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

1 files changed, 31 insertions, 31 deletions

diff --git a/source/know/concept/wkb-approximation/index.md b/source/know/concept/wkb-approximation/index.md
index efeeca6..ef57a3b 100644
--- a/source/know/concept/wkb-approximation/index.md
+++ b/source/know/concept/wkb-approximation/index.md
@@ -9,13 +9,13 @@ layout: "concept"
 ---
 
 In quantum mechanics, the **Wentzel-Kramers-Brillouin** or simply the **WKB
-approximation** is a technique to approximate the wave function $\psi(x)$ of
+approximation** is a technique 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
 balance between classical and quantum physics.
 
-In classical mechanics, a particle travelling in a potential $V(x)$
-along a path $x(t)$ has a total energy $E$ as follows, which we
+In classical mechanics, a particle travelling in a potential $$V(x)$$
+along a path $$x(t)$$ has a total energy $$E$$ as follows, which we
 rearrange:
 
 $$\begin{aligned}
@@ -25,13 +25,13 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The left-hand side of the rearranged version is simply the momentum squared,
-so we define the magnitude of the momentum $p(x)$ accordingly:
+so we define the magnitude of the momentum $$p(x)$$ accordingly:
 
 $$\begin{aligned}
     p(x) = \sqrt{2 m (E - V(x))}
 \end{aligned}$$
 
-Note that this is under the assumption that $E > V$,
+Note that this is under the assumption that $$E > V$$,
 which is always true in classical mechanics,
 but not necessarily in quantum mechanics.
 We rewrite the Schrödinger equation:
@@ -42,7 +42,7 @@ $$\begin{aligned}
     = \dvn{2}{\psi}{x} + \frac{p^2}{\hbar^2} \psi
 \end{aligned}$$
 
-If $V(x)$ were constant, and by extension $p(x)$ too, then the solution
+If $$V(x)$$ were constant, and by extension $$p(x)$$ too, then the solution
 is easy:
 
 $$\begin{aligned}
@@ -51,9 +51,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This form is reminiscent of the generator of translations. In practice,
-$V(x)$ and $p(x)$ vary with $x$, but we can still salvage this solution
-by assuming that $V(x)$ varies slowly compared to the wavelength
-$\lambda(x) = 2 \pi / k(x)$, where $k(x) = p(x) / \hbar$ is the
+$$V(x)$$ and $$p(x)$$ vary with $$x$$, but we can still salvage this solution
+by assuming that $$V(x)$$ varies slowly compared to the wavelength
+$$\lambda(x) = 2 \pi / k(x)$$, where $$k(x) = p(x) / \hbar$$ is the
 wavenumber. The solution then takes the following form:
 
 $$\begin{aligned}
@@ -61,9 +61,9 @@ $$\begin{aligned}
     = \psi(0) \exp\!\Big(\!\pm\! \frac{i}{\hbar} \int_0^x \chi(\xi) \dd{\xi} \Big)
 \end{aligned}$$
 
-$\chi(\xi)$ is an unknown function, which intuitively should be related
-to $p(x)$. The purpose of the integral is to accumulate the change of
-$\chi$ from the initial point $0$ to the current position $x$.
+$$\chi(\xi)$$ is an unknown function, which intuitively should be related
+to $$p(x)$$. The purpose of the integral is to accumulate the change of
+$$\chi$$ from the initial point $$0$$ to the current position $$x$$.
 Let us write this as an indefinite integral for convenience:
 
 $$\begin{aligned}
@@ -71,8 +71,8 @@ $$\begin{aligned}
     = \psi(0) \exp\!\bigg( \!\pm\! \frac{i}{\hbar} \Big( \int \chi(x) \dd{x} - C \Big) \bigg)
 \end{aligned}$$
 
-Where $C = \int \chi(x) \dd{x} |_{x = 0}$ is the initial point of the definite integral.
-For simplicity, we absorb the constant $C$ into $\psi(0)$.
+Where $$C = \int \chi(x) \dd{x} |_{x = 0}$$ is the initial point of the definite integral.
+For simplicity, we absorb the constant $$C$$ into $$\psi(0)$$.
 We can now clearly see that:
 
 $$\begin{aligned}
@@ -81,7 +81,7 @@ $$\begin{aligned}
     \chi(x) = \pm \frac{\hbar}{i} \frac{\psi'(x)}{\psi(x)}
 \end{aligned}$$
 
