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-rw-r--r--source/know/concept/lindhard-function/index.md23
1 files changed, 9 insertions, 14 deletions
diff --git a/source/know/concept/lindhard-function/index.md b/source/know/concept/lindhard-function/index.md
index 4033148..fd620df 100644
--- a/source/know/concept/lindhard-function/index.md
+++ b/source/know/concept/lindhard-function/index.md
@@ -149,11 +149,8 @@ $$\begin{aligned}
= \hat{n}^\dagger(\vb{q})
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-density"/>
-<label for="proof-density">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-density">Proof.</label>
+
+{% include proof/start.html id="proof-density" -%}
Starting from the general definition of $$\hat{n}$$,
we write out the field operators $$\hat{\Psi}(\vb{r})$$,
and insert the known non-interacting single-electron orbitals
@@ -210,8 +207,8 @@ $$\begin{aligned}
The summation variable $$\vb{k}$$ has an associated spin $$\sigma$$,
and $$\hat{n}$$ does not carry any spin.
-</div>
-</div>
+{% include proof/end.html id="proof-density" %}
+
When neglecting interactions, it is tradition to rename $$\chi$$ to $$\chi_0$$.
We insert $$\hat{n}$$, suppressing spin:
@@ -290,12 +287,10 @@ $$\begin{aligned}
= \hat{c}_{\vb{k}}^\dagger \hat{c}_{\vb{k}} - \hat{c}_{\vb{k} + \vb{q}}^\dagger \hat{c}_{\vb{k} + \vb{q}}
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-commutator"/>
-<label for="proof-commutator">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-commutator">Proof.</label>
+
+{% include proof/start.html id="proof-commutator" -%}
In general, for any single-particle states labeled by $$m$$, $$n$$, $$o$$ and $$p$$, we have:
+
$$\begin{aligned}
\comm{\hat{c}_m^\dagger \hat{c}_n}{\hat{c}_o^\dagger \hat{c}_p}
&= \hat{c}_m^\dagger \hat{c}_n \hat{c}_o^\dagger \hat{c}_p - \hat{c}_o^\dagger \hat{c}_p \hat{c}_m^\dagger \hat{c}_n
@@ -319,8 +314,8 @@ $$\begin{aligned}
In this case, $$m = p = \vb{k}$$ and $$n = o = \vb{k} \!+\! \vb{q}$$,
so the Kronecker deltas are unnecessary.
-</div>
-</div>
+{% include proof/end.html id="proof-commutator" %}
+
We substitute this result into $$\chi_0$$,
and reintroduce the spin index $$\sigma$$ associated with $$\vb{k}$$: