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diff --git a/source/know/concept/wkb-approximation/index.md b/source/know/concept/wkb-approximation/index.md
index ef57a3b..fb04414 100644
--- a/source/know/concept/wkb-approximation/index.md
+++ b/source/know/concept/wkb-approximation/index.md
@@ -8,24 +8,25 @@ categories:
 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
-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 quantum mechanics, the **Wentzel-Kramers-Brillouin**
+or simply the **WKB approximation**
+is a technique to approximate the wavefunction $$\psi(x)$$
+of the 1D 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
 rearrange:
 
 $$\begin{aligned}
-    E = \frac{1}{2} m \dot{x}^2 + V(x)
-    \quad \implies \quad
+    E = \frac{1}{2} m (x')^2 + V(x)
+    \qquad \implies \qquad
     m^2 (x')^2 = 2 m (E - V(x))
 \end{aligned}$$
 
 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 know that the magnitude of the momentum $$p(x)$$ is:
 
 $$\begin{aligned}
     p(x) = \sqrt{2 m (E - V(x))}
@@ -38,8 +39,9 @@ We rewrite the Schrödinger equation:
 
 $$\begin{aligned}
     0
-    = \dvn{2}{\psi}{x} + \frac{2 m}{\hbar^2} (E - V) \psi
-    = \dvn{2}{\psi}{x} + \frac{p^2}{\hbar^2} \psi
+    &= \dvn{2}{\psi}{x} + \frac{2 m}{\hbar^2} (E - V) \psi
+    \\
+    &= \dvn{2}{\psi}{x} + \frac{p^2}{\hbar^2} \psi
 \end{aligned}$$
 
 If $$V(x)$$ were constant, and by extension $$p(x)$$ too, then the solution
@@ -50,20 +52,20 @@ $$\begin{aligned}
     = \psi(0) \exp(\pm i p x / \hbar)
 \end{aligned}$$
 
-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
+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
-wavenumber. The solution then takes the following form:
+$$2 \pi / k(x)$$, where $$k(x) = p(x) / \hbar$$ is the wavenumber.
+The solution then takes the following form:
 
 $$\begin{aligned}
     \psi(x)
     = \psi(0) \exp\!\Big(\!\pm\! \frac{i}{\hbar} \int_0^x \chi(\xi) \dd{\xi} \Big)
 \end{aligned}$$
 
-$$\chi(\xi)$$ 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:
 
 $$\begin{aligned}
@@ -71,111 +73,118 @@ $$\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.
+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:
 
 $$\begin{aligned}
-    \psi'(x) = \pm \frac{i}{\hbar} \chi(x) \psi(x)
-    \quad \implies \quad
-    \chi(x) = \pm \frac{\hbar}{i} \frac{\psi'(x)}{\psi(x)}
+    \psi'(x)
+    = \pm \frac{i}{\hbar} \chi(x) \psi(x)
 \end{aligned}$$
 
-Next, we insert this ansatz for $$\psi(x)$$ into the Schrödinger equation
-to get:
+We insert this ansatz for $$\psi(x)$$ into the Schrödinger equation to get:
 
 $$\begin{aligned}
     0
     &= \pm \frac{i}{\hbar} \dv{(\chi \psi)}{x} + \frac{p^2}{\hbar^2} \psi
-    = \pm \frac{i}{\hbar} \chi' \psi \pm \frac{i}{\hbar} \chi \psi' + \frac{p^2}{\hbar^2} \psi
-    = \pm \frac{i}{\hbar} \chi' \psi - \frac{1}{\hbar^2} \chi^2 \psi + \frac{p^2}{\hbar^2} \psi
+    \\
+    &= \pm \frac{i}{\hbar} \chi' \psi \pm \frac{i}{\hbar} \chi \psi' + \frac{p^2}{\hbar^2} \psi
+    \\
+    &= \pm \frac{i}{\hbar} \chi' \psi - \frac{1}{\hbar^2} \chi^2 \psi + \frac{p^2}{\hbar^2} \psi
 \end{aligned}$$
 
-Dividing out $$\psi$$ and rearranging gives us the following, which is
-still exact:
+Dividing out $$\psi$$ and rearranging gives us the following, which is still exact:
 
 $$\begin{aligned}
     \pm \frac{\hbar}{i} \chi'
     = p^2 - \chi^2
 \end{aligned}$$
 
-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
-strength of quantum effects:
+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 strength of quantum effects:
 
 $$\begin{aligned}
-    \chi(x) = \chi_0(x) + \frac{\hbar}{i} \chi_1(x) + \frac{\hbar^2}{i^2} \chi_2(x) + ...
+    \chi(x)
+    = \chi_0(x) + \frac{\hbar}{i} \chi_1(x) + \frac{\hbar^2}{i^2} \chi_2(x) + \cdots
+\end{aligned}$$
+
+The heart of the WKB approximation is its assumption that quantum effects
+are sufficiently weak that we only need to consider
+the first two terms of this expansion,
+i.e. $$\hbar^2$$ is so small that it is negligible.
+Therefore, our approximated wavefunction $$\psi(x)$$ now looks like this:
+
+$$\begin{aligned}
+    \psi(x)
+    &\approx \psi(0) \exp\!\Big( \!\pm\! \frac{i}{\hbar} \int \chi_0(x) \dd{x} \Big) \exp\!\Big( \!\pm\! \int \chi_1(x) \dd{x} \Big)
 \end{aligned}$$
 
