From 7c2d27ca89c5b096694b950c766e50df2dc87001 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 8 Jan 2022 14:09:13 +0100
Subject: Minor fixes, rename "Ion Sound Wave" and "Ito Process"

---
 content/know/concept/electric-dipole-approximation/index.pdc | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

(limited to 'content/know/concept/electric-dipole-approximation')

diff --git a/content/know/concept/electric-dipole-approximation/index.pdc b/content/know/concept/electric-dipole-approximation/index.pdc
index 96b4fed..265babf 100644
--- a/content/know/concept/electric-dipole-approximation/index.pdc
+++ b/content/know/concept/electric-dipole-approximation/index.pdc
@@ -46,7 +46,7 @@ In an electromagnetic wave,
 $\vec{A}$ is oscillating sinusoidally in time and space as follows:
 
 $$\begin{aligned}
-    \vec{A}(\vec{r}, t) = \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t)
+    \vec{A}(\vec{r}, t) = - i \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t)
 \end{aligned}$$
 
 The corresponding perturbative
@@ -59,7 +59,7 @@ $$\begin{aligned}
     = \vec{E}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t)
 \end{aligned}$$
 
-Where $\vec{E}_0 = i \omega \vec{A}_0$.
+Where $\vec{E}_0 = \omega \vec{A}_0$.
 Let us restrict ourselves to visible light,
 whose wavelength $2 \pi / k \approx 10^{-6} \:\mathrm{m}$.
 Meanwhile, an atomic orbital is on the order of $10^{-10} \:\mathrm{m}$,
@@ -82,7 +82,7 @@ and the electron quantum-mechanically.
 Next, we want to convert $\hat{H}_1$
 to use the electric field $\vec{E}$ instead of the potential $\vec{A}$.
 To do so, we rewrite the momemtum $\vec{P} = m \: \dv*{\vec{r}}{t}$
-and evaluate this in the [Heisenberg picture](/know/concept/heisenberg-picture/):
+and evaluate this in the [interaction picture](/know/concept/interaction-picture/):
 
 $$\begin{aligned}
     \matrixel{2}{\dv*{\vec{r}}{t}}{1}
@@ -95,15 +95,15 @@ $$\begin{aligned}
 
 Therefore, $\vec{P} / m = i \omega_0 \vec{r}$,
 where $\omega_0 \equiv (E_2 - E_1) / \hbar$ is the resonance frequency of the transition,
-close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating.
+close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating, i.e. $\omega \approx \omega_0$.
 We thus get:
 
 $$\begin{aligned}
     \hat{H}_1(t)
     &= - \frac{q}{m} \vec{P} \cdot \vec{A}
-    = - i q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t)
+    = - (- i i) q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t)
     \\
-    &= - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t)
+    &\approx - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t)
     = - \vec{d} \cdot \vec{E}_0 \exp\!(- i \omega t)
 \end{aligned}$$
 
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