-Next, we insert this ansatz for $\psi(x)$ into the Schrödinger equation
+Next, we insert this ansatz for $$\psi(x)$$ into the Schrödinger equation
 to get:
 
 $$\begin{aligned}
@@ -91,7 +91,7 @@ $$\begin{aligned}
     = \pm \frac{i}{\hbar} \chi' \psi - \frac{1}{\hbar^2} \chi^2 \psi + \frac{p^2}{\hbar^2} \psi
 \end{aligned}$$
 
-Dividing out $\psi$ and rearranging gives us the following, which is
+Dividing out $$\psi$$ and rearranging gives us the following, which is
 still exact:
 
 $$\begin{aligned}
@@ -99,9 +99,9 @@ $$\begin{aligned}
     = p^2 - \chi^2
 \end{aligned}$$
 
-Next, we expand this as a power series of $\hbar$. This is why it is
+Next, we expand this as a power series of $$\hbar$$. This is why it is
 called *semiclassical*: so far we have been using full quantum mechanics,
-but now we are treating $\hbar$ as a parameter which controls the
+but now we are treating $$\hbar$$ as a parameter which controls the
 strength of quantum effects:
 
 $$\begin{aligned}
@@ -109,9 +109,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The heart of the WKB approximation is its assumption that quantum effects are
-sufficiently weak (i.e. $\hbar$ is small enough) that we only need to
+sufficiently weak (i.e. $$\hbar$$ is small enough) that we only need to
 consider the first two terms, or, more specifically, that we only go up to
-$\hbar$, not $\hbar^2$ or higher. Inserting the first two terms of this
+$$\hbar$$, not $$\hbar^2$$ or higher. Inserting the first two terms of this
 expansion into the equation:
 
 $$\begin{aligned}
@@ -119,8 +119,8 @@ $$\begin{aligned}
     &= p^2 - \chi_0^2 - 2 \frac{\hbar}{i} \chi_0 \chi_1
 \end{aligned}$$
 
-Where we have discarded all terms containing $\hbar^2$. At order
-$\hbar^0$, we then get the expected classical result for $\chi_0(x)$:
+Where we have discarded all terms containing $$\hbar^2$$. At order
+$$\hbar^0$$, we then get the expected classical result for $$\chi_0(x)$$:
 
 $$\begin{aligned}
     0 = p^2 - \chi_0^2
@@ -128,7 +128,7 @@ $$\begin{aligned}
     \chi_0(x) = p(x)
 \end{aligned}$$
 
-While at order $\hbar$, we get the following quantum-mechanical
+While at order $$\hbar$$, we get the following quantum-mechanical
 correction:
 
 $$\begin{aligned}
@@ -138,7 +138,7 @@ $$\begin{aligned}
     \chi_1(x) = \mp \frac{1}{2} \frac{\chi_0'(x)}{\chi_0(x)}
 \end{aligned}$$
 
-Therefore, our approximated wave function $\psi(x)$ currently looks like
+Therefore, our approximated wave function $$\psi(x)$$ currently looks like
 this:
 
 $$\begin{aligned}
@@ -158,7 +158,7 @@ $$\begin{aligned}
     = \frac{1}{\sqrt{p(x)}}
 \end{aligned}$$
 
-In the WKB approximation for $E > V$, the solution $\psi(x)$ is thus
+In the WKB approximation for $$E > V$$, the solution $$\psi(x)$$ is thus
 given by:
 
 $$\begin{aligned}
@@ -167,7 +167,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-What if $E < V$? In classical mechanics, this is just not allowed; a ball
+What if $$E < V$$? In classical mechanics, this is just not allowed; a ball
 cannot simply go through a potential bump without the necessary energy.
 On the other hand, in quantum physics, particles can **tunnel** through barriers.
 
@@ -178,7 +178,7 @@ $$\begin{aligned}
     p(x) = \sqrt{2 m (E - V(x))} = i \sqrt{2 m (V(x) - E)}
 \end{aligned}$$
 
-And then take the absolute value in the appropriate place in front of $\psi(x)$:
+And then take the absolute value in the appropriate place in front of $$\psi(x)$$:
 
 $$\begin{aligned}
     \boxed{
@@ -186,10 +186,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-In the classical region ($E > V$), the wave function oscillates, and
-in the quantum-physical region ($E < V$) it is exponential.
-Note that for $E \approx V$ the approximation breaks down,
-because of the appearance of $p(x)$ in the denominator.
+In the classical region ($$E > V$$), the wave function oscillates, and
+in the quantum-physical region ($$E < V$$) it is exponential.
+Note that for $$E \approx V$$ the approximation breaks down,
+because of the appearance of $$p(x)$$ in the denominator.
 
 
 ## References