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

1 files changed, 28 insertions, 28 deletions

diff --git a/source/know/concept/kubo-formula/index.md b/source/know/concept/kubo-formula/index.md
index 80309c5..4cb39ac 100644
--- a/source/know/concept/kubo-formula/index.md
+++ b/source/know/concept/kubo-formula/index.md
@@ -10,19 +10,19 @@ layout: "concept"
 ---
 
 Consider the following quantum Hamiltonian,
-split into a main time-independent term $\hat{H}_{0,S}$
-and a small time-dependent perturbation $\hat{H}_{1,S}$,
-which is turned on at $t = t_0$:
+split into a main time-independent term $$\hat{H}_{0,S}$$
+and a small time-dependent perturbation $$\hat{H}_{1,S}$$,
+which is turned on at $$t = t_0$$:
 
 $$\begin{aligned}
     \hat{H}_S(t)
     = \hat{H}_{0,S} + \hat{H}_{1,S}(t)
 \end{aligned}$$
 
-And let $\Ket{\psi_S(t)}$ be the corresponding solutions to the Schrödinger equation.
-Then, given a time-independent observable $\hat{A}$,
-its expectation value $\expval{\hat{A}}$ evolves like so,
-where the subscripts $S$ and $I$
+And let $$\Ket{\psi_S(t)}$$ be the corresponding solutions to the Schrödinger equation.
+Then, given a time-independent observable $$\hat{A}$$,
+its expectation value $$\expval{\hat{A}}$$ evolves like so,
+where the subscripts $$S$$ and $$I$$
 respectively refer to the Schrödinger
 and [interaction pictures](/know/concept/interaction-picture/):
 
@@ -34,7 +34,7 @@ $$\begin{aligned}
     &= \matrixel{\psi_I(t_0)\,}{\,\hat{K}_I^\dagger(t, t_0) \hat{A}_I(t) \hat{K}_I(t, t_0)\,}{\,\psi_I(t_0)}
 \end{aligned}$$
 
-Where the time evolution operator $\hat{K}_I(t, t_0)$ is as follows,
+Where the time evolution operator $$\hat{K}_I(t, t_0)$$ is as follows,
 which we Taylor-expand:
 
 $$\begin{aligned}
@@ -56,7 +56,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Where we have dropped the last term,
-because $\hat{H}_{1}$ is assumed to be so small
+because $$\hat{H}_{1}$$ is assumed to be so small
 that it only matters to first order.
 Here, we notice a commutator, so we can rewrite:
 
@@ -65,10 +65,10 @@ $$\begin{aligned}
     &= \hat{A}_I(t) - \frac{i}{\hbar} \int_{t_0}^t \Comm{\hat{A}_I(t)}{\hat{H}_{1,I}(t')} \dd{t'}
 \end{aligned}$$
 
-Returning to $\expval{\hat{A}}$,
+Returning to $$\expval{\hat{A}}$$,
 we have the following formula,
-where $\Expval{}$ is the expectation value for $\Ket{\psi(t)}$,
-and $\Expval{}_0$ is the expectation value for $\Ket{\psi_I(t_0)}$:
+where $$\Expval{}$$ is the expectation value for $$\Ket{\psi(t)}$$,
+and $$\Expval{}_0$$ is the expectation value for $$\Ket{\psi_I(t_0)}$$:
 
 $$\begin{aligned}
     \expval{\hat{A}}(t)
@@ -76,9 +76,9 @@ $$\begin{aligned}
     = \expval{\hat{A}_I(t)}_0 - \frac{i}{\hbar} \int_{t_0}^t \Expval{\Comm{\hat{A}_I(t)}{\hat{H}_{1,I}(t')}}_0 \dd{t'}
 \end{aligned}$$
 
-Now we define $\delta\!\expval{\hat{A}}\!(t)$
-as the change of $\expval{\hat{A}}$ due to the perturbation $\hat{H}_1$,
-and insert $\expval{\hat{A}}(t)$:
+Now we define $$\delta\!\expval{\hat{A}}\!(t)$$
+as the change of $$\expval{\hat{A}}$$ due to the perturbation $$\hat{H}_1$$,
+and insert $$\expval{\hat{A}}(t)$$:
 
 $$\begin{aligned}
     \delta\!\expval{\hat{A}}\!(t)
@@ -87,10 +87,10 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Finally, we introduce
-a [Heaviside step function](/know/concept/heaviside-step-function) $\Theta$
+a [Heaviside step function](/know/concept/heaviside-step-function) $$\Theta$$
 and change the integration limit accordingly,
 leading to the **Kubo formula**
-describing the response of $\expval{\hat{A}}$ to first order in $\hat{H}_1$:
+describing the response of $$\expval{\hat{A}}$$ to first order in $$\hat{H}_1$$:
 
 $$\begin{aligned}
     \boxed{
@@ -99,7 +99,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where we have defined the **retarded correlation function** $C^R_{A H_1}(t, t')$ as follows:
+Where we have defined the **retarded correlation function** $$C^R_{A H_1}(t, t')$$ as follows:
 
 $$\begin{aligned}
     \boxed{
@@ -115,10 +115,10 @@ of particle creation/annihiliation operators.
 Therefore, this correlation function
 is a two-particle [Green's function](/know/concept/greens-functions/).
 
-A common situation is that $\hat{H}_1$ consists of
-a time-independent operator $\hat{B}$
-and a time-dependent function $f(t)$,
-allowing us to split $C^R_{A H_1}$ as follows:
+A common situation is that $$\hat{H}_1$$ consists of
+a time-independent operator $$\hat{B}$$
+and a time-dependent function $$f(t)$$,
+allowing us to split $$C^R_{A H_1}$$ as follows:
 
 $$\begin{aligned}
     \hat{H}_{1,S}(t)
@@ -128,10 +128,10 @@ $$\begin{aligned}
     = C^R_{A B}(t, t') f(t')
 \end{aligned}$$
 
-Since $C_{AB}^R$ is a Green's function,
-we know that it only depends on the difference $t - t'$,
+Since $$C_{AB}^R$$ is a Green's function,
+we know that it only depends on the difference $$t - t'$$,
 as long as the system was initially in thermodynamic equilibrium,
-and $\hat{H}_{0,S}$ is time-independent:
+and $$\hat{H}_{0,S}$$ is time-independent:
 
 $$\begin{aligned}
     C^R_{A B}(t, t')
@@ -139,7 +139,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 With this, the Kubo formula can be written as follows,
-where we have set $t_0 = - \infty$:
+where we have set $$t_0 = - \infty$$:
 
 $$\begin{aligned}
     \delta\!\expval{A}\!(t)
@@ -150,8 +150,8 @@ $$\begin{aligned}
 This is a convolution,
 so the [convolution theorem](/know/concept/convolution-theorem/)
 states that the [Fourier transform](/know/concept/fourier-transform/)
-of $\delta\!\expval{\hat{A}}\!(t)$ is simply the product
-of the transforms of $C^R_{AB}$ and $f$:
+of $$\delta\!\expval{\hat{A}}\!(t)$$ is simply the product
+of the transforms of $$C^R_{AB}$$ and $$f$$:
 
 $$\begin{aligned}
     \boxed{