Various improvements to knowledge base

1 files changed, 13 insertions, 10 deletions

diff --git a/latex/know/concept/pauli-exclusion-principle/source.md b/latex/know/concept/pauli-exclusion-principle/source.md
index 8870c0c..d1c2149 100644
--- a/latex/know/concept/pauli-exclusion-principle/source.md
+++ b/latex/know/concept/pauli-exclusion-principle/source.md
@@ -3,7 +3,7 @@
 
 # Pauli exclusion principle
 
-In quantum mechanics, the *Pauli exclusion principle* is a theorem that
+In quantum mechanics, the **Pauli exclusion principle** is a theorem that
 has profound consequences for how the world works.
 
 Suppose we have a composite state
@@ -26,14 +26,17 @@ $$\begin{aligned}
 Therefore, $\ket{a}\ket{b}$ is an eigenvector of $\hat{P}^2$ with
 eigenvalue $1$. Since $[\hat{P}, \hat{P}^2] = 0$, $\ket{a}\ket{b}$
 must also be an eigenket of $\hat{P}$ with eigenvalue $\lambda$,
-satisfying $\lambda^2 = 1$, so we know that $\lambda = 1$ or
-$\lambda = -1$.
+satisfying $\lambda^2 = 1$, so we know that $\lambda = 1$ or $\lambda = -1$:
+
+$$\begin{aligned}
+    \hat{P} \ket{a}\ket{b} = \lambda \ket{a}\ket{b}
+\end{aligned}$$
 
 As it turns out, in nature, each class of particle has a single
 associated permutation eigenvalue $\lambda$, or in other words: whether
 $\lambda$ is $-1$ or $1$ depends on the species of particle that $x_1$
 and $x_2$ represent. Particles with $\lambda = -1$ are called
-*fermions*, and those with $\lambda = 1$ are known as *bosons*. We
+**fermions**, and those with $\lambda = 1$ are known as **bosons**. We
 define $\hat{P}_f$ with $\lambda = -1$ and $\hat{P}_b$ with
 $\lambda = 1$, such that:
 
@@ -80,14 +83,14 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Where $C$ is a normalization constant. As expected, this state is
-*symmetric*: switching $a$ and $b$ gives the same result. Meanwhile, for
+**symmetric**: switching $a$ and $b$ gives the same result. Meanwhile, for
 fermions ($\lambda = -1$), we find that $\alpha = -\beta$:
 
 $$\begin{aligned}
     \ket{\Psi(a, b)}_f = C \big( \ket{a}\ket{b} - \ket{b}\ket{a} \big)
 \end{aligned}$$
 
-This state is called *antisymmetric* under exchange: switching $a$ and $b$
+This state is called **antisymmetric** under exchange: switching $a$ and $b$
 causes a sign change, as we would expect for fermions.
 
 Now, what if the particles $x_1$ and $x_2$ are in the same state $a$?
@@ -106,7 +109,7 @@ $$\begin{aligned}
     = 0
 \end{aligned}$$
 
-At last, this is the Pauli exclusion principle: fermions may never
-occupy the same quantum state. One of the many notable consequences of
-this is that the shells of an atom only fit a limited number of
-electrons, since each must have a different quantum number.
+At last, this is the Pauli exclusion principle: **fermions may never
+occupy the same quantum state**. One of the many notable consequences of
+this is that the shells of atoms only fit a limited number of
+electrons (which are fermions), since each must have a different quantum number.