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

1 files changed, 34 insertions, 34 deletions

diff --git a/source/know/concept/path-integral-formulation/index.md b/source/know/concept/path-integral-formulation/index.md
index b92cf5f..a8dcc76 100644
--- a/source/know/concept/path-integral-formulation/index.md
+++ b/source/know/concept/path-integral-formulation/index.md
@@ -14,10 +14,10 @@ which is equivalent to the "traditional" Schrödinger equation.
 Whereas the latter is based on [Hamiltonian mechanics](/know/concept/hamiltonian-mechanics/),
 the former comes from [Lagrangian mechanics](/know/concept/lagrangian-mechanics/).
 
-It expresses the [propagator](/know/concept/propagator/) $K$
-using the following sum over all possible paths $x(t)$,
-which all go from the initial position $x_0$ at time $t_0$
-to the destination $x_N$ at time $t_N$:
+It expresses the [propagator](/know/concept/propagator/) $$K$$
+using the following sum over all possible paths $$x(t)$$,
+which all go from the initial position $$x_0$$ at time $$t_0$$
+to the destination $$x_N$$ at time $$t_N$$:
 
 $$\begin{aligned}
     \boxed{
@@ -26,32 +26,32 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $A$ normalizes.
-$S[x]$ is the classical action of the path $x$, whose minimization yields
+Where $$A$$ normalizes.
+$$S[x]$$ is the classical action of the path $$x$$, whose minimization yields
 the Euler-Lagrange equation from Lagrangian mechanics.
 Note that each path is given an equal weight,
 even unrealistic paths that make big detours.
 
 This apparent problem solves itself,
-thanks to the fact that paths close to the classical optimum $x_c(t)$
-have an action close to $S_c = S[x_c]$,
+thanks to the fact that paths close to the classical optimum $$x_c(t)$$
+have an action close to $$S_c = S[x_c]$$,
 while the paths far away have very different actions.
-Since $S[x]$ is inside a complex exponential,
-this means that paths close to $x_c$ add contructively,
+Since $$S[x]$$ is inside a complex exponential,
+this means that paths close to $$x_c$$ add contructively,
 and the others add destructively and cancel out.
 
-An interesting way too look at it is by varying $\hbar$:
+An interesting way too look at it is by varying $$\hbar$$:
 as its value decreases, minor action differences yield big phase differences,
-which make the quantum wave function stay closer to $x_c$.
-In the limit $\hbar \to 0$, quantum mechanics thus turns into classical mechanics.
+which make the quantum wave function stay closer to $$x_c$$.
+In the limit $$\hbar \to 0$$, quantum mechanics thus turns into classical mechanics.
 
 ## Time-slicing derivation
 
 The most popular way to derive the path integral formulation proceeds as follows:
-starting from the definition of the propagator $K$,
-we divide the time interval $t_N - t_0$ into $N$ "slices"
-of equal width $\Delta t = (t_N - t_0) / N$,
-where $N$ is large:
+starting from the definition of the propagator $$K$$,
+we divide the time interval $$t_N - t_0$$ into $$N$$ "slices"
+of equal width $$\Delta t = (t_N - t_0) / N$$,
+where $$N$$ is large:
 
 $$\begin{aligned}
     K(x_N, t_N; x_0, t_0)
@@ -59,9 +59,9 @@ $$\begin{aligned}
     = \matrixel{x_N}{e^{- i \hat{H} \Delta t / \hbar} \cdots e^{- i \hat{H} \Delta t / \hbar}}{x_0}
 \end{aligned}$$
 
-Between the exponentials we insert $N\!-\!1$ identity operators
-$\hat{I} = \int \Ket{x} \Bra{x} \dd{x}$,
-and define $x_j = x(t_j)$ for an arbitrary path $x(t)$:
+Between the exponentials we insert $$N\!-\!1$$ identity operators
+$$\hat{I} = \int \Ket{x} \Bra{x} \dd{x}$$,
+and define $$x_j = x(t_j)$$ for an arbitrary path $$x(t)$$:
 
 $$\begin{aligned}
     K
@@ -69,10 +69,10 @@ $$\begin{aligned}
     \dd{x_1} \cdots \dd{x_{N - 1}}
 \end{aligned}$$
 
