From 06e2d1f11d2d390c3f31e4ad9cfe28ff039d075f Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 30 Mar 2021 17:17:39 +0200
Subject: Expand knowledge base

---
 content/know/concept/wkb-approximation/index.pdc | 27 ++++++++++++------------
 1 file changed, 13 insertions(+), 14 deletions(-)

(limited to 'content/know/concept/wkb-approximation')

diff --git a/content/know/concept/wkb-approximation/index.pdc b/content/know/concept/wkb-approximation/index.pdc
index 985bcec..b8ace29 100644
--- a/content/know/concept/wkb-approximation/index.pdc
+++ b/content/know/concept/wkb-approximation/index.pdc
@@ -29,17 +29,17 @@ $$\begin{aligned}
     m^2 (x')^2 = 2 m (E - V(x))
 \end{aligned}$$
 
-The left-hand side of the rearrangement is simply the momentum squared,
+The left-hand side of the rearranged version is simply the momentum squared,
 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 $E > V$, which is always the
-case in classical mechanics, but not necessarily so in quantum
-mechanics, but we stick with it for now. We rewrite the Schrödinger
-equation:
+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:
 
 $$\begin{aligned}
     0
@@ -172,19 +172,18 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-What if $E < V$? In classical mechanics, this is 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.
-However, in quantum mechanics, particles can **tunnel** through barriers.
+On the other hand, in quantum physics, particles can **tunnel** through barriers.
 
-Conveniently, all we need to change for the WKB approximation is to let
-the momentum take imaginary values:
+Luckily, the only thing we need to change for the WKB approximation
+is to let the momentum take imaginary values:
 
 $$\begin{aligned}
     p(x) = \sqrt{2 m (E - V(x))} = i \sqrt{2 m (V(x) - E)}
 \end{aligned}$$
 
-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 $\psi(x)$:
 
 $$\begin{aligned}
     \boxed{
@@ -193,9 +192,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 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.
+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
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