From e71c14aa725d71a2ea7310c69b3d11a8bc12c0b0 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sat, 20 Feb 2021 13:46:18 +0100 Subject: Add latex/ directory + fix MathJax renderer --- latex/Makefile | 13 +++++ latex/know/concept/blochs-theorem/source.md | 81 +++++++++++++++++++++++++++ latex/main.css | 24 ++++++++ latex/template.html | 36 ++++++++++++ static/know/concept/blochs-theorem/index.html | 2 +- 5 files changed, 155 insertions(+), 1 deletion(-) create mode 100644 latex/Makefile create mode 100644 latex/know/concept/blochs-theorem/source.md create mode 100644 latex/main.css create mode 100644 latex/template.html diff --git a/latex/Makefile b/latex/Makefile new file mode 100644 index 0000000..590d33d --- /dev/null +++ b/latex/Makefile @@ -0,0 +1,13 @@ + +CONV = pandoc +CONVFLAGS = --from markdown --to html --standalone --template template.html --mathjax=/mathjax/tex-svg.js + +know: $(shell find ./know/concept -mindepth 1 -maxdepth 1 -type d -printf "%p/index.html\\n") + for file in $^; do \ + mkdir -p ../static/`dirname $$file` ; \ + cp -v $$file ../static/`dirname $$file`/ ; \ + done + +know/concept/%/index.html: know/concept/%/source.md template.html main.css + $(CONV) $(CONVFLAGS) $< -o $@ + diff --git a/latex/know/concept/blochs-theorem/source.md b/latex/know/concept/blochs-theorem/source.md new file mode 100644 index 0000000..2307d2e --- /dev/null +++ b/latex/know/concept/blochs-theorem/source.md @@ -0,0 +1,81 @@ +% Bloch's theorem + + +# Bloch's theorem +In quantum mechanics, *Bloch's theorem* states that, +given a potential $V(\vec{r})$ which is periodic on a lattice, +i.e. $V(\vec{r}) = V(\vec{r} + \vec{a})$ +for a primitive lattice vector $\vec{a}$, +then it follows that the solutions $\psi(\vec{r})$ +to the time-independent Schrödinger equation +take the following form, +where the function $u(\vec{r})$ is periodic on the same lattice, +i.e. $u(\vec{r}) = u(\vec{r} + \vec{a})$: + +$$ +\begin{aligned} + \boxed{ + \psi(\vec{r}) = u(\vec{r}) e^{i \vec{k} \cdot \vec{r}} + } +\end{aligned} +$$ + +In other words, in a periodic potential, +the solutions are simply plane waves with a periodic modulation, +known as *Bloch functions* or *Bloch states*. + +This is suprisingly easy to prove: +if the Hamiltonian $\hat{H}$ is lattice-periodic, +then it will commute with the unitary translation operator $\hat{T}(\vec{a})$, +i.e. $\comm{\hat{H}}{\hat{T}(\vec{a})} = 0$. +Therefore $\hat{H}$ and $\hat{T}(\vec{a})$ must share eigenstates $\psi(\vec{r})$: + +$$ +\begin{aligned} + \hat{H} \:\psi(\vec{r}) = E \:\psi(\vec{r}) + \qquad + \hat{T}(\vec{a}) \:\psi(\vec{r}) = \tau \:\psi(\vec{r}) +\end{aligned} +$$ + +Since $\hat{T}$ is unitary, +its eigenvalues $\tau$ must have the form $e^{i \theta}$, with $\theta$ real. +Therefore a translation by $\vec{a}$ causes a phase shift, +for some vector $\vec{k}$: + +$$ +\begin{aligned} + \psi(\vec{r} + \vec{a}) + = \hat{T}(\vec{a}) \:\psi(\vec{r}) + = e^{i \theta} \:\psi(\vec{r}) + = e^{i \vec{k} \cdot \vec{a}} \:\psi(\vec{r}) +\end{aligned} +$$ + +Let us now define the following function, +keeping our arbitrary choice of $\vec{k}$: + +$$ +\begin{aligned} + u(\vec{r}) + = e^{- i \vec{k} \cdot \vec{r}} \:\psi(\vec{r}) +\end{aligned} +$$ + +As it turns out, this function is guaranteed to be lattice-periodic for any $\vec{k}$: + +$$ +\begin{aligned} + u(\vec{r} + \vec{a}) + &= e^{- i \vec{k} \cdot (\vec{r} + \vec{a})} \:\psi(\vec{r} + \vec{a}) + \\ + &= e^{- i \vec{k} \cdot \vec{r}} e^{- i \vec{k} \cdot \vec{a}} e^{i \vec{k} \cdot \vec{a}} \:\psi(\vec{r}) + \\ + &= e^{- i \vec{k} \cdot \vec{r}} \:\psi(\vec{r}) + \\ + &= u(\vec{r}) +\end{aligned} +$$ + +Then Bloch's theorem follows from +isolating the definition of $u(\vec{r})$ for $\psi(\vec{r})$. diff --git a/latex/main.css b/latex/main.css new file mode 100644 index 0000000..390858b --- /dev/null +++ b/latex/main.css @@ -0,0 +1,24 @@ +body { + background:#ddd; + color:#222; + max-width:80ch; + text-align:justify; + margin:auto; + padding:1em 0; + font-family:sans-serif; + line-height:1.3; +} +a {text-decoration:none;color:#00f;} +h1,h2,h3 {text-align:center} +h1 {font-size:200%;} +h2 {font-size:160%;} +h3 {font-size:120%;} +.nav {height:3rem;font-size:250%;} +.nav a:link,a:visited {color:#222;} +.nav a:hover,a:focus,a:active {color:#00f;} +.navl {width:30%;float:left;text-align:left;} +.navr {width:70%;float:left;text-align:right;} +pre {filter:invert(100%);} +@media (prefers-color-scheme: dark) { + body {background:#222;filter:invert(100%);} +} diff --git a/latex/template.html b/latex/template.html new file mode 100644 index 0000000..3764f3a --- /dev/null +++ b/latex/template.html @@ -0,0 +1,36 @@ + + + + + + + Prefetch | $pagetitle$ + + + + $if(math)$ + $math$ + $endif$ + + +
+
PREFETCH
+
+blog  +code  +know +
+
+
+$body$ +
+© "Prefetch". Licensed under CC BY-SA 4.0. + + diff --git a/static/know/concept/blochs-theorem/index.html b/static/know/concept/blochs-theorem/index.html index d900aba..6e5767c 100644 --- a/static/know/concept/blochs-theorem/index.html +++ b/static/know/concept/blochs-theorem/index.html @@ -37,7 +37,7 @@ tex: {packages: {"[+]": ["physics"]}} }; - +
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