Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

1 files changed, 32 insertions, 32 deletions

diff --git a/source/know/concept/second-quantization/index.md b/source/know/concept/second-quantization/index.md
index 975921c..e446557 100644
--- a/source/know/concept/second-quantization/index.md
+++ b/source/know/concept/second-quantization/index.md
@@ -20,13 +20,13 @@ known as the **Fock states**, which are specific members of a **Fock space**,
 a special kind of [Hilbert space](/know/concept/hilbert-space/),
 with a well-defined number of particles.
 
-For a set of $N$ single-particle energy eigenstates
-$\psi_n(x)$ and $N$ identical particles $x_n$, the Fock states are
-all the wave functions which contain $n$ particles, for $n$ going from $0$ to $N$.
+For a set of $$N$$ single-particle energy eigenstates
+$$\psi_n(x)$$ and $$N$$ identical particles $$x_n$$, the Fock states are
+all the wave functions which contain $$n$$ particles, for $$n$$ going from $$0$$ to $$N$$.
 
-So for $n = 0$, there is one basis vector with $0$ particles,
-for $n = 1$, there are $N$ basis vectors with $1$ particle each,
-for $n = 2$, there are $N (N \!-\! 1)$ basis vectors with $2$ particles,
+So for $$n = 0$$, there is one basis vector with $$0$$ particles,
+for $$n = 1$$, there are $$N$$ basis vectors with $$1$$ particle each,
+for $$n = 2$$, there are $$N (N \!-\! 1)$$ basis vectors with $$2$$ particles,
 etc.
 
 In this basis, we define the **particle creation operators**
@@ -60,10 +60,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The notation $\Ket{N_\alpha, N_\beta, ...}$ is shorthand for
+The notation $$\Ket{N_\alpha, N_\beta, ...}$$ is shorthand for
 the appropriate [Slater determinants](/know/concept/slater-determinant/).
-As an example, take $\Ket{0, 1, 0, 1, 1}$,
-which contains three particles $a$, $b$ and $c$
+As an example, take $$\Ket{0, 1, 0, 1, 1}$$,
+which contains three particles $$a$$, $$b$$ and $$c$$
 in states 2, 4 and 5:
 
 $$\begin{aligned}
@@ -77,9 +77,9 @@ $$\begin{aligned}
     \end{bmatrix}
 \end{aligned}$$
 
-The creation operator $\hat{c}_\alpha^\dagger$ and annihilation
-operator $\hat{c}_\alpha$ are defined to live up to their name:
-they create or destroy a particle in the state $\psi_\alpha$:
+The creation operator $$\hat{c}_\alpha^\dagger$$ and annihilation
+operator $$\hat{c}_\alpha$$ are defined to live up to their name:
+they create or destroy a particle in the state $$\psi_\alpha$$:
 
 $$\begin{aligned}
     \boxed{
@@ -93,16 +93,16 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The factor $J_\alpha$ is sometimes known as the **Jordan-Wigner string**,
+The factor $$J_\alpha$$ is sometimes known as the **Jordan-Wigner string**,
 and is necessary here to enforce the fermionic antisymmetry,
-when creating or destroying a particle in the $\alpha$th state:
+when creating or destroying a particle in the $$\alpha$$th state:
 
 $$\begin{aligned}
     J_\alpha = (-1)^{\sum_{j < \alpha} N_j}
 \end{aligned}$$
 
 So, for example, when creating a particle in state 4
-of $\Ket{0, 1, 1, 0, 1}$, we get the following:
+of $$\Ket{0, 1, 1, 0, 1}$$, we get the following:
 
 $$\begin{aligned}
     \hat{c}_4^\dagger \Ket{0, 1, 1, 0, 1}
@@ -122,8 +122,8 @@ $$\begin{aligned}
     = - \Ket{1, 0}
 \end{aligned}$$
 
-In other words, $\hat{c}_1^\dagger \hat{c}_2 = - \hat{c}_2 \hat{c}_1^\dagger$,
-meaning that the anticommutator $\{\hat{c}_2, \hat{c}_1^\dagger\} = 0$.
+In other words, $$\hat{c}_1^\dagger \hat{c}_2 = - \hat{c}_2 \hat{c}_1^\dagger$$,
+meaning that the anticommutator $$\{\hat{c}_2, \hat{c}_1^\dagger\} = 0$$.
 You can verify for youself that
 the general anticommutators of these operators are given by:
 
