This is an example of what LaTeX can look like:
When I drop it (the image, not the LaTeX code) onto ChatGPT 4o, I get the following text:
“This image contains mathematical equations written in LaTeX format, specifically related to quantum mechanics. The equations involve wavefunctions (ψm\psi_mψm), coefficients (Cn(t)C_n(t)Cn(t)), Hamiltonians (H1H_1H1), and time evolution expressions.
Key elements:
- The Schrödinger equation in the interaction picture.
- Expansion in terms of basis states ψn\psi_nψn.
- The time evolution of coefficients Cn(t)C_n(t)Cn(t).
- Matrix elements of the perturbation Hamiltonian H1H_1H1.
- The final boxed equation represents the differential equation governing Cn(t)C_n(t)Cn(t), which is common in time-dependent perturbation theory.
Would you like me to extract and format this in LaTeX for better readability?”
and when I answer yes, I get this:
“Here is the LaTeX code for the equations in your image:
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
\langle \psi_m | \times \left( i\hbar \sum_n \dot{C}_n(t)e^{-iE_n^{(0)}t/\hbar} | \psi_n \rangle = \sum_n H_1 C_n(t)e^{-iE_n^{(0)}t/\hbar} | \psi_n \rangle \right)
\end{equation}
\begin{equation}
i\hbar \sum_n \dot{C}_n(t)e^{-iE_n^{(0)}t/\hbar} \delta_{m,n} = \sum_n C_n(t)e^{-iE_n^{(0)}t/\hbar} \langle \psi_m | H_1 | \psi_n \rangle
\end{equation}
\begin{equation}
i\hbar \dot{C}_m(t) = \sum_n C_n(t)e^{i(E_m^{(0)} – E_n^{(0)})t/\hbar} \langle \psi_m | H_1 | \psi_n \rangle
\end{equation}
\begin{equation}
\boxed{ \dot{C}_m(t) = -\frac{i}{\hbar} \sum_n C_n(t)e^{i\omega_{mn}t} \langle \psi_m | H_1 | \psi_n \rangle }
\end{equation}
\end{document}
Explanation:
- The equations describe the time evolution of coefficients Cn(t)C_n(t)Cn(t) in quantum mechanics.
- The boxed equation is a common result in time-dependent perturbation theory.
- The notation ⟨ψm∣H1∣ψn⟩\langle \psi_m | H_1 | \psi_n \rangle⟨ψm∣H1∣ψn⟩ represents the matrix elements of the perturbation Hamiltonian.
You can compile this LaTeX code in Overleaf or any LaTeX editor to reproduce the formatted output. Let me know if you need modifications! 😊”
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To go into more detail about producing this type of content in a manner that is most friendly to screen readers:
When it comes to LaTeX accessibility, the most recommended tool is Pandoc, which allows you to convert your LaTeX documents into HTML with MathML support, ensuring screen readers can interpret mathematical content effectively; this is considered the best way to make LaTeX documents accessible, especially for complex math equations.
Key points about using Pandoc for LaTeX accessibility:
-
Conversion to HTML:
Pandoc can convert your LaTeX file directly into HTML, which is generally better supported by screen readers than native PDF output from LaTeX.
-
MathML support:
When converting to HTML, Pandoc can translate your LaTeX math expressions into MathML, a standard format for representing mathematical notation that is well-read by screen readers.
-
Cross-platform compatibility:
Pandoc is available across various operating systems, making it widely accessible.
Other considerations for LaTeX accessibility:
-
Alt text:
Always provide descriptive alt text for images and figures within your LaTeX document.
-
Semantic structure:
Use appropriate LaTeX commands for headings, lists, and other structural elements to improve screen reader navigation.
-
Metadata inclusion:
Add relevant metadata like document title, author, and keywords to enhance accessibility.
Given this information, I went back to ChatGPT and asked if it could render it into html for me. It gave me this code:
”
<!DOCTYPE html>
<html lang=”en”>
<head>
<meta charset=”UTF-8″>
<meta name=”viewport” content=”width=device-width, initial-scale=1.0″>
<title>Quantum Mechanics Equations</title>
<script type=”text/javascript” async
src=”https://polyfill.io/v3/polyfill.min.js?features=es6″>
</script>
<script type=”text/javascript” async
src=”https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js”>
</script>
</head>
<body>
<h2>Quantum Mechanics Equations</h2>
<p>
\[
\langle \psi_m | \times \left( i\hbar \sum_n \dot{C}_n(t)e^{-iE_n^{(0)}t/\hbar} | \psi_n \rangle = \sum_n H_1 C_n(t)e^{-iE_n^{(0)}t/\hbar} | \psi_n \rangle \right)
\]
</p>
<p>
\[
i\hbar \sum_n \dot{C}_n(t)e^{-iE_n^{(0)}t/\hbar} \delta_{m,n} = \sum_n C_n(t)e^{-iE_n^{(0)}t/\hbar} \langle \psi_m | H_1 | \psi_n \rangle
\]
</p>
<p>
\[
i\hbar \dot{C}_m(t) = \sum_n C_n(t)e^{i(E_m^{(0)} – E_n^{(0)})t/\hbar} \langle \psi_m | H_1 | \psi_n \rangle
\]
</p>
<p>
\[
\boxed{ \dot{C}_m(t) = -\frac{i}{\hbar} \sum_n C_n(t)e^{i\omega_{mn}t} \langle \psi_m | H_1 | \psi_n \rangle }
\]
\]
</p>
</body>
</html>”
which when copied into a index.html file looks like this:
Which I believe looks just like the original image:
What is the point? This started with a picture and ended with screen readable html.
We can do this lift.
one more thing….
I tried to find handwritten math notes online and had a really difficult time. Most everything that came up was stock images. So I ended up drawing out the last line of the above formula.
Dropping this onto ChatGPT gave me this:
The handwritten equation in the image appears to be:
which also gave me the html code resulting in:
Feel free to send me handwritten notes to play with!
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