Fix "Wentzel-Kramers-Brillouin approximation"

1 files changed, 8 insertions, 8 deletions

diff --git a/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md b/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md
index f862004..a50302c 100644
--- a/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md
+++ b/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md
@@ -5,8 +5,8 @@
 
 In quantum mechanics, the *Wentzel-Kramers-Brillouin* or simply the *WKB
 approximation* is a method to approximate the wave function $\psi(x)$ of
-the one-dimensional time-independent Schrödinger equation. It is also
-known as the *semiclassical approximation*, because it tries to find a
+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 $V(x)$
@@ -49,7 +49,7 @@ 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
-wavenumber. The solution is then of the form:
+wavenumber. The solution then takes the following form:
 
 $$\begin{aligned}
     \psi(x)
@@ -95,7 +95,7 @@ $$\begin{aligned}
 \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 quatum mechanics,
+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:
 
@@ -103,9 +103,9 @@ $$\begin{aligned}
     \chi(x) = \chi_0(x) + \frac{\hbar}{i} \chi_1(x) + \frac{\hbar^2}{i^2} \chi_2(x) + ...
 \end{aligned}$$
 
-The heart of the WKB approximation is to assume that quantum effects are
+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 generally, that we only go up 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:
 
@@ -182,7 +182,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-In the classical region ($E > V$), the wave function oscillates, whereas
-in the quantum region ($E < V$) it is exponential. Note that for
+In the classical region ($E > V$), the wave function oscillates, and
+in the quantum-mechanical region ($E < V$) it is exponential. Note that for
 $E \approx V$ the approximation breaks down, due to the appearance of
 $p(x)$ in the denominator.