}
\end{equation}
<hr>
Tuesday, September 17, 2019
MathJax example 2
Monday, September 16, 2019
MathJax Example using Latex and Automatic Equation Numbering
The commutator:
The Ladder Operators Construction
For the Harmonic Oscillator, we form the two operators<html>
<head>
<title>The Harmonic Oscillator</title>
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script>
MathJax = {
tex: {
tags: 'ams' // should be 'ams', 'none', or 'all'
}
};
</script>
<script async="" id="MathJax-script" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']],
tags:'ams'
},
svg: {
fontCache: 'global'
}
};
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</head>
<body>
The Hamiltonian for the Harmonic Oscillator is
\begin{equation}\frac{p^2}{2\mu} + \frac{k}{2} x^2
\end{equation}
where p is the momentum operator and x is the postition operator. We know that the Schrödinger prescription is
\begin{equation}p \rightarrow - \boldsymbol{i} \hbar \frac{\partial}{\partial x}
\end{equation}
while $x \rightarrow x$, as usual. This means, as we well know, that the commutator of $x$ and $p$ is non-zero.
(Note that we've dropped the operator subscript.)
<h2>
The commutator: $[\alpha,\beta]$</h2>
\begin{equation*}
[p,x]f =( px - xp)f = -\boldsymbol{i} \hbar \frac{\partial}{\partial x}xf -
x(-\boldsymbol{i} \hbar \frac{\partial}{\partial x}f)
\end{equation*}
where $f(x) \mapsto f$,
which leads to
\begin{equation*}
[p,x]f =( px - xp)f = -f\boldsymbol{i} \hbar \frac{\partial x}{\partial x} +
x \left (-\boldsymbol{i} \hbar \frac{\partial}{\partial x}\right )f -
x\left (-\boldsymbol{i} \hbar \frac{\partial}{\partial x}f\right )
\end{equation*}
which means, of course,
\begin{equation}
[p,x] = px - xp = -\boldsymbol{i} \hbar
\end{equation}
<br />
<h2>
The Ladder Operators Construction</h2>
For the Harmonic Oscillator, we form the two operators
\begin{equation}
a^+ = p + \boldsymbol{i} \mu\omega x
\end{equation}
and
\begin{equation}
a^- = p - \boldsymbol{i} \mu\omega x
\end{equation}
which differ solely by that intervening sign
(Remember that $\omega = \sqrt{\frac{k}{\mu}}$).
The entire derivation now hinges on the properties of these two operators.
We start with the elementary question, what is the commutator of $a^+$ and
$a^-$?
</body>
</html>
Wednesday, September 11, 2019
MathJax example with Latex
<html>
<head>
<title>MathJax TeX Test Page</title>
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']]
},
svg: {
fontCache: 'global'
}
};
</script>
<script type="text/javascript" id="MathJax-script" async
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-svg.js">
</script>
</head>
<body>
This is the text
\begin{equation}
\sqrt{b^2}
\end{equation}
</body>
</html>
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