Improve knowledge baseHEADmaster

1 files changed, 15 insertions, 10 deletions

diff --git a/source/know/concept/blochs-theorem/index.md b/source/know/concept/blochs-theorem/index.md
index d7fcf90..c6278f3 100644
--- a/source/know/concept/blochs-theorem/index.md
+++ b/source/know/concept/blochs-theorem/index.md
@@ -12,14 +12,14 @@ given a potential $$V(\vb{r})$$ which is periodic on a lattice,
 i.e. $$V(\vb{r}) = V(\vb{r} + \vb{a})$$
 for a primitive lattice vector $$\vb{a}$$,
 then it follows that the solutions $$\psi(\vb{r})$$
-to the time-independent Schrödinger equation
-take the following form,
+to the time-independent Schrödinger equation take the following form,
 where the function $$u(\vb{r})$$ is periodic on the same lattice,
 i.e. $$u(\vb{r}) = u(\vb{r} + \vb{a})$$:
 
 $$\begin{aligned}
 	\boxed{
-		\psi(\vb{r}) = u(\vb{r}) e^{i \vb{k} \cdot \vb{r}}
+		\psi(\vb{r})
+		= u(\vb{r}) e^{i \vb{k} \cdot \vb{r}}
 	}
 \end{aligned}$$
 
@@ -33,9 +33,11 @@ then both $$\psi(\vb{r})$$ and $$\psi(\vb{r} + \vb{a})$$
 are eigenstates with the same energy:
 
 $$\begin{aligned}
-	\hat{H} \psi(\vb{r}) = E \psi(\vb{r})
-	\qquad
-	\hat{H} \psi(\vb{r} + \vb{a}) = E \psi(\vb{r} + \vb{a})
+	\hat{H} \psi(\vb{r})
+	= E \psi(\vb{r})
+	\qquad \qquad
+	\hat{H} \psi(\vb{r} + \vb{a})
+	= E \psi(\vb{r} + \vb{a})
 \end{aligned}$$
 
 Now define the unitary translation operator $$\hat{T}(\vb{a})$$ such that
@@ -52,18 +54,21 @@ $$\begin{aligned}
 In other words, if $$\hat{H}$$ is lattice-periodic,
 then it will commute with $$\hat{T}(\vb{a})$$,
 i.e. $$[\hat{H}, \hat{T}(\vb{a})] = 0$$.
-Consequently, $$\hat{H}$$ and $$\hat{T}(\vb{a})$$ must share eigenstates $$\psi(\vb{r})$$:
+Consequently, $$\hat{H}$$ and $$\hat{T}(\vb{a})$$
+must share eigenstates $$\psi(\vb{r})$$:
 
 $$\begin{aligned}
-	\hat{H} \:\psi(\vb{r}) = E \:\psi(\vb{r})
+	\hat{H} \psi(\vb{r})
+	= E \psi(\vb{r})
 	\qquad \qquad
-	\hat{T}(\vb{a}) \:\psi(\vb{r}) = \tau \:\psi(\vb{r})
+	\hat{T}(\vb{a}) \psi(\vb{r})
+	= \tau \psi(\vb{r})
 \end{aligned}$$
 
 Since $$\hat{T}$$ is unitary,
 its eigenvalues $$\tau$$ must have the form $$e^{i \theta}$$, with $$\theta$$ real.
 Therefore a translation by $$\vb{a}$$ causes a phase shift,
-for some vector $$\vb{k}$$:
+so there exists a vector $$\vb{k}$$ such that:
 
 $$\begin{aligned}
 	\psi(\vb{r} + \vb{a})