-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
-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
-expansion into the equation:
+Inserting the expansion's first two terms into our equation for $$\chi(x)$$ gives:
 
 $$\begin{aligned}
     \pm \frac{\hbar}{i} \chi_0'
     &= p^2 - \chi_0^2 - 2 \frac{\hbar}{i} \chi_0 \chi_1
 \end{aligned}$$
 
-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 $$\chi_0(x)$$:
 
 $$\begin{aligned}
     0 = p^2 - \chi_0^2
-    \quad \implies \quad
-    \chi_0(x) = p(x)
+    \qquad \implies \qquad
+    \chi_0(x)
+    = p(x)
 \end{aligned}$$
 
-While at order $$\hbar$$, we get the following quantum-mechanical
-correction:
+While at order $$\hbar$$, we get the following quantum-mechanical correction:
 
 $$\begin{aligned}
     \pm \frac{\hbar}{i} \chi_0'
     = - 2 \frac{\hbar}{i} \chi_0 \chi_1
-    \quad \implies \quad
-    \chi_1(x) = \mp \frac{1}{2} \frac{\chi_0'(x)}{\chi_0(x)}
+    \qquad \implies \qquad
+    \chi_1(x)
+    = \mp \frac{1}{2} \frac{\chi_0'(x)}{\chi_0(x)}
 \end{aligned}$$
 
-Therefore, our approximated wave function $$\psi(x)$$ currently looks like
-this:
-
-$$\begin{aligned}
-    \psi(x)
-    &\approx \psi(0) \exp\!\Big( \!\pm\! \frac{i}{\hbar} \int \chi_0(x) \dd{x} \Big) \exp\!\Big( \!\pm\! \int \chi_1(x) \dd{x} \Big)
-\end{aligned}$$
-
-We can reduce the latter exponential using integration by substitution:
+We can use this to simplify the latter exponential in $$\psi(x)$$
+using integration by substitution:
 
 $$\begin{aligned}
     \exp\!\Big( \!\pm\! \int \chi_1(x) \dd{x} \Big)
     &= \exp\!\Big( \!-\! \frac{1}{2} \int \frac{\chi_0'(x)}{\chi_0(x)} \dd{x} \Big)
-    = \exp\!\Big( \!-\! \frac{1}{2} \int \frac{1}{\chi_0}\:d\chi_0 \Big)
+    \\
+    &= \exp\!\Big( \!-\! \frac{1}{2} \int \frac{1}{\chi_0}\:d\chi_0 \Big)
     \\
     &= \exp\!\Big( \!-\! \frac{1}{2} \ln\!\big(\chi_0(x)\big) \Big)
-    = \frac{1}{\sqrt{\chi_0(x)}}
-    = \frac{1}{\sqrt{p(x)}}
+    \\
+    &= \frac{1}{\sqrt{\chi_0(x)}}
 \end{aligned}$$
 
-In the WKB approximation for $$E > V$$, the solution $$\psi(x)$$ is thus
-given by:
+In the WKB approximation for $$E > V$$,
+the solution $$\psi(x)$$ is therefore given by:
 
 $$\begin{aligned}
     \boxed{
-        \psi(x) \approx \frac{A}{\sqrt{p(x)}} \exp\!\Big( \!\pm\! \frac{i}{\hbar} \int p(x) \dd{x} \Big)
+        \psi(x)
+        \approx \frac{A}{\sqrt{p(x)}} \exp\!\Big( \!\pm\! \frac{i}{\hbar} \int p(x) \dd{x} \Big)
     }
 \end{aligned}$$
 
-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.
+What if $$E < V$$? In classical mechanics, this is not allowed:
+a ball cannot simply go through or over a potential bump without the necessary energy.
 On the other hand, in quantum physics, particles can **tunnel** through barriers.
 
 Luckily, the only thing we need to change for the WKB approximation
 is to let the momentum take imaginary values:
 
 $$\begin{aligned}
-    p(x) = \sqrt{2 m (E - V(x))} = i \sqrt{2 m (V(x) - E)}
+    p(x)
+    = \sqrt{2 m (E - V(x))}
+    = i \sqrt{2 m (V(x) - E)}
 \end{aligned}$$
 
 And then take the absolute value in the appropriate place in front of $$\psi(x)$$:
@@ -186,12 +195,13 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-In the classical region ($$E > V$$), the wave function oscillates, and
-in the quantum-physical region ($$E < V$$) it is exponential.
+In the classical region ($$E > V$$), the wavefunction 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
 1.  D.J. Griffiths, D.F. Schroeter,
     *Introduction to quantum mechanics*, 3rd edition,