Various improvements to knowledge base

6 files changed, 59 insertions, 29 deletions

diff --git a/latex/know/concept/blochs-theorem/source.md b/latex/know/concept/blochs-theorem/source.md
index 528c218..79ee9a6 100644
--- a/latex/know/concept/blochs-theorem/source.md
+++ b/latex/know/concept/blochs-theorem/source.md
@@ -2,7 +2,7 @@
 
 
 # Bloch's theorem
-In quantum mechanics, *Bloch's theorem* states that,
+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 $\vec{a}$,
@@ -22,13 +22,38 @@ $$
 
 In other words, in a periodic potential,
 the solutions are simply plane waves with a periodic modulation,
-known as *Bloch functions* or *Bloch states*.
+known as **Bloch functions** or **Bloch states**.
 
 This is suprisingly easy to prove:
 if the Hamiltonian $\hat{H}$ is lattice-periodic,
-then it will commute with the unitary translation operator $\hat{T}(\vec{a})$,
+then both $\psi(\vec{r})$ and $\psi(\vec{r} + \vec{a})$
+are eigenstates with the same energy:
+
+$$
+\begin{aligned}
+	\hat{H} \psi(\vec{r}) = E \psi(\vec{r})
+	\qquad
+	\hat{H} \psi(\vec{r} + \vec{a}) = E \psi(\vec{r} + \vec{a})
+\end{aligned}
+$$
+
+Now define the unitary translation operator $\hat{T}(\vec{a})$ such that
+$\psi(\vec{r} + \vec{a}) = \hat{T}(\vec{a}) \psi(\vec{r})$.
+From the previous equation, we then know that:
+
+$$
+\begin{aligned}
+	\hat{H} \hat{T}(\vec{a}) \psi(\vec{r})
+	= E \hat{T}(\vec{a}) \psi(\vec{r})
+	= \hat{T}(\vec{a}) \big(E \psi(\vec{r})\big)
+	= \hat{T}(\vec{a}) \hat{H} \psi(\vec{r})
+\end{aligned}
+$$
+
+In other words, if $\hat{H}$ is lattice-periodic,
+then it will commute with $\hat{T}(\vec{a})$,
 i.e. $[\hat{H}, \hat{T}(\vec{a})] = 0$.
-Therefore $\hat{H}$ and $\hat{T}(\vec{a})$ must share eigenstates $\psi(\vec{r})$:
+Consequently, $\hat{H}$ and $\hat{T}(\vec{a})$ must share eigenstates $\psi(\vec{r})$:
 
 $$
 \begin{aligned}
diff --git a/latex/know/concept/dirac-notation/source.md b/latex/know/concept/dirac-notation/source.md
index 47aa370..f34047d 100644
--- a/latex/know/concept/dirac-notation/source.md
+++ b/latex/know/concept/dirac-notation/source.md
@@ -3,18 +3,18 @@
 
 # Dirac notation
 
-*Dirac notation* is a notation to do calculations in a Hilbert space
+**Dirac notation** is a notation to do calculations in a Hilbert space
 without needing to worry about the space's representation. It is
 basically the *lingua franca* of quantum mechanics.
 
-In Dirac notation there are *kets* $\ket{V}$ from the Hilbert space
-$\mathbb{H}$ and *bras* $\bra{V}$ from a dual $\mathbb{H}'$ of the
+In Dirac notation there are **kets** $\ket{V}$ from the Hilbert space
+$\mathbb{H}$ and **bras** $\bra{V}$ from a dual $\mathbb{H}'$ of the
 former. Crucially, the bras and kets are from different Hilbert spaces
 and therefore cannot be added, but every bra has a corresponding ket and
 vice versa.
 