@@ -154,7 +154,7 @@ $$\begin{aligned}
     = \matrixel{...(N_\alpha\!=\!0) ...}{\hat{c}_\alpha}{... (N_\alpha\!=\!1) ...}
 \end{aligned}$$
 
-Let us now use these operators to define the **number operator** $\hat{N}_\alpha$ as follows:
+Let us now use these operators to define the **number operator** $$\hat{N}_\alpha$$ as follows:
 
 $$\begin{aligned}
     \boxed{
@@ -162,7 +162,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Its eigenvalue is the number of particles residing in state $\psi_\alpha$
+Its eigenvalue is the number of particles residing in state $$\psi_\alpha$$
 (look at the hats):
 
 $$\begin{aligned}
@@ -198,8 +198,8 @@ They must be symmetric under the exchange of two bosons.
 To achieve this, the Fock states are represented by Slater *permanents*
 rather than determinants.
 
-The boson creation and annihilation operators $\hat{c}_\alpha^\dagger$ and
-$\hat{c}_\alpha$ are straightforward:
+The boson creation and annihilation operators $$\hat{c}_\alpha^\dagger$$ and
+$$\hat{c}_\alpha$$ are straightforward:
 
 $$\begin{gathered}
     \boxed{
@@ -212,8 +212,8 @@ $$\begin{gathered}
         \end{aligned}
 }\end{gathered}$$
 
-Applying the annihilation operator $\hat{c}_\alpha$ when there are zero
-particles in $\alpha$ will quench the state:
+Applying the annihilation operator $$\hat{c}_\alpha$$ when there are zero
+particles in $$\alpha$$ will quench the state:
 
 $$\begin{aligned}
     \boxed{
@@ -232,8 +232,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The constant factors applied by $\hat{c}_\alpha^\dagger$ and $\hat{c}_\alpha$
-ensure that $\hat{N}_\alpha$ keeps the same nice form:
+The constant factors applied by $$\hat{c}_\alpha^\dagger$$ and $$\hat{c}_\alpha$$
+ensure that $$\hat{N}_\alpha$$ keeps the same nice form:
 
 $$\begin{aligned}
     \boxed{
@@ -244,8 +244,8 @@ $$\begin{aligned}
 
 ## Operators
 
-Traditionally, an operator $\hat{V}$ simultaneously acting on $N$ indentical particles
-is the sum of the individual single-particle operators $\hat{V}_1$ acting on the $n$th particle:
+Traditionally, an operator $$\hat{V}$$ simultaneously acting on $$N$$ indentical particles
+is the sum of the individual single-particle operators $$\hat{V}_1$$ acting on the $$n$$th particle:
 
 $$\begin{aligned}
     \hat{V}
@@ -261,7 +261,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where the matrix element $\matrixel{\alpha}{\hat{V}_1}{\beta}$ is to be
+Where the matrix element $$\matrixel{\alpha}{\hat{V}_1}{\beta}$$ is to be
 evaluated in the normal way:
 
 $$\begin{aligned}
@@ -269,7 +269,7 @@ $$\begin{aligned}
     = \int \psi_\alpha^*(\vec{r}) \: \hat{V}_1(\vec{r}) \: \psi_\beta(\vec{r}) \dd{\vec{r}}
 \end{aligned}$$
 
-Similarly, given some two-particle operator $\hat{V}$ in first-quantized form:
+Similarly, given some two-particle operator $$\hat{V}$$ in first-quantized form:
 
 $$\begin{aligned}
     \hat{V}
@@ -287,7 +287,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where the constant $v_{\alpha \beta \gamma \delta}$ is defined from the
+Where the constant $$v_{\alpha \beta \gamma \delta}$$ is defined from the
 single-particle wave functions:
 
 $$\begin{aligned}
@@ -306,9 +306,9 @@ $$\begin{aligned}
     = \sum_{\alpha} \Inprod{\alpha}{b} \hat{c}_\alpha^\dagger \Ket{0}
 \end{aligned}$$
 
-Where $\alpha$ and $b$ need not be in the same basis.
+Where $$\alpha$$ and $$b$$ need not be in the same basis.
 With this, we can define the **field operators**,
-which create or destroy a particle at a given position $\vec{r}$:
+which create or destroy a particle at a given position $$\vec{r}$$:
 
 $$\begin{aligned}
     \boxed{