From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 .../concept/electric-dipole-approximation/index.md | 48 +++++++++++-----------
 1 file changed, 24 insertions(+), 24 deletions(-)

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

diff --git a/source/know/concept/electric-dipole-approximation/index.md b/source/know/concept/electric-dipole-approximation/index.md
index 66501e2..7c710ec 100644
--- a/source/know/concept/electric-dipole-approximation/index.md
+++ b/source/know/concept/electric-dipole-approximation/index.md
@@ -22,11 +22,11 @@ $$\begin{aligned}
     &= \frac{\vu{P}{}^2}{2 m} - \frac{q}{2 m} (\vb{A} \cdot \vu{P} + \vu{P} \cdot \vb{A}) + \frac{q^2 \vb{A}^2}{2m} + q \varphi
 \end{aligned}$$
 
-With charge $q = - e$,
-canonical momentum operator $\vu{P} = - i \hbar \nabla$,
-and magnetic vector potential $\vb{A}(\vb{x}, t)$.
-We reduce this by fixing the Coulomb gauge $\nabla \cdot \vb{A} = 0$,
-so that $\vb{A} \cdot \vu{P} = \vu{P} \cdot \vb{A}$:
+With charge $$q = - e$$,
+canonical momentum operator $$\vu{P} = - i \hbar \nabla$$,
+and magnetic vector potential $$\vb{A}(\vb{x}, t)$$.
+We reduce this by fixing the Coulomb gauge $$\nabla \cdot \vb{A} = 0$$,
+so that $$\vb{A} \cdot \vu{P} = \vu{P} \cdot \vb{A}$$:
 
 $$\begin{aligned}
     \comm{\vb{A}}{\vu{P}} \psi
@@ -36,9 +36,9 @@ $$\begin{aligned}
     = 0
 \end{aligned}$$
 
-Where $\psi$ is an arbitrary test function.
-Assuming $\vb{A}$ is so small that $\vb{A}{}^2$ is negligible, we split $\hat{H}$ as follows,
-where $\hat{H}_1$ can be regarded as a perturbation to $\hat{H}_0$:
+Where $$\psi$$ is an arbitrary test function.
+Assuming $$\vb{A}$$ is so small that $$\vb{A}{}^2$$ is negligible, we split $$\hat{H}$$ as follows,
+where $$\hat{H}_1$$ can be regarded as a perturbation to $$\hat{H}_0$$:
 
 $$\begin{aligned}
     \hat{H}
@@ -51,7 +51,7 @@ $$\begin{aligned}
     \equiv - \frac{q}{m} \vu{P} \cdot \vb{A}
 \end{aligned}$$
 
-In an electromagnetic wave, $\vb{A}$ is oscillating sinusoidally in time and space:
+In an electromagnetic wave, $$\vb{A}$$ is oscillating sinusoidally in time and space:
 
 $$\begin{aligned}
     \vb{A}(\vb{x}, t) = \vb{A}_0 \sin(\vb{k} \cdot \vb{x} - \omega t)
@@ -64,7 +64,7 @@ $$\begin{aligned}
     \vb{A}(\vb{x}, t) = - i \vb{A}_0 \exp(i \vb{k} \cdot \vb{x} - i \omega t)
 \end{aligned}$$
 
-The corresponding perturbative [electric field](/know/concept/electric-field/) $\vb{E}$ is then given by:
+The corresponding perturbative [electric field](/know/concept/electric-field/) $$\vb{E}$$ is then given by:
 
 $$\begin{aligned}
     \vb{E}(\vb{x}, t)
@@ -72,11 +72,11 @@ $$\begin{aligned}
     = \vb{E}_0 \exp(i \vb{k} \cdot \vb{x} - i \omega t)
 \end{aligned}$$
 
-Where $\vb{E}_0 = \omega \vb{A}_0$.
+Where $$\vb{E}_0 = \omega \vb{A}_0$$.
 Let us restrict ourselves to visible light,
-whose wavelength $2 \pi / |\vb{k}| \sim 10^{-6} \:\mathrm{m}$.
-Meanwhile, an atomic orbital is several Bohr $\sim 10^{-10} \:\mathrm{m}$,
-so $\vb{k} \cdot \vb{x}$ is negligible:
+whose wavelength $$2 \pi / |\vb{k}| \sim 10^{-6} \:\mathrm{m}$$.
+Meanwhile, an atomic orbital is several Bohr $$\sim 10^{-10} \:\mathrm{m}$$,
+so $$\vb{k} \cdot \vb{x}$$ is negligible:
 
 $$\begin{aligned}
     \boxed{
@@ -86,15 +86,15 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This is the **electric dipole approximation**:
-we ignore all spatial variation of $\vb{E}$,
+we ignore all spatial variation of $$\vb{E}$$,
 and only consider its temporal oscillation.
 Also, since we have not used the word "photon",
 we are implicitly treating the radiation classically,
 and the electron quantum-mechanically.
 
-Next, we want to rewrite $\hat{H}_1$
-to use the electric field $\vb{E}$ instead of the potential $\vb{A}$.
-To do so, we use that $\vu{P} = m \: \idv{\vu{x}}{t}$
+Next, we want to rewrite $$\hat{H}_1$$
+to use the electric field $$\vb{E}$$ instead of the potential $$\vb{A}$$.
+To do so, we use that $$\vu{P} = m \: \idv{\vu{x}}{t}$$
 and evaluate this in the [interaction picture](/know/concept/interaction-picture/):
 
 $$\begin{aligned}
@@ -105,7 +105,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Taking the off-diagonal inner product with
-the two-level system's states $\Ket{1}$ and $\Ket{2}$ gives:
+the two-level system's states $$\Ket{1}$$ and $$\Ket{2}$$ gives:
 
 $$\begin{aligned}
     \matrixel{2}{\vu{P}}{1}
@@ -113,9 +113,9 @@ $$\begin{aligned}
     = m i \omega_0 \matrixel{2}{\vu{x}}{1}
 \end{aligned}$$
 
-Therefore, $\vu{P} / m = i \omega_0 \vu{x}$,
-where $\omega_0 \equiv (E_2 \!-\! E_1) / \hbar$ is the resonance of the energy gap,
-close to which we assume that $\vb{A}$ and $\vb{E}$ are oscillating, i.e. $\omega \approx \omega_0$.
+Therefore, $$\vu{P} / m = i \omega_0 \vu{x}$$,
+where $$\omega_0 \equiv (E_2 \!-\! E_1) / \hbar$$ is the resonance of the energy gap,
+close to which we assume that $$\vb{A}$$ and $$\vb{E}$$ are oscillating, i.e. $$\omega \approx \omega_0$$.
 We thus get:
 
 $$\begin{aligned}
@@ -127,7 +127,7 @@ $$\begin{aligned}
     = - \vu{d} \cdot \vb{E}_0 \exp(- i \omega t)
 \end{aligned}$$
 
-Where $\vu{d} \equiv q \vu{x} = - e \vu{x}$ is
+Where $$\vu{d} \equiv q \vu{x} = - e \vu{x}$$ is
 the **transition dipole moment operator** of the electron,
 hence the name **electric dipole approximation**.
 Finally, we take the real part, yielding:
@@ -141,7 +141,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 If this approximation is too rough,
-$\vb{E}$ can always be Taylor-expanded in $(i \vb{k} \cdot \vb{x})$:
+$$\vb{E}$$ can always be Taylor-expanded in $$(i \vb{k} \cdot \vb{x})$$:
 
 $$\begin{aligned}
     \vb{E}(\vb{x}, t)
-- 
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