-Bras and kets can only be combined in two ways: the *inner product*
-$\braket{V}{W}$, which returns a scalar, and the *outer product*
+Bras and kets can be combined in two ways: the **inner product**
+$\braket{V}{W}$, which returns a scalar, and the **outer product**
 $\ket{V} \bra{W}$, which returns a mapping $\hat{L}$ from kets $\ket{V}$
 to other kets $\ket{V'}$, i.e. a linear operator. Recall that the
 Hilbert inner product must satisfy:
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.
diff --git a/latex/know/concept/probability-current/source.md b/latex/know/concept/probability-current/source.md
index bffc599..bb84121 100644
--- a/latex/know/concept/probability-current/source.md
+++ b/latex/know/concept/probability-current/source.md
@@ -3,7 +3,7 @@
 
 # Probability current
 
-In quantum mechanics, the *probability current* describes the movement
+In quantum mechanics, the **probability current** describes the movement
 of the probability of finding a particle at given point in space.
 In other words, it treats the particle as a heterogeneous fluid with density $|\psi|^2$.
 Now, the probability of finding the particle within a volume $V$ is:
diff --git a/latex/know/concept/time-independent-perturbation-theory/source.md b/latex/know/concept/time-independent-perturbation-theory/source.md
index a3167cd..48504f4 100644
--- a/latex/know/concept/time-independent-perturbation-theory/source.md
+++ b/latex/know/concept/time-independent-perturbation-theory/source.md
@@ -3,8 +3,8 @@
 
 # Time-independent perturbation theory
 
-*Time-independent perturbation theory*, sometimes also called
-*stationary state perturbation theory*, is a specific application of
+**Time-independent perturbation theory**, sometimes also called
+**stationary state perturbation theory**, is a specific application of
 perturbation theory to the time-independent Schrödinger
 equation in quantum physics, for
 Hamiltonians of the following form:
@@ -27,8 +27,8 @@ $$\begin{aligned}
     &= E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + ...
 \end{aligned}$$
 
-Where $E_n^{(1)}$ and $\ket*{\psi_n^{(1)}}$ are called the *first-order
-corrections*, and so on for higher orders. We insert this into the
+Where $E_n^{(1)}$ and $\ket*{\psi_n^{(1)}}$ are called the **first-order
+corrections**, and so on for higher orders. We insert this into the
 Schrödinger equation:
 
 $$\begin{aligned}
@@ -72,6 +72,7 @@ $$\begin{aligned}
 The approach to solving the other two equations varies depending on
 whether this $\hat{H}_0$ has a degenerate spectrum or not.
 
+
 ## Without degeneracy
 
 We start by assuming that there is no degeneracy, in other words, each
@@ -178,6 +179,7 @@ $$\begin{aligned}
 
 In practice, it is not particulary useful to calculate more corrections.
 
+
 ## With degeneracy
 
 If $\varepsilon_n$ is $D$-fold degenerate, then its eigenstate could be
@@ -295,7 +297,7 @@ The trick is to find a Hermitian operator $\hat{L}$ (usually using
 symmetries of the system) which commutes with both $\hat{H}_0$ and $\hat{H}_1$:
 
 $$\begin{aligned}
-    [\hat{L}, \hat{H}_0] = [\hat{L}, \hat{H}_1] = 0
+    \comm*{\hat{L}}{\hat{H}_0} = \comm{\hat{L}}{\hat{H}_1} = 0
 \end{aligned}$$
 
 So that it shares its eigenstates with $\hat{H}_0$ (and $\hat{H}_1$),
diff --git a/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md b/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md
index a50302c..79f344a 100644
--- a/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md
+++ b/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md
@@ -3,10 +3,10 @@
 
 # Wentzel-Kramers-Brillouin approximation
 
-In quantum mechanics, the *Wentzel-Kramers-Brillouin* or simply the *WKB
-approximation* is a method to approximate the wave function $\psi(x)$ of
+In quantum mechanics, the **Wentzel-Kramers-Brillouin** or simply the **WKB
+approximation** is a method to approximate the wave function $\psi(x)$ of
 the one-dimensional time-independent Schrödinger equation. It is an example
-of a *semiclassical approximation*, because it tries to find a
+of a **semiclassical approximation**, because it tries to find a
 balance between classical and quantum physics.
 
 In classical mechanics, a particle travelling in a potential $V(x)$
@@ -164,7 +164,7 @@ $$\begin{aligned}
 
 What if $E < V$? In classical mechanics, this is not allowed; a ball
 cannot simply go through a potential bump without the necessary energy.
-However, in quantum mechanics, particles can *tunnel* through barriers.
+However, in quantum mechanics, particles can **tunnel** through barriers.
 
 Conveniently, all we need to change for the WKB approximation is to let
 the momentum take imaginary values: