From 6e70f28ccbd5afc1506f71f013278a9d157ef03a Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 27 Oct 2022 20:40:09 +0200
Subject: Optimize last images, add proof template, improve CSS

---
 source/know/concept/selection-rules/index.md | 46 +++++++++++-----------------
 1 file changed, 18 insertions(+), 28 deletions(-)

(limited to 'source/know/concept/selection-rules')

diff --git a/source/know/concept/selection-rules/index.md b/source/know/concept/selection-rules/index.md
index 373486e..620e345 100644
--- a/source/know/concept/selection-rules/index.md
+++ b/source/know/concept/selection-rules/index.md
@@ -25,6 +25,7 @@ between $$\ell_i$$, $$\ell_f$$, $$m_i$$ and $$m_f$$, which, if not met,
 guarantee that the above matrix element is zero.
 
 
+
 ## Parity rules
 
 Let $$\hat{O}$$ denote any operator which is odd under spatial inversion
@@ -73,6 +74,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 
+
 ## Dipole rules
 
 Arguably the most common operator found in such matrix elements
@@ -87,11 +89,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-<div class="accordion">
-<input type="checkbox" id="proof-dipole-m"/>
-<label for="proof-dipole-m">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-dipole-m">Proof.</label>
+
+{% include proof/start.html id="proof-dipole-m" -%}
 We know that the angular momentum $$z$$-component operator $$\hat{L}_z$$ satisfies:
 
 $$\begin{aligned}
@@ -166,8 +165,8 @@ whenever $$\matrixel{f}{\hat{z}}{i} \neq 0$$.
 Only if $$\matrixel{f}{\hat{z}}{i} = 0$$
 does the previous rule $$\Delta m = \pm 1$$ hold,
 in which case the inner products of $$\hat{x}$$ and $$\hat{y}$$ are nonzero.
-</div>
-</div>
+{% include proof/end.html id="proof-dipole-m" %}
+
 
 Meanwhile, for the total angular momentum $$\ell$$ we have the following:
 
@@ -177,11 +176,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-<div class="accordion">
-<input type="checkbox" id="proof-dipole-l"/>
-<label for="proof-dipole-l">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-dipole-l">Proof.</label>
+
+{% include proof/start.html id="proof-dipole-l" -%}
 We start from the following relation
 (which is already quite a chore to prove):
 
@@ -190,11 +186,8 @@ $$\begin{aligned}
     = 2 \hbar^2 (\vu{r} \hat{L}^2 + \hat{L}^2 \vu{r})
 \end{aligned}$$
 
-<div class="accordion">
-<input type="checkbox" id="proof-dipole-l-comm"/>
-<label for="proof-dipole-l-comm">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-dipole-l-comm">Proof.</label>
+
+{% include proof/start.html id="proof-dipole-l-commutator" -%}
 To begin with, we want to find the commutator of $$\hat{L}^2$$ and $$\hat{x}$$:
 
 $$\begin{aligned}
@@ -364,8 +357,8 @@ $$\begin{aligned}
 
 At last, this brings us to the desired equation for $$\comm{\hat{L}^2}{\comm{\hat{L}^2}{\vu{r}}}$$,
 with $$\vu{r} = (\hat{x}, \hat{y}, \hat{z})$$.
-</div>
-</div>
+{% include proof/end.html id="proof-dipole-l-commutator" %}
+
 
 We then multiply this relation by $$\Bra{f} = \Bra{\ell_f m_f}$$ on the left
 and $$\Ket{i} = \Ket{\ell_i m_i}$$ on the right,
@@ -458,9 +451,8 @@ $$\begin{aligned}
     (\ell_f - \ell_i)^2
     = 1
 \end{aligned}$$
+{% include proof/end.html id="proof-dipole-l" %}
 
-</div>
-</div>
 
 
 ## Rotational rules
@@ -502,11 +494,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-<div class="accordion">
-<input type="checkbox" id="proof-rot-scalar"/>
-<label for="proof-rot-scalar">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-rot-scalar">Proof.</label>
+
+{% include proof/start.html id="proof-rotation-scalar" -%}
 Firstly, we look at the commutator of $$\hat{s}$$ with the $$z$$-component $$\hat{L}_z$$:
 $$\begin{aligned}
     0
@@ -578,8 +567,8 @@ $$\begin{aligned}
 
 Which means that the value of the matrix element
 does not depend on $$m_i$$ (or $$m_f$$) at all.
-</div>
-</div>
+{% include proof/end.html id="proof-rotation-scalar" %}
+
 
 Similarly, given a general (pseudo)vector operator $$\vu{V}$$,
 which, by nature, must satisfy the following commutation relations,
@@ -631,6 +620,7 @@ $$\begin{gathered}
 \end{gathered}$$
 
 
+
 ## Superselection rule
 
 Selection rules are not always about atomic electron transitions, or angular momenta even.
-- 
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