-For sufficiently small time steps $\Delta t$ (i.e. large $N$
+For sufficiently small time steps $$\Delta t$$ (i.e. large $$N$$
 we make the following approximation
 (which would be exact, were it not for the fact that
-$\hat{T}$ and $\hat{V}$ are operators):
+$$\hat{T}$$ and $$\hat{V}$$ are operators):
 
 $$\begin{aligned}
     e^{- i \hat{H} \Delta t / \hbar}
@@ -80,7 +80,7 @@ $$\begin{aligned}
     \approx e^{- i \hat{T} \Delta t / \hbar} e^{- i \hat{V} \Delta t / \hbar}
 \end{aligned}$$
 
-Since $\hat{V} = V(x_j)$,
+Since $$\hat{V} = V(x_j)$$,
 we can take it out of the inner product as a constant factor:
 
 $$\begin{aligned}
@@ -89,15 +89,15 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Here we insert the identity operator
-expanded in the momentum basis $\hat{I} = \int \Ket{p} \Bra{p} \dd{p}$,
-and commute it with the kinetic energy $\hat{T} = \hat{p}^2 / (2m)$ to get:
+expanded in the momentum basis $$\hat{I} = \int \Ket{p} \Bra{p} \dd{p}$$,
+and commute it with the kinetic energy $$\hat{T} = \hat{p}^2 / (2m)$$ to get:
 
 $$\begin{aligned}
     \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j}
     = \int_{-\infty}^\infty \Inprod{x_{j+1}}{p} \exp\!\Big(\!-\! i \frac{p^2 \Delta t}{2 m \hbar}\Big) \Inprod{p}{x_j} \dd{p}
 \end{aligned}$$
 
-In the momentum basis $\Ket{p}$,
+In the momentum basis $$\Ket{p}$$,
 the position basis vectors
 are represented by plane waves:
 
@@ -119,7 +119,7 @@ $$\begin{aligned}
     &= \frac{1}{2 \pi \hbar} \sqrt{\frac{2 \pi m \hbar}{i \Delta t}} \exp\!\Big( i \frac{m (x_{j+1} - x_j)^2}{2 \hbar \Delta t} \Big)
 \end{aligned}$$
 
-Inserting this back into the definition of the propagator $K(x_N, t_N; x_0, t_0)$ yields:
+Inserting this back into the definition of the propagator $$K(x_N, t_N; x_0, t_0)$$ yields:
 
 $$\begin{aligned}
     K
@@ -129,7 +129,7 @@ $$\begin{aligned}
     \dd{x_1} \cdots \dd{x_{N-1}}
 \end{aligned}$$
 
-For large $N$ and small $\Delta t$, the sum in the exponent becomes an integral:
+For large $$N$$ and small $$\Delta t$$, the sum in the exponent becomes an integral:
 
 $$\begin{aligned}
     \frac{i}{\hbar} \sum_{j = 0}^{N - 1} \Big( \frac{m (x_{j+1} \!-\! x_j)^2}{2 \Delta t^2} - V(x_j) \Big) \Delta t
@@ -137,8 +137,8 @@ $$\begin{aligned}
     \frac{i}{\hbar} \int_{t_0}^{t_N} \Big( \frac{1}{2} m \dot{x}^2 - V(x) \Big) \dd{\tau}
 \end{aligned}$$
 
-Upon closer inspection, this integral turns out to be the classical action $S[x]$,
-with the integrand being the Lagrangian $L$:
+Upon closer inspection, this integral turns out to be the classical action $$S[x]$$,
+with the integrand being the Lagrangian $$L$$:
 
 $$\begin{aligned}
     S[x(t)]
@@ -146,7 +146,7 @@ $$\begin{aligned}
     = \int_{t_0}^{t_N} \Big( \frac{1}{2} m \dot{x}^2 - V(x) \Big) \dd{\tau}
 \end{aligned}$$
 
-The definition of the propagator $K$ is then further reduced to the following:
+The definition of the propagator $$K$$ is then further reduced to the following:
 
 $$\begin{aligned}
     K
@@ -155,8 +155,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Finally, for the purpose of normalization,
-we define the integral over all paths $x(t)$ as follows,
-where we write $D[x]$ instead of $\dd{x}$:
+we define the integral over all paths $$x(t)$$ as follows,
+where we write $$D[x]$$ instead of $$\dd{x}$$:
 
 $$\begin{aligned}
     \int D[x]
@@ -164,7 +164,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We thus arrive at **Feynman's path integral**,
-which sums over all possible paths $x(t)$:
+which sums over all possible paths $$x(t)$$:
 
 $$\begin{aligned}
     K