From f7d88d0ae1c5544cc517ae6b25970f82d82b1496 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 20 Feb 2021 13:35:51 +0100
Subject: Complete switch to MathJax
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content/_index.md | 6 +++---
static/know/concept/blochs-theorem/index.html | 13 +++++++++----
2 files changed, 12 insertions(+), 7 deletions(-)
diff --git a/content/_index.md b/content/_index.md
index 2e49ddb..155442f 100644
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Welcome to my personal website.
When the stars align, I'll post something here
related to my areas of interest:
-open-source software, programming, optimization, server management,
-mathematics, physics and sometimes even linguistics.
+open-source software, server management, programming, optimization,
+mathematics, physics and even linguistics.
This website is served by [nginx](https://nginx.org/),
and built from scratch using the awesome
[Zola](https://www.getzola.org/) static generation tool.
-I also aim to keep it forever free of advertising, trackers, and JavaScript,
+I also aim to keep it forever free of advertising and trackers,
and to maintain my A+ score for [TLS quality](https://www.ssllabs.com/ssltest/analyze.html?d=prefetch.eu).
diff --git a/static/know/concept/blochs-theorem/index.html b/static/know/concept/blochs-theorem/index.html
index 9710710..d900aba 100644
--- a/static/know/concept/blochs-theorem/index.html
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Prefetch | Bloch’s theorem
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-
+
+
@@ -42,7 +49,6 @@
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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. \(u(\vec{r}) = u(\vec{r} + \vec{a})\):
\[
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\end{aligned}
\]
In other words, in a periodic potential, the solutions are simply plane waves with a periodic modulation, known as Bloch functions or Bloch states.
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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. \([\hat{H}, \hat{T}(\vec{a})] = 0\). Therefore \(\hat{H}\) and \(\hat{T}(\vec{a})\) must share eigenstates \(\psi(\vec{r})\):
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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 \(\psi(\vec{